Calculate the area of the shaded region, which is a circular segment with radius 35 cm and central angle…

Question

Calculate the area of the shaded region, which is a circular segment with radius 35 cm and central angle 65.76 degrees.

ScanSolve AI Tutor · Accepted Answer

To calculate the shaded area, we need to find the area of the circular segment in the circle with center R. The shaded area is the region bounded by the arc AB and the chord AB in the circle with center R.

The formula for the area of a circular segment is:

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

For the circle with center R:

Radius, $r = 35 \text{ cm}$
Central angle, $\theta = \angle ARB = 65.76^\circ$

Calculate the area of the sector ARB.
The formula for the area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$ .
$A_{\text{sector}} = \frac{65.76^\circ}{360^\circ} \times \pi (35 \text{ cm})^2$ $A_{\text{sector}} = \frac{65.76}{360} \times \pi \times 1225$ $A_{\text{sector}} \approx 0.1826666667 \times 1225 \times 3.1415926535$ $A_{\text{sector}} \approx 703.9510 \text{ cm}^2$
Calculate the area of the triangle ARB.
The formula for the area of a triangle given two sides and the included angle is $\frac{1}{2}ab\sin C$ . Here, $a=b=r$ .
$A_{\text{triangle}} = \frac{1}{2} r^2 \sin(\theta)$ $A_{\text{triangle}} = \frac{1}{2} (35 \text{ cm})^2 \sin(65.76^\circ)$ $A_{\text{triangle}} = \frac{1}{2} \times 1225 \times \sin(65.76^\circ)$ $A_{\text{triangle}} = 612.5 \times 0.9117601$ $A_{\text{triangle}} \approx 558.4561 \text{ cm}^2$
Calculate the shaded area (Area of segment).
$A_{\text{shaded}} = A_{\text{sector}} - A_{\text{triangle}}$ $A_{\text{shaded}} \approx 703.9510 \text{ cm}^2 - 558.4561 \text{ cm}^2$ $A_{\text{shaded}} \approx 145.4949 \text{ cm}^2$
Rounding to two decimal places, the shaded area is $145.49 \text{ cm}^2$ .

Calculate the area of the shaded region, which is a circular segment with radius 35 cm and central angle 65.76 degrees.

Answer

Step-by-step solution

Calculate the area of the sector ARB.

Calculate the area of the triangle ARB.

Calculate the shaded area (Area of segment).

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