Find the area of segment of circle with radius 14cm and central angle 60 degrees

Question

Find the area of the minor segment of a circle with radius 14 cm and central angle 60°. (Use $\pi = \frac{22}{7}$ )

Solution — Step by Step

The area of a minor segment = Area of the corresponding sector − Area of the triangle formed by the two radii and the chord.

\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}

\text{Area of sector} = \frac{\theta}{360°} \times \pi r^2 = \frac{60}{360} \times \frac{22}{7} \times 14^2

= \frac{1}{6} \times \frac{22}{7} \times 196 = \frac{1}{6} \times \frac{22 \times 196}{7} = \frac{1}{6} \times \frac{4312}{7}

= \frac{1}{6} \times 616 = \frac{616}{6} = \frac{308}{3}\ \text{cm}^2

The triangle is formed by two radii (each 14 cm) and the chord. The angle between the radii is 60°.

Since two sides are equal (both radii = 14 cm) and the angle between them is 60°, this is an equilateral triangle (all three angles = 60°, all three sides = 14 cm).

\text{Area of equilateral triangle} = \frac{\sqrt{3}}{4} \times a^2 = \frac{\sqrt{3}}{4} \times 14^2 = \frac{\sqrt{3}}{4} \times 196 = 49\sqrt{3}\ \text{cm}^2

Using $\sqrt{3} \approx 1.732$ :

= 49 \times 1.732 = 84.868 \approx 84.87\ \text{cm}^2

\text{Area of segment} = \frac{308}{3} - 49\sqrt{3}

= 102.67 - 84.87 = \mathbf{17.8\ \text{cm}^2}

(approximately, using $\sqrt{3} = 1.732$ )

Or exactly: $\text{Area} = \frac{308}{3} - 49\sqrt{3}$ cm²

Why This Works

A segment is the region between a chord and the arc it subtends. The area of the full sector (the “pie slice”) includes the triangular region. Subtracting the triangle gives us only the curved segment.

The 60° angle is a gift here: an isoceles triangle with two equal sides and the apex angle = 60° is automatically equilateral. This simplifies the triangle area calculation significantly compared to, say, a 120° problem.

Common Mistake

Students often use the formula $\frac{1}{2}r^2\sin\theta$ for the triangle area (which is correct — $\frac{1}{2}(14)(14)\sin 60° = 98 \times \frac{\sqrt{3}}{2} = 49\sqrt{3}$ ) but forget that this formula gives area in terms of the included angle, not the base and height. Both approaches are valid, but if you use the equilateral triangle formula $\frac{\sqrt{3}}{4}a^2$ , make sure you recognise first that the triangle IS equilateral (angle = 60°, equal sides). Don’t use $\frac{\sqrt{3}}{4}a^2$ for non-equilateral triangles.

For board exams: Always keep the answer in exact form first ( $\frac{308}{3} - 49\sqrt{3}$ ), then substitute numerical values. Writing the exact form first shows understanding and earns step marks even if arithmetic errors occur later.

Question

Solution — Step by Step

Why This Works

Common Mistake

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