How to find the area of a circular segment
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Video: How to find the Area of Segment of a Circle 2024, July
One of the common geometric problems is to calculate the area of a circular segment - the part of a circle bounded by a chord and the corresponding chord of an arc of a circle.
The area of the circular segment is equal to the difference between the area of the corresponding circular sector and the area of the triangle formed by the radii of the corresponding segment of the sector and the chord bounding the segment.
Example 1
The length of the chord contracting a circle is equal to the value of a. The degree measure of the arc corresponding to the chord is 60 °. Find the area of the circular segment.
Decision
A triangle formed by two radii and a chord is isosceles, so the height drawn from the top of the central angle to the side of the triangle formed by the chord will also be the bisector of the central angle, halving it and the median, halving the chord. Knowing that the sine of the angle in a right-angled triangle is equal to the ratio of the opposite side to the hypotenuse, we can calculate the radius:
Sin 30 ° = a / 2: R = 1/2;
R = a.
The area of the sector corresponding to a given angle can be calculated by the following formula:
Sc = πR² / 360 ° * 60 ° = πa² / 6
The area of the triangle corresponding to the sector is calculated as follows:
S ▲ = 1/2 * ah, where h is the height drawn from the top of the central angle to the chord. By the Pythagorean theorem, h = √ (R²-a² / 4) = √3 * a / 2.
Accordingly, S ▲ = √3 / 4 * a².
The area of the segment, calculated as Sseg = Sc - S ▲, is equal to:
Sseg = πa² / 6 - √3 / 4 * a²
Substituting a numerical value instead of a, you can easily calculate the numerical value of the segment area.
Example 2
The radius of the circle is equal to a. The degree measure of the arc corresponding to the segment is 60 °. Find the area of the circular segment.