Final answer:
The area of the segment in a circle with a 60-degree angle at the center and a radius of 21 cm is approximately 231 cm², by subtracting the area of an equilateral triangle from the area of the sector.
Step-by-step explanation:
The area of the segment formed in a circle by an arc subtending a 60-degree angle at the center can be calculated using two areas: the area of the sector (part of the circle) and the area of the triangle formed by the two radii and the arc. The radius of the circle is given as 21 cm. First, we calculate the area of the sector which is a fraction of the circle's area, proportional to the angle subtended by the arc. Given that a full circle subtends 360 degrees, for a 60-degree angle, the area of the sector is ⅖ of the circle's area.
The area of the circle is A = π×r² (where r is the radius), which equates to A = π×(21 cm)². Thus, the area of the sector is (60°/360°)×π×(21 cm)². Next, we calculate the area of the triangle formed by the radii and the base length equal to the arc length. The arc length for a 60-degree angle in a circle with radius 21 cm is (60°/360°)×2π×21 cm. The area of the triangle can be found as ⅖ of the area of the rectangle formed by the radius and the arc length because it is an equilateral triangle (since all angles are 60 degrees and the sides are equal).Taking these into consideration, the final area of the segment is found by subtracting the area of the triangle from the area of the sector. After calculating, the closest match from the given options is 231 cm².