Question

whats the approximate area of a segment of a circle with a height 6 m and the length of the chord is 20 m? round your answer to the nearest whole number.
a) 85.4 m²
b) 74.6 m²
c) 746.67 m²
d) 8.54 m²

Explanation:

Step1: Find the radius of the circle

Let the radius of the circle be \(r\). Using the formula \(r=\frac{h}{2}+\frac{c^{2}}{8h}\), where \(h = 6\) (height of the segment) and \(c=20\) (length of the chord).

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Step2: Find the central angle \(\theta\)

Using the formula \(\sin\frac{\theta}{2}=\frac{c}{2r}\), substitute \(c = 20\) and \(r=\frac{34}{3}\)

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\(\frac{\theta}{2}\approx\sin^{- 1}(0.8824)\approx61.93^{\circ}\), so \(\theta\approx123.86^{\circ}\)

Step3: Find the area of the sector

The area of the sector \(A_{sector}=\frac{\theta}{360}\times\pi r^{2}\), substitute \(\theta = 123.86^{\circ}\) and \(r=\frac{34}{3}\)

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Step4: Find the area of the triangle

The area of the triangle \(A_{triangle}=\frac{1}{2}\times c\times(r - h)\), substitute \(c = 20\), \(r=\frac{34}{3}\) and \(h = 6\)

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Step5: Find the area of the segment

The area of the segment \(A = A_{sector}-A_{triangle}\)

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Answer:

A. \(85.4\ m^{2}\)