The height of a circular segment is $6\text{cm}$ and the radius of the circle is $13\text{cm}$. Find the area of the segment (correct to one decimal place).

Step-by-step solution

1. For a segment, let $\theta$ be the central angle in radians. The height $h$ relates to $\theta$ by $h=r(1-\cos \frac{\theta }{2})$.
2. Thus $\cos \frac{\theta }{2}=\frac{r-h}{r}=\frac{13-6}{13}=\frac{7}{13}$.
3. Therefore $\frac{\theta }{2}={\cos }^{-1}\left(\frac{7}{13}\right)\approx 57.5^{\circ }=1.0035\text{rad}$, so $\theta \approx 2.007\text{rad}$.
4. Sector area = $\frac{1}{2}{r}^2\theta =\frac{1}{2}\times 13^2\times 2.007\approx 169.5{\text{cm}}^2$.
5. Triangle area = $\frac{1}{2}{r}^2\sin \theta =\frac{1}{2}\times 13^2\times \sin (2.007)\approx 76.6{\text{cm}}^2$.
6. Segment area = sector area $-$ triangle area $\approx 169.5-76.6=92.9{\text{cm}}^2$.

Concept tested: Area of a circular segment

Published 21 August 2026 · How this solution was prepared and checked

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