Circle Minor Segment Area Calculation

This solution explains how to find the area of a minor segment of a circle based on the given radius and chord length.

Given Information

• Radius of the circle, $R = 12$ cm.
• Length of the chord, $L = 12$ cm.

Identifying the Central Angle

The two radii drawn to the endpoints of the chord, along with the chord itself, form a triangle. In this case, the lengths of the two radii ($12$ cm) are equal to the length of the chord ($12$ cm). This signifies that the triangle formed is an equilateral triangle.

• Sides of the triangle: $R, R, L$.
• Given values: $12 \text{ cm}, 12 \text{ cm}, 12 \text{ cm}$.
• Since all sides are equal, the triangle is equilateral.
• The angle ($\theta$) subtended by the chord at the center of the circle is $60^\circ$.
• Convert the angle to radians for area calculation: $\theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}$ radians.

Calculating Segment Area

The area of a minor segment is found by subtracting the area of the triangle (formed by the radii and chord) from the area of the circular sector (formed by the radii).

Formula: Area of Minor Segment = Area of Sector - Area of Triangle

Area of the Sector

• Formula: Area$_{sector} = \frac{1}{2} R^2 \theta$ (where $\theta$ is in radians).
• Calculation: Area$_{sector} = \frac{1}{2} \times (12 \text{ cm})^2 \times \frac{\pi}{3} = \frac{1}{2} \times 144 \text{ cm}^2 \times \frac{\pi}{3} = 72 \times \frac{\pi}{3} \text{ cm}^2 = 24\pi \text{ cm}^2$.

Area of the Triangle

• Formula: Area$_{triangle} = \frac{1}{2} R^2 \sin \theta$.
• Calculation: Area$_{triangle} = \frac{1}{2} \times (12 \text{ cm})^2 \times \sin(\frac{\pi}{3}) = \frac{1}{2} \times 144 \text{ cm}^2 \times \frac{\sqrt{3}}{2} = 72 \times \frac{\sqrt{3}}{2} \text{ cm}^2 = 36\sqrt{3} \text{ cm}^2$.

Area of the Minor Segment

• Calculation: Area$_{segment} = \text{Area}_{sector} - \text{Area}_{triangle} = 24\pi \text{ cm}^2 - 36\sqrt{3} \text{ cm}^2$.
• Result: Area$_{segment} = (24\pi - 36\sqrt{3}) \text{ cm}^2$.

The area of the minor segment is $(24\pi - 36\sqrt{3})$ square centimeters.