What is the conditional statement of a triangle is an isosceles when the base angles are equal?

Picture yourself in Paris, France in front of the Eiffel Tower and observe its structure.

Eiffel Tower and isosceles triangle, StudySmarter Originals

From above, we see that the structure of the Eiffel Tower represents a triangle. Now further inspect the dimensions of this triangle. Notice how the two opposite sides are equal while the base is different. This means we can rule out the Eiffel Tower being the shape of an equilateral triangle, which you recall is a triangle of three equal sides. So what kind of triangle might this be? To answer your question, this is called an isosceles triangle.

An isosceles triangle is a triangle with two equal sides.

Components of an Isosceles Triangle

Consider the isosceles triangle ABC below.

Isosceles triangle, StudySmarter Originals

Here are the most important components of an isosceles triangle:

Sides and vertices

• The legs of the isosceles triangle are represented by the variable a.

• The base is defined by the variable b.

• The vertex is the topmost section of the isosceles triangle. This is also called the apex.

• The altitudeis a perpendicular line segment drawn from the vertex to the base of an isosceles triangle.

• The length of the altitude is called height and is described by the variable h.

Angles

• The angle between the legs of the isosceles triangle is called the vertex angle (or apex angle).

• Each of the two angles and between a leg and the base of the isosceles triangle is called base angle.

There are several significant properties of isosceles triangles you should familiarise yourself with in order to fully understand the composition of an isosceles triangle. The table below describes this in detail.

Say we are given a triangle with three sides. We are told that the triangle is indeed an isosceles triangle. However, we need to determine which sides are the legs of the isosceles triangle and which side is the base.

To determine the legs of the isosceles triangle, take note of the characteristics below:

• Exactly two equal sides are present, which are the legs

• Both legs branch out from the vertex of the triangle

• The altitude is adjacent to the two legs

In contrast, the base should fulfil the following properties.

• The two angles at the ends of the base are equal

• An altitude drawn from the vertex is perpendicular to the base

• A perpendicular line segment through the vertex bisects the base, i.e. cuts the base into two equal halves

Isosceles Triangle Theorems

With that in mind, let us now discuss two notable theorems involving isosceles triangles that take a closer look into the two main properties as described above.

The angles opposite the equal sides of an isosceles triangle are equal.

Proof of Theorem 1

Consider the isosceles triangle ABC below where AC = BC. Draw a bisector passing through ∠C. We shall name this line segment CD.

Isosceles triangle theorem 1, StudySmarter Originals

We aim to prove that the angles opposite the sides AC and BC are equal.

Essentially, we want to show that ∠A = ∠B.

Notice that in triangles ACD and BCD:

1. AC = BC

2. ∠ACD = ∠BCD

3. CD = CD

SAS Congruence

If two sides and an included angle of one triangle is equal to the two sides and included angle of the second triangle, then the two triangles are said to be congruent.

By the SAS Congruence rule above, triangles ACD and BCD must be congruent. As the two triangles are congruent, the corresponding angles must also be congruent. Thus, ∠A must be equal to ∠B.

The sides opposite the equal angles of an isosceles triangle are equal.

Proof of Theorem 2

Consider the isosceles triangle ABC below where ∠A = ∠B. We shall construct a bisector CD that meets the side AB at right angles.

Isosceles triangle theorem 2, StudySmarter Originals

We aim to prove that AC = BC to show that triangle ABC is indeed an isosceles triangle.

Notice that in triangles ACD and BCD:

1. ∠ACD = ∠BCD

2. CD = CD

3. ∠ADC = ∠BDC = 90o

ASA Congruence

If two angles and an included side between the angles of one triangle are equal to the corresponding two angles and included side between the angles of the second triangle, then the two triangles are said to be congruent.

By the ASA Congruence rule above, triangles ACD and BCD must be congruent. As the two triangles are congruent, the corresponding sides must also be congruent. Thus, AC must be equal to BC and so triangle ABC is an isosceles triangle.

There are three types of isosceles triangles to consider, namely

1. Isosceles acute;

2. Isosceles right;

3. Isosceles obtuse.

The table below compares each of these types of isosceles triangles.

Type of Isosceles Triangle	Diagram	Description
Isosceles Acute	Acute isosceles triangle, StudySmarter Originals	Consists of two equal sides and one unequal side. The two angles opposite the sides are equal. If each of the equal angles is more than 45o and less than 90o, the apex angle will be an acute angle.
Isosceles Right	Right isosceles triangle, StudySmarter Originals	Consist of two equal sides: one of which act as the perpendicular and the other as the base of the triangle. The third unequal side acts as the hypotenuse of the triangle. If each of the equal angles is exactly 45o, then the apex angle is a right angle. By Pythagoras Theoremh2 = a2 + a2 = 2a2
Isosceles Obtuse	Obtuse isosceles triangle, StudySmarter Originals	Consists of two equal sides and one obtuse angle. If each of the equal angles is less than 45o, then the vertex angle is an obtuse angle.

Formulas of Isosceles Triangles

In this section, we shall look at three important formulas involving isosceles triangles, namely

1. The height of an isosceles triangle;

2. The perimeter of an isosceles triangle;

3. The area of an isosceles triangle.

The Height of an Isosceles Triangle

The height of an isosceles triangle can be found by applying Pythagoras Theorem. Say we have the isosceles triangle ABC below where the measures of a leg a and the base b are given.

Height of an isosceles triangle, StudySmarter Originals

We know that the altitude (line segment CD) from the vertex angle bisects the base of the isosceles triangle. This means that

.

Furthermore, ADC and BDC are right-angled triangles where a is the hypotenuse. Thus, to find the height, we can simply adopt Pythagoras Theorem as

The Perimeter of an Isosceles Triangle

The perimeter of an isosceles triangle is given by the following formula.

where a is the length of the two equal sides and b is the base of the isosceles triangle. Let us demonstrate this with a worked example.

Given the triangle below, calculate its perimeter.

Example 1, StudySmarter Originals

Solution

By the perimeter formula, we find that the perimeter of this isosceles triangle is

The Area of an Isosceles Triangle

Once you know the height of an isosceles triangle, calculating the area is a breeze. The formula for this is

,

where b is the base and h is the height of the isosceles triangle. Below is a worked example applying this method.

Find the area of an isosceles triangle whose base is 6 units and side is 13 units.

Solution

Let us begin by making a sketch of this isosceles triangle. Construct an altitude from the vertex angle of this isosceles triangle to the base.

Example 2, StudySmarter Originals

We know that the altitude bisects the base of the isosceles triangle and creates two congruent right-angled triangles. Since the base is 6 units, then AD = BD = 3 units. The height is found by applying Pythagoras Theorem as

Now that we have the height of the isosceles triangle, we can use the area formula. We find that the area of this isosceles triangle is

Let us now define an altitude of a triangle.

An altitude is a line that passes through the vertex of a triangle that is perpendicular to the opposite side.

Do not confuse this term with perpendicular bisectors! A perpendicular bisector divides a segment into two equal parts and is perpendicular to that segment.

Now that we have established the definition of an altitude, we shall now link this idea with our subject at hand. The following are two theorems that relate the altitude to isosceles triangles.

Theorem 1

The altitude to the base of an isosceles triangle bisects the vertex angle.

Theorem 2

The altitude to the base of an isosceles triangle bisects the base.

Proof of Theorem 1 and 2

Consider the isosceles triangle shown below.

The altitude of an isosceles triangle, StudySmarter Originals

Say we draw an altitude to the base of the isosceles triangle. We find that two congruent triangles are formed. The altitude creates two right-angled triangles ADC and BDC and becomes the shared side between the two triangles. The congruent sides of the triangle become the hypotenuse for triangles ADC and BDC and are of equal length.

Since constructing an altitude to the base of the isosceles triangle forms two congruent right-angled triangles, we conclude that the altitude bisects both the base and vertex of the isosceles triangle.

Given the triangle, ABC below, determine lengths AC and BC if ∠A = ∠B.

Example 3, StudySmarter Originals

Solution

Since the two angles of the triangle above are congruent, the sides opposite them are also congruent. In other words, as ∠A = ∠B, then AC = BC.

Given the triangles ABD and BDC below, determine the value of ∠X if AB = BD = CD and ∠C is 23o.

Example 4, StudySmarter Originals

Solution

We know that if two sides of a triangle are equal then the angles opposite them are also equal. This means that since BD = CD then ∠C = ∠CBD = 23o.

As the sum of the interior angles of a triangle is 180o, the ∠BDC is 130o, for triangle BDC.

The ∠ADB is the exterior angle of triangle BDC. The sum of the exterior angle and its adjacent interior angle of a triangle is 180o. Thus ∠ADB is 50o.

As AB = BD, ∠A = ∠ADB = 50o. As before, since the sum of the interior angles of a triangle, is 180o, the ∠X is 80o, for triangle ABD.

Given the triangles ACB and DCE below, determine the value of angles X, Y and Z if AC = BC, DC = EC and ∠ACB = 31o.

Example 5, StudySmarter Originals

Solution

As ∠Y and ∠ACB are vertical angles then ∠Y = ∠ACB = 31o.

We know that if two sides of a triangle are congruent the angles opposite them are also congruent. ∠X = ∠B = ∠D = ∠Z since the vertex angle for triangles ACB and DCE are equal. Noting that the sum of the interior angles of a triangle is 180o, we obtain

Thus, ∠X = ∠Z = 74.5o.

There are three types of triangles we shall often see throughout this syllabus, namely

1. Isosceles triangle

2. Equilateral triangle

3. Scalene triangle

In this final section, we shall look at the differences between these three triangles. By familiarising ourselves with these contrasts, we can properly distinguish each type we are dealing with and perform the correct calculations. The table below compares these three triangles with respect to sides, angles and altitudes.

Property	Isosceles Triangle	Equilateral Triangle	Scalene Triangle
Diagram	Isosceles triangle, StudySmarter Originals	Equilateral triangle, StudySmarter Originals	Scalene triangle, StudySmarter Originals
Sides	Two sides of equal length	Three sides of equal length	Three sides of different length
Angles	Two angles of equal value	Three angles of equal value	Three angles of different value
	An altitude drawn from the vertex angle bisects that angle and the unequal side of the triangle.	An altitude drawn from any angle bisects that angle and the opposite side of the triangle.	No special criteria

Isosceles Triangles - Key takeaways

• An isosceles triangle
  • Consists of two equal sides and two equal angles
  • Have equal base angles
  • The altitude from the vertex angle bisects the base and the vertex angle
  • The altitude drawn from the apex angle divides the isosceles triangle into two congruent triangles
• There are three types of isosceles triangles: Acute, Right and Obtuse
• The area is given by
• The perimeter is given by

An isosceles triangle is a triangle with two equal sides 

We can find the height of an isosceles triangle by using Pythagoras Theorem

The area of an isosceles triangle is the product of the base and height multiplied by half

The base angles of an isosceles triangle are equal

Acute, right and obtuse isosceles triangles

Question

What is an isosceles triangle?

Answer

An isosceles triangle is a triangle with two equal sides.  

Question

"An isosceles triangle consists of two equal sides and 3 equal angles"

Is the statement above true or false? 

Answer

Question

"The base angles of an isosceles triangle are equal"

Is the statement above true or false? 

Answer

Question

"The altitude from the vertex angle of an isosceles triangle bisects its base and the vertex angle "

Is the statement above true or false? 

Answer

Question

"The altitude drawn from the apex angle divides the isosceles triangle into two similar triangles "

Is the statement above true or false? 

Answer

Question

Given the isosceles triangle ABC, how do you write the notation for the statement:

"The angles opposite the equal sides of an isosceles triangle are equal"

Answer

Question

Given the isosceles triangle ABC, how do you write the notation for the statement:

"The sides opposite the equal angles of an isosceles triangle are equal"

Answer

Question

What does the SAS Congruence rule tell us?

Answer

If two sides and an included angle of one triangle is equal to the two sides and included angle of the second triangle, then the two triangles are said to be congruent. 

Question

What does the ASA Congruence rule tell us?

Answer

If two angles and an included side between the angles of one triangle are equal to the corresponding two angles and included side between the angles of the second triangle, then the two triangles are said to be congruent.

Question

What are the three types of isosceles triangles?

Answer

1. Isosceles Acute
2. Isosceles Right
3. Isosceles Obtuse

Question

What is the altitude of a triangle?

Answer

An altitude is a line that passes through the vertex of a triangle that is perpendicular to the opposite side.  

Question

Say we have an isosceles triangle ABC. If the altitude bisects the base of the triangle at point P, what is the relationship between points A, B and P?

Answer

Question

What is the formula for the perimeter of an isosceles triangle given the sides a and base b? 

Answer

Question

Say we have an isosceles triangle ABC. If altitude to the base of an isosceles triangle bisects the vertex angle at point P, what is the relationship between angles ACP and BCP? 

Answer

Question

For an isosceles triangle ABC, the altitude to the base bisects the base at point P. Two triangles, ACP and BCP, are formed due to this bisection. What is the relationship between these two triangles?

Answer

Triangle ACP is congruent to Triangle BCP

Question

State the differences in the length of the sides and measures of the angles between the isosceles triangle and equilateral triangle.

Answer

1. An isosceles triangle has 2 equal sides while an equilateral triangle has 3 equal sides
2. An isosceles triangle has 2 equal angles while an equilateral triangle has 3 equal angles

Question

State the differences in the length of the sides and measures of the angles between the isosceles triangle and scalene triangle.

Answer

1. An isosceles triangle has 2 equal sides while a scalene  triangle has 3 unequal sides
2. An isosceles triangle has 2 equal angles while a scalene  triangle has 3 unequal angles

Page 2

What is a two-dimensional figure in math? Maybe the term 2D sounds familiar and makes you think of a video game, like Tetris, or a movie. This concept is similar to the understanding of 2D that we have in math. Two-dimensional (2D) figures are shapes formed by closed lines in a plane, and they have two dimensions: length and width. As they do not exist in three-dimensional (3D) space, these shapes do not have depth.

Definition of 2-dimensional figures

Two-dimensional figures are the flat plane shapes or figures that have two dimensions (length and width) in the same plane.

For example, if we drew three lines on a 2D plane surface, like a piece of paper, we could obtain a triangle, which is an example of a 2D shape. We just need one plane to show these 2D figures, as they do not have depth. In math, there are as many 2D shapes as you can imagine, as you just have to link one line with another in a plane.

These lines that form the shapes are called the sides of the plane figure. All sides do not have to be connected, as we can distinguish between closed shapes or open shapes, depending on whether they form vertices or not. We will mainly focus on closed shapes, as they are the most common in math.

Examples of 2-dimensional figures

Now, think about the popular game, Tetris, which is played in 2D. All of the shapes we can see in this game are two-dimensional figures that have lengths and widths. In Tetris, there are numerous two-dimensional shapes, but in math, there are four distinguished two-dimensional figures we work often with:

• Triangle
• Square
• Rectangle
• Circle

Let's consider each of these four two-dimensional shapes in more detail.

Triangle

As a 2D shape, the triangle consists of three sides and three vertices. The summation of all the internal angles in a triangle is equal to 180º. We can distinguish between different types of triangles depending on whether the sides are equal or not. We can also distinguish types of triangles by the angles they form with one another.

For example, the triangles with all sides of the same length are called equilateral triangles, while if they have just two equal sides, they are called isosceles triangles. If none of the sides are the same length, the triangle is called a scalene triangle. On the other hand, an example of a triangle classified by its internal angles is a right triangle, which has an angle of 90º.

Square

As 2-dimensional figures, squares are formed by four equal sides with four vertices. All of the internal angles formed by the vertices are equal to 90º. We can label a 2D shape as a square only if all four sides are of the same length.

Rectangle

These shapes are formed by four sides, with each side equal only to its opposite side. Therefore, both of these two pairs have the same length between them. In a rectangle, all internal angles formed by the vertices are equal to 90º, like in the square. If all the sides' lengths were equal as well, the 2-dimensional figure would be a square.

Circle

In a 2D plane, the circle consists of points that are all equally distanced with respect to one point in the shape's center. This means that it has no vertices. In other words, we can also understand a circle as a uniquely curved line that is equally distanced from the center at all of its points.

The distance from the circle's points to its center is called the radius. Also, if we measure from one point of the circle to another, passing through the circle's center, the distance is called the diameter. The diameter is always twice the length of the radius.

There are more two-dimensional shapes in math that we can classify based on aspects like the number of sides and vertices as well as their structure.

Perimeter of a 2-dimensional figure

In math, the perimeter of a 2D figure is the total sum of the length of all of its sides. Therefore, if the sides of the plane figure are expressed in the length unit of meters, for example, the perimeter of the shape is also expressed with meters. We can express the perimeter with the following formula:

P=a1+a2+a3+...+an=∑ani=1n

In the perimeter formula, the terms a1+a2+a3 (and so on) represent the different sides in the two-dimensional figure. In the second part of the equation is a symbol (∑) which indicates that all these side lengths should be summed up.

Perimeter of a triangle

Let’s take a look at the 2D shapes with the lowest number of sides: the triangles. The triangle has three sides; therefore, the perimeter of the triangle is equal to the sum of those three sides. Let's take a look at an example of the perimeter calculation below.

In the picture above, we have an isosceles triangle in 2D. This type of triangle has two sides of the same length and a third side with a different length. If we compute the perimeter of this 2D figure, we obtain:

P = a + b + c = 3m + 3m + 1m = 7m

Perimeter of squares and rectangles

Even though a triangle, a square, and a rectangle are not the same, we can still calculate their perimeters with the same formula given above. And if we have any other 2D figure, this process of summing up all sides remains the same as well.

For squares and rectangles, we have to sum up four sides to calculate the perimeter. The perimeter of the square is a+a+a+a, where a is the side length of all four sides. The perimeter of a rectangle is a+a+b+b, where a and b are the two different side lengths of the equal opposite pairs. Let's see some examples.

Eva has a whiteboard that measures 46 cm by 60 cm. what is the perimeter of this board?

Solution: Two different side lengths are given, and we know that a whiteboard has four sides. So, the figure will be a rectangle. The perimeter of this rectangle =46+46+60+60 =212 cm

Find the perimeter of the given figure.

Solution: The perimeter of the above square figure is:

Perimeter=a+a+a+a =25+25+25+25 =100 cm

Now you may be wondering, "But what about the circle?" Calculating a circle's perimeter of course cannot be done with side lengths! We defined the circle as a 2D shape formed by points that are all equally distanced from the center. To calculate the perimeter of a circle in 2D (also called the circumference), we use a different formula:

P = 2πr

In this formula, r is equal to the radius of the circle and π is the number pi, which has a fixed value. From this formula, we see that the perimeter of a circle is proportional to its radius. So, if we increase the radius of a circle, we also increase its perimeter.

The diameter of a circle is given as 14 cm. What is the perimeter or circumference of this circle?

Solution: The circle's diameter was given as d=14 cm.To calculate the perimeter, we need to find the radius. And we know that the diameter is twice the length of the radius.

⇒d=2r⇒r=d2 =142 =7 cm

So, the perimeter of a circle is:

P=2πr =2×π×7 =44 cm

Hence, the perimeter of the circle is 44 cm.

In math, the area of a two-dimensional figure is the quantity of surface delimited by the perimeter of a figure in a plane. In other words, the area in 2D is the space inside the lines we use to draw a figure. We use square units to describe area, like square meters (m2) or square feet (ft2).

Now, take a look out from your computer at the floor of the room. Imagine the walls as lines of a shape in 2D. The surface of the floor you are observing is its area because it is the space inside the perimeter (in this case, the room's walls).

Depending on two-dimensional figure and its shape, we have different formulas to compute area.

Area of a triangle

Starting again with the 2D shape with the lowest number of vertices, the area of the triangle is calculated with the following math formula:

A = 12bh

The area of the triangle depends on the base b of the triangle and its height h, which is the distance from the middle of the base to the opposite vertex. The base of the triangle does not need to be its shortest side: it can be any side. However, we then need to measure the height from the side chosen as the base to the opposite vertex.

A triangle has a base of 13 inches and a height of 6 inches. What is the area of this triangle?

Solution: Here, base b=13 inches and height h=6 inches. So the area is:

A=12×b×h =12×13×6 =13×3 =39

So, the area of the given triangle is 39 inches2.

Area of squares and rectangles

The area measurement for the square and the rectangle are the same, but we will describe the area of the rectangle first, as it is more general with this math formula:

A= bh

In this case, b is one side and h is another side with a different value. This area computation works for any 2D figure with four sides that are parallel to each other, called a parallelogram. Therefore, it also works for the square, but as all the sides have the same length in a square, we can also calculate its area as:

A = b2=b×b

Where b is the length of any side.

We have a tablecloth of size 70 inches by 70 inches. What is its area?

Solution: Here, both sides are the same length, so it is a square with length b=70 inches. The area of the square tablecloth is:

A=b2 =(70)2 =4900

The area of the table cloth is 4900 inches2.

Lastly, we have the area of the circle. As with the perimeter, its area also depends on the radius. The area of a circle can be calculated with the following equation:

A=πr2

Again, the r corresponds to the radius of the circle, and π is the number pi. From the formula, we see that if we make the radius bigger and bigger, the area of the circle also grows (in this case, by the power of two).

For example, you could see how this relationship works in real life in a garden. Imagine you attach a rope to some point and spin it in circles around that point. This motion would describe the shape of a 2D circle. If you moved the spinning rope further away from its center point, increasing the radius of the circle, you would see that the area of the spinning rope is now bigger.

Find the area of a circle with radius r=5.2 cm and round it to the nearest tenth.

Solution: The area of the circle is:

A=πr2 =3.14×5.22 =3.14×5.2×5.2 =84.9056 ≈84.9 cm2

Further representations of 2-dimensional figures

We have previously seen some 2D shapes such as the triangle, the square, the rectangle, and the circle. But an infinite number of figures exist that you could describe. In general, we classify two-dimensional figures by their number of sides and vertices as well as their internal angles (formed by the vertices).

If we increased a rectangle's sides by one, it would have five sides, making it a pentagon. With six sides, it'd be a hexagon, and so on.

There are also different types of four-sided two-dimensional figures. Apart from the rectangle and the square, if a 2D shape has at least two equal sides and its angles are not 90º, it is a rhombus, with a shape similar to a diamond.

There are a lot of different 2D shapes, with regular sides, irregular sides, equal angles, etc. Now you just have to use some imagination and try to search for examples of them!

2 Dimensional Figures - Key takeaways

• In math, two-dimensional figures consist of figures with two dimensions: length and width. They are also called polygons.
• We can classify two-dimensional figures by the number of sides and vertices, the sides' lengths, and the internal angles they form.
• Some of the most-used shapes in math are the triangle, the square, the rectangle, and the circle.
• The circle consists of points that are all equally distanced with respect to one point in the shape's center. This means it has no vertices.
• The perimeter is the sum of all side lengths of the shape. For the circle, it is directly proportional to its radius.
• The area of the figure is the 2D surface delimited by its sides. Depending on the figure, we use different math formulas to compute its area.
• There are shapes with five sides called pentagons, six sides called hexagons, and more. Also, there are more examples of shapes with four sides, such as the rhombus.

Page 3

Maybe you are reading this in front of your computer. Or maybe you have a glass of water next to you.

If you look at any of these objects that surround you, it is clear that they are objects in 3d. But, what is the math definition for a three-dimensional figure?

In this article, we will learn more about 3-dimensional figures and their applications.

What is a 3-dimensional figure?

A three-dimensional shape is a geometric body with 3 dimensions of space that are, length, width, and depth. Sometimes the depth is referred to as height.

For example, imagine you grab a box from a certain delivery company.

If you put the box in a way that you can only observe one of its faces, you will be observing a plane surface in 2d, and then you will be observing just the length and width of that face.

But if you turn it a little bit you will see that the box also has some depth. That is what we refer to with three-dimensional figures.

As you may have observed with the box, these three-dimensional shapes have volume. In math, we define volume as the quantity of space inside a closed surface.

Grabbing the box again and if you open it now, the volume would be the quantity of space inside the box. We will learn later how to compute this volume.

These geometric shapes generally, except for some exceptions we will use, have faces which are the surfaces with a certain surface area that delimitate the figure. These faces join in vertices, which are points of union.

Finally, the lines that delimitate these surfaces and the contour of the geometric figure are named edges. We would compare them with the sides of the 2-dimensional shapes.

3-dimensional figures examples

Taking the look away from this article and looking around you, you will probably identify a lot of three-dimensional figures with different structures. From the bed to the chair, to the table or even to the books you use to study. All of them are 3d shapes as they have the 3 dimensions we mentioned before; length, width, and depth, and also because they have volume.

We distinguish between regular and irregular 3d shapes. We will focus on the regular three-dimensional figures, as they are more common in math.

Cone

A cone is a three-dimensional figure that we would obtain if we make a right triangle (that has one angle equal to 90º) turn around with one of its sides fixed, so we get a shape in 3d. This figure normally has a circular base and a vertex where the lateral surface of the cone tapers to.

The base does not have to necessarily be a circle, it can also be another two-dimensional circular figure such as an oval. You can observe this shape in the real world when you look at the traffic cones.

Pyramid

This figure is similar to the cone, but in this case, the base does not have a circular form. The base is a two-dimensional figure with three or more sides such as a triangle, square, rectangle, etc.

As the geometric form of the base can vary, it also changes the number of edges. All of its surfaces, no matter how much it has, taper to a vertex.

The famous pyramids of Egypt are one example of these geometric shapes, in this case, they have a squared base.

Cube

This geometric figure consists of six faces of the equal-area meeting three of them in a single vertex, with a total of eight vertices and a total of twelve edges.

An example of a cube is a dice. If you observe it, all of the faces of a regular dice have the same surface and each vertex of it works as a union for three different faces.

Rectangular prism

It is similar to the cube, as it also has eight vertices, twelve edges, and six faces, but in this case, all of the faces are not equal. Each face is equal to its opposite, therefore we have pairs of equal faces.

An example of a rectangular prism could be a drawer or even a box, although sometimes they have the shape of a cube.

There exist other kinds of prisms, regarding the shape of its base and the opposite face. For example, if these faces have the shape of a triangle, it is a triangular prism that will have five faces in total instead of the six faces that the rectangular prism has. But this base (and the opposite face) can have another 2-dimensional figure that gives different types of prisms: pentagonal prisms, hexagonal prisms, etc.

Cylinder

The shape of this figure can remind you of a rectangular prism, but in this case, it has two surfaces, which are called the top and bottom (or base) of the figure, that consist of two-dimensional circular figures.

This figure does not have any vertex. The surface that connects those two faces is essentially a rectangle but curved.

You can find these kinds of geometric shapes in cans or some glasses.

Sphere

A football, a basketball, or maybe, if we do not want to just limit ourselves to the sports world: a bubble. All of these objects share one common thing: they are spheres.

These geometric shapes are obtained if we make a circle, which is a two-dimensional figure, turned around its diameter. The volume this revolution describes is defined as a sphere.

As it happens with the circle in two dimensions, all of the points of the surface are equally distanced from the point in the center of the figure. This distance is called the radius. If we trace a distance between two points of the surface of the sphere that goes through the center of it, this distance is called the diameter of the sphere, which corresponds to two times the radius.

Formulas of 3-dimensional figures

When working with 3d shapes, there are some things we might want to know about them. In particular, there are two characteristics we are interested in.

The first one is the area of the figure.

The area of the figure is the quantity of surface that the faces of the figure occupy. The units for the surface area of the figure are the units of area, being the square meter the standard one (m2).

To obtain the total surface area of the figure we have to sum the areas of each face of the shape. We should not confuse the surface area of the figure with its volume. The area consists only of the surface of the faces, independently of what is inside of them.

On the other hand, we have the volume of the figure.

The volume of a figure is the quantity of space there is inside the surface delimited by the faces of the figure. The units for the volume are the units of volume, being the cube meter as the standard one.

If we grab again the box we have talked about in this article, you can see that the surface of the cardboard used for all of the faces corresponds to the surface area of the box, but the space there which is inside the box corresponds to its volume.

Let’s see how some of the math equations for the 3d shapes we have seen before.

Area and volume of a cone

The surface area of a three-dimensional figure is the sum of the areas of its faces.

For a cone, the surface area of its base is , where r is the radius of the circle. The area of the lateral face is , being g the distance between any point of the edge of the base to the vertex. Therefore, the surface area of a cone can be generally expressed as,

.

The volume for a cone is given by the following formula,

,

where h is the distance from the center of the base to the vertex.

Area and volume of a pyramid

In this case, the formulas of the area and volume will depend on the number of edges the base has.

For example, if the pyramid has a squared base, the surface area of the pyramid will be the sum of the area of the square with the sum of the areas of each triangle that connects the vertices . In general, we can express the surface area of a pyramid as,

Be careful, as the base does not have to be regular, and the surface area of the triangles that connect with the vertex does not have to be either.

The volume of a pyramid will also depend on the base it has. For a square pyramid, the volume follows the formula,

being

• h the distance from the center of the base to the vertex

• l the length of the edges of the base.

Area and volume of a rectangular prism and a cube

In this case, as the rectangular prism and the cube are formed by six faces, to obtain the total surface area of the figure we just have to sum the areas of each face.

For the cube, all six faces will have the same area, but for the rectangular prism, as each face is equal to its opposite, there are three different values. A general math expression for the surface area of a rectangular prism is,

where A1 , A2 , and A3 are the three different values of those areas. The area of a rectangle is .

The volume for those shapes is the multiplication of the three edges; the length, the width, and the depth of the prism, such as,

In the case of the cube, as all of the sides are equally long, we have,

Area and volume of a cylinder

The cylinder consists of two circles that are the top and bottom of the figure and a curved rectangle. Therefore, if the area for a circle is , the sum of all the areas is,

where h is the height from one point of the bottom to the point in the top at the same position.

The volume for the cylinder is described by the following equation,

Area and volume of a sphere

The sphere we know is a different type of geometrical figure, as it is not formed by the union of different faces. That is why we need a math expression to compute its surface area,

And the volume for the sphere is determined by the following formula,

.

Examples of problems on 3-dimensional figures

Now, let us look into some examples of problems you may encounter on 3-dimensional figures.

Find the volume of water that is needed to fill a cylindrical glass cup of height 12cm and radius 7cm. Take .

Solution

Using

then,

Kohe wishes to make a conical cap of a radius of 14cm and a height of 20cm for 8 friends ahead of his birthday party. What is the total area of the cardboard paper does he need to make all 8 for his friends?

Solution

First we find the total surface area of one conical cap. Using

In this case, g is the height of the cone which is 20cm and r is 14cm. Hence,

But this is just the area of 1onecone, you need to find the area of 8 cones. Thus,

Hence Kohe would need a cardboard with a total surface area of 11,968cm2 to successfully make 8 conical caps for his friends ahead of his birthday party.

3-Dimensional Figures - Key takeaways

• Three-dimensional figures consist of shapes with three dimensions; length, width, and depth. Sometimes depth is referred to as height.
• These figures have surfaces that formed them called faces. The faces join themselves in vertices. And the lines that delimit these faces are called edges.
• There are lots of different examples of 3d shapes. Some of the most used figures are the cone, the pyramid, the cube, the prisms, the cylinder, and the sphere.
• Some 3-dimensional figures such as the cone, the pyramid, or the sphere are obtained if you make a two-dimensional figure revolute around one of its axes or edges.
• The area of a three-dimensional figure is the surface occupied by its faces. Generally, the area of a three-dimensional figure is obtained by summing the surface areas of all of its faces. The volume of the 3d shapes is the space that is inside of the surface delimited by its faces. To obtain it we use different formulas regarding the figure we want to calculate the volume of.

Page 4

When you look at a plain sheet of paper, you would only take notice of its 2 dimensions, i.e. looking only at the length and breadth possibly because it is so flat. However, what happens when a box is now placed in front of you? Your vision seems to have upgraded to 3 dimensions because you are not just considering the length and breadth but the height or perhaps the thickness of the box. This article will explore 3-dimensional vectors.

What are 3 Dimensional Vectors?

3-dimensional or 3D vectors are vectors that are represented on a three-dimensional plane or space to have three coordinates such as the x, y and z.

If we imagine a 3D plane with axis i, j and k, (which represents the x, y, and z-axis respectively) we can write a 3D vector as the sum of its i, j and k component.

Imagine a vector which travels from the origin (0,0,0) and goes to the coordinates (3,2,5). We could write that vector as

For this vector, the i component would be 3, the j component would be 2 and the k component would be 5.

The three-dimensional vector has three coordinates which are represented in the x, y and z-axis. Recall that in a two-dimensional plane, you have coordinates only on the x and y-axis. Thus, in a 2D vector coordinates are given in the form (x, y). However, the coordinates of 3D vectors are given in the form (x, y, z)

How do you plot a 3D vector?

Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y-axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.

3-Dimensional axis (math.brown.edu)

After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.

Vector can also be written in matrix form. In this form, we can write the vector as three rows by one column matrix. The first row is the i component, the second row is the j component and the third row is the k component.

We do not write the x, y, and z terms in matrix form.

If we use the vector above as an example, we get:

We can combine two vectors to find the dot product of these vectors.

Suppose we have vector and vector , the dot product can be found by following the method below:

Step 1: Transpose vector , that is, convert it from a 3 rows by 1 column vector to a 1 row by 3 column vector.

For vector , vector

Step 2: Write the dot product of both vectors as the multiplication of both matrices.

Step 3: Perform the matrix multiplication:

Step 4: Simplify the matrix. You should end up with a 1-by-1 matrix.

Let vector , and vector . Find the dot product of vectors and .

Solution:

Writing both vectors in matrix form, we get:

and

Step 1:

Step 2:

Step 3:

Step 4:

Essentially, there are two main 3D equations. However, a third equation which is the angle between 3D vectors is derived from these two main equations. The two main equations are the dot product and the magnitude of a 3D vector equation.

Dot product of 3D vectors

For two certain 3D vectors A (x1, y1, z1) and B (x2, y2, z2) which are represented in the vector form

and

The dot product is

Find the product of Vector G and K located (-1, 2, 3) and (0, 5, 1) of a plane.

Solution:

By applying the dot product formula

Then,

The magnitude of a three-dimensional vector is derived using the extended Pythagoras theorem. Recall that the Pythagoras theorem is applied knowing the x and y-axis are perpendicular, note that the additional z-axis in 3D is perpendicular to both the x and y-axis. Hence, in order to calculate the magnitude of a certain 3D vector A (x1, y1, z1) which is represented in the vector form.

apply

Find the magnitude of vector C given by

Solution:

Since the magnitude of a vector is calculated as

Then the magnitude of vector C is

To find the angle between two corresponding 3D vectors, use the formula below:

Where is the angle between vectors a and b, is the dot product of vectors a and b, and where and are the magnitudes of vector a and vector b respectively.

Find the magnitude of the vector traveling from the origin to the coordinates (2,1,2).

Solution:

The vector can be written as

Using the equation above:

Therefore:

The magnitude of the vector is 3 units.

We can now combine all that we have learned to find the angle between two vectors!

Find the angle between vectors and vector .

Solution:

Writing the matrix form of these vectors:

and

Writing vector in transcript form:

Therefore:

The magnitude of vector is:

The magnitude of vector is:

Since:

Hence:

3-Dimensional Vectors - Key takeaways

• 3D vectors have values i, j, and k for their x, y, and z-axis respectively.
• 3D vectors can be written in matrix form.
• In this form, we can find the dot product of two vectors by performing matrix multiplication.
• By also finding the magnitude of those vectors through an extended version of Pythagoras' theorem, we can find the angle between those vectors.
• Graphing vectors comprise of drawing the axes, the coordinates where the vector ends and begins, and sketching a line connecting both points.

It is a line segment in three-dimensional space from Point A to Point B.

1. Write vector A in transpose form.

2. Write the dot product as a multiplication of the transposed matrix and the matrix for vector B.

3. Perform the vector multiplication.

4. Simplify.

Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.

After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.

Generally, slope is only applicable to 2D vectors and as such the slope of vectors in 3D is undefined.

Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.

After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.

3 dimensional vector problems are questions about 3D vectors which may require you to determine the magnitude of 3D vectors, angle between two 3D vectors or anything related to vectors in 3D.

Question

Answer

3D vectors are vectors which are represented on a three dimensional plane or space to have three coordinates such as the x, y and z.

Question

What are the coordinates of a 3D vector? 

Answer

Question

How do you plot a 3D vector? 

Answer

Begin by drawing a set of axis. Firstly, draw the vertical z-axis. Perpendicular to that, draw a y-axis. In between the z and y axis, draw the x-axis. Note that all 3 axes are perpendicular to each other.

After that, place a scale on each axis and mark the point where the head of the vector arrives. Then draw an arrow between the origin and the head of the vector. Finally, mark the coordinates of the head of the arrow.

Question

Can vectors be written in matrix form?

Question

P(1,2,-4) and Q(0,1,0) what is the dot product of P and Q.

Question

Q(0,1,0), what is the magnitude of Q.

Question

What is the difference between a 2D and 3D coordinate system?

Answer

A 2D has only x and y coordinates while a 3D has x, y and z coordinates.