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Read on to learn more about how to find the length of a circle.
Radius r, diameter d, and circumference c are lengths related to each other. The higher the radius or the diameter, the higher the circumference. The circle length formula connects these variables in a single equation: If you only know the diameter, you can find the length of the radius of the circle by simply remembering the diameter is twice the radius: Reordering the equations, you can also calculate the length of the circle diameter in terms of the circumference: Similarly, you can know what the length of the radius of the circle is:
Let's suppose you want to know what is the length of the radius of a circle of 10 cm circumference. You can follow these steps:
The length of half a circle is π (pi) times radius, expressed with the formula as chalf = πr or chalf = πd/2. It is the length of an ordinary circle, but in this case, divided by two.
18.8496 inches. To find the circumference of any circle, multiply its diameter by π. In this instance:
The measurement of angles in degrees goes back to antiquity. It may have arisen from the idea that there were roughly 360 days in a year, or it may have arisen from the Babylonian penchant for base 60 numerals. In any event, both the Greeks and the Indians divided the angle in a circle into 360 equal parts, which we now call degrees. They further divided each degree into 60 equal parts called minutes and divided each minute into 60 seconds. An example would be \(15^\circ22'16''\). This way of measuring angles is very inconvenient and it was realised in the 16th century (or even before) that it is better to measure angles via arc length. We define one radian, written as \(1^c\) (where the \(c\) refers to circular measure), to be the angle subtended at the centre of a unit circle by a unit arc length on the circumference. Definition of one radian. Since the full circumference of a unit circle is \(2\pi\) units, we have the conversion formula So one radian is equal to \(\dfrac{180}{\pi}\) degrees, which is approximately \(57.3^\circ\). Since many angles in degrees can be expressed as simple fractions of 180, we use \(\pi\) as a basic unit in radians and often express angles as fractions of \(\pi\). The commonly occurring angles \(30^\circ\), \(45^\circ\) and \(60^\circ\) are expressed in radians respectively as \(\dfrac{\pi}{6}\), \(\dfrac{\pi}{4}\) and \(\dfrac{\pi}{3}\).
Express in radians:
Solution
Students should have a deal of practice in finding the trigonometric functions of angles expressed in radians. Since students are more familiar with degrees, it is often best to convert back to degrees.
Find
Solution
Arc lengths, sectors and segmentsMeasuring angles in radians enables us to write down quite simple formulas for the arc length of part of a circle and the area of a sector of a circle. In any circle of radius \(r\), the ratio of the arc length \(\ell\) to the circumference equals the ratio of the angle \(\theta\) subtended by the arc at the centre and the angle in one revolution. Thus, measuring the angles in radians, \begin{alignat*}{2} & & \dfrac{\ell}{2\pi r} &= \dfrac{\theta}{2\pi}\\ &\implies\ & \ell &= r\theta. \end{alignat*}It should be stressed again that, to use this formula, we require the angle to be in radians.
In a circle of radius 12 cm, find the length of an arc subtending an angle of \(60^\circ\) at the centre. SolutionWith \(r=12\) and \(\theta = 60^\circ = \dfrac{\pi}{3}\), we have It is often best to leave your answer in terms of \(\pi\) unless otherwise stated. We use the same ratio idea to obtain the area of a sector in a circle of radius \(r\) containing an angle \(\theta\) at the centre. The ratio of the area \(A\) of the sector to the total area of the circle equals the ratio of the angle in the sector to one revolution. Thus, with angles measured in radians, \begin{alignat*}{2} & & \dfrac{A}{\pi r^2} &= \dfrac{\theta}{2\pi}\\ &\implies\ & A &= \dfrac{1}{2} r^2\theta. \end{alignat*}The arc length and sector area formulas given above should be committed to memory.
In a circle of radius 36 cm, find the area of a sector subtending an angle of \(30^\circ\) at the centre. SolutionWith \(r=36\) and \(\theta = 30^\circ = \dfrac{\pi}{6}\), we have \[ A = \dfrac{1}{2}\times 36^2\times \dfrac{\pi}{6} = 108\pi\text{ cm}^2. \]As mentioned above, in problems such as these it is best to leave your answer in terms of \(\pi\) unless otherwise stated. The area \(A_s\) of a segment of a circle is easily found by taking the difference of two areas. In a circle of radius \(r\), consider a segment that subtends an angle \(\theta\) at the centre. We can find the area of the segment by subtracting the area of the triangle (using \(\dfrac{1}{2}ab\,\sin\theta\)) from the area of the sector. Thus \[ A_s = \dfrac{1}{2}r^2\theta - \dfrac{1}{2}r^2\sin \theta = \dfrac{1}{2}r^2(\theta - \sin \theta). \]
Find the area of the segment shown. SolutionWith \(r=4\) and \(\theta = 18^\circ = \dfrac{18\pi}{180} = \dfrac{\pi}{10}\), we have \[ A_s = \dfrac{1}{2}\times 4^2\times \Bigl(\dfrac{\pi}{10} - \sin\dfrac{\pi}{10}\Bigr)\approx 0.041. \]
Exercise 15 Around a circle of radius \(r\), draw an inner and outer hexagon as shown in the diagram. By considering the perimeters of the two hexagons, show that \(3 \leq \pi \leq 2\sqrt{3}\).
Solving trigonometric equations in radiansThe basic steps for solving trigonometric equations, when the solution is required in radians rather than degrees, are unchanged. Indeed, it is sometimes best to find the solution(s) in degrees and convert to radians at the end of the problem.
Solve each equation for \(0 \leq x \leq 2\pi\):
Solution
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