A circular segment refers to the region enclosed between a chord and an arc of a circle. Calculating the area of a circular segment involves a straightforward formula that requires knowing two key parameters: the radius of the circle and the measure of the central angle subtended by the segment. This article will guide you through the step-by-step process of calculating the area of a circular segment.

To begin, let’s define the necessary terms. The radius of a circle is the distance from the center of the circle to any point on its circumference. The central angle is the angle formed by two radii from the center of the circle, extending to the endpoints of the chord that forms the segment. Understanding these terms is essential for accurately calculating the area of a circular segment.

The formula used to calculate the area of a circular segment can be derived from the area of a sector of a circle. Remember that a sector of a circle is a region bounded by two radii and an arc, while a segment is bounded by a chord and an arc. The formula for the area of a sector is (θ/360)πr², where θ represents the central angle and r is the radius of the circle.

Using the formula for the area of a sector, we can deduce the formula for the area of a circular segment. First, calculate the area of the entire sector using the given angle and radius. Next, determine the area of the triangle formed by the sector. Finally, subtract the area of the triangle from the area of the sector to obtain the area of the circular segment.

Let’s now break down the step-by-step process to calculate the area of a circular segment:

1. Measure the central angle (θ) in degrees and record the value.
2. Measure the radius (r) of the circle and record the value.
3. Use the formula for the area of a sector: sector_area = (θ/360)πr². Calculate this value.
4. Calculate the area of the triangle formed by the sector: triangle_area = (1/2) * r * r * sin(θ). Use the sine function to find the value of sin(θ) and calculate the triangle’s area.
5. Subtract the area of the triangle from the area of the sector to obtain the area of the circular segment: segment_area = sector_area – triangle_area.

It’s worth noting that the angle measurement must be in degrees, as the formula uses the degree measure. If the angle is given in radians, it can be converted to degrees by multiplying it by (180/π).

In conclusion, calculating the area of a circular segment involves understanding the radius, central angle, and applying a simple formula derived from the area of a sector. By following the step-by-step process outlined in this article, you can accurately determine the area of a circular segment using just the given radius and central angle.

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