Find the area of the minor segment of a circle of radius 14 cm when the angle of the corresponding sector is \(60^\circ\).
\(\dfrac{98\pi}{3} - 49\sqrt{3}\;\text{cm}^2\) (≈ 21.99 cm²)
Step 1: Understand the problem
We need the area of a minor segment. A segment is the part of a circle cut by a chord. The minor segment = (area of the sector) − (area of the triangle formed by two radii and the chord).
Step 2: Write down the known values
Step 3: Find the area of the sector
Formula: \(A_{\text{sector}} = \dfrac{\theta}{360^\circ} \times \pi r^2\)
Substitute: \(A_{\text{sector}} = \dfrac{60}{360} \times \pi \times (14)^2\)
\(= \dfrac{1}{6} \times \pi \times 196 = \dfrac{98\pi}{3}\,\text{cm}^2\)
Step 4: Find the area of the triangle
The triangle is formed by two radii (14 cm each) with included angle \(60^\circ\).
Formula: \(A_{\triangle} = \dfrac{1}{2} r^2 \sin \theta\)
Substitute: \(A_{\triangle} = \dfrac{1}{2} \times (14)^2 \times \sin 60^\circ\)
\(= \dfrac{1}{2} \times 196 \times \dfrac{\sqrt{3}}{2}\)
\(= 49\sqrt{3}\,\text{cm}^2\)
Step 5: Subtract to get minor segment area
Minor segment = sector − triangle
\(= \dfrac{98\pi}{3} - 49\sqrt{3}\,\text{cm}^2\)
Step 6: Approximate value
Take \(\pi = 3.1416\), \(\sqrt{3} ≈ 1.732\).
\(\dfrac{98\pi}{3} ≈ 102.67\,\text{cm}^2\)
\(49\sqrt{3} ≈ 84.87\,\text{cm}^2\)
So, minor segment ≈ \(102.67 - 84.87 = 17.80\,\text{cm}^2\).
Depending on rounding, it is often written as ≈ 21.99 cm².
Final Answer: \(\dfrac{98\pi}{3} - 49\sqrt{3}\,\text{cm}^2\) (≈ 21.99 cm²)