Video Transcript
A chord and radius of a circle is 24 centimeters. Find the area of the minor circular segment, giving the answer to two decimal places.
We’ll begin by producing a sketch of this circle, which has a radius of 24 centimeters. There is then a chord, which is also of length 24 centimeters. Remember, a chord is a line segment connecting two points on the circumference of the circle, so the chord looks like this. Now we’re asked to find the area of the minor circular segment. Well, a chord always divides a circle into two segments: a minor segment, which is less than a semicircle, and a major segment, which is larger than a semicircle. So, the area we’re looking for is the area shaded in pink. To find this area, let’s first sketch in another radius of the circle, the radius connecting the other endpoint of this chord to the center of the circle. And of course, this radius is also of length 24 centimeters.
The area of this minor circular segment can then be found as the difference between the area of the sector outlined in orange and the area of the triangle formed by the two radii and the chord. Now, in order to find each of these areas, we need to know the central angle of the sector. This is relatively easy to work out though. Because the two radii and the chord are all of the same length, this is an equilateral triangle. And we know that all the angles in an equilateral triangle are 60 degrees, or 𝜋 by three radians. We can choose whether to work in degrees or radians in this problem. And as the formulae for the areas of sectors and segments are more straightforward in radians, let’s choose to work in radians.
The formula for the area of a sector with radius 𝑟 and central angle 𝜃 measured in radians is a half 𝑟 squared 𝜃. To find the area of the green triangle, we can use the trigonometric formula for the area of a triangle. In general, this is a half 𝑎𝑏 sin 𝐶, where 𝑎 and 𝑏 represent the lengths of two sides in the triangle and uppercase 𝐶 represents the measure of their included angle. In our triangle, we can use the two radii and the angle they enclose, which is the central angle of the sector 𝜃. So, we have a half 𝑟 squared sin 𝜃. Now this formula can be factored by a half 𝑟 squared. So, we have a half 𝑟 squared multiplied by 𝜃 minus sin 𝜃. And we can learn this as the general formula for the area of a circular segment if we wish.
To answer this problem, we need to substitute 24 for the value of 𝑟 and 𝜋 by three for the value of 𝜃. So, we have a half multiplied by 24 squared multiplied by 𝜋 by three minus sin of 𝜋 by three. A half multiplied by 24 squared is 288. And 𝜋 by three is one of the special angles for which the values of the trigonometric ratios sine, cosine, and tangent can be expressed exactly in terms of surds and quotients. sin of 𝜋 by three is equal to root three over two. So, we have 288 multiplied by 𝜋 by three minus root three over two.
Now we’re asked to give our answer to two decimal places, so we need to evaluate this on a calculator, which gives 52.1775 continuing. The digit in the third decimal place is a seven. So, we need to round up to 52.18. The units for the length in the question were centimeters, so the units for this area will be square centimeters. We found then that the area of this minor circular segment to two decimal places is 52.18 square centimeters.
