Question

OBC is a sector of a circle with centre o and radius 9cm work out the area of the shaded region as a percentage of the area of sector obc to 1 decimal place

Answer 1

Answer:

83.7 %

Step-by-step explanation:

To get the area of the shaded region, we will following these steps.

Step one: Get the area of the sector of the circle.

The area of the sector of the circle can be obtained using the formula:

$Area = \frac{\theta}{360}\times \pi r^{2}$

Theta will be 60 degrees because triangle OAD is an equilateral triangle

Hence, $Area = \frac{60}{360}\times \pi \times 9^{2}= 42.42cm^{2}$

Step Two: Get the area of the equilateral triangle

The area of the equilateral triangle can be got using the formula

$Area = \frac{1}{2} Base \times Height$

The height of the triangle can be got by drawing an imaginary line bisecting the triangle into two parts and applying the Pythagoras' theorem to any of the resulting right-angled triangles formed.

This will be $height =\sqrt{4^{2}-2^{2}} =3.46cm$

Thus $Area = \frac{1}{2}\times 4\times 3.46=6.93cm^{2}$

Step Three: Subtract the area of the triangle from the area of the sector

Area of the shaded portion = $42.42cm^{2}-6.93cm^{2}=35.49cm^{2}$

Step Four: Divide the area of the shaded portion by the area of the entire shape and multiply the result by 100:

Expressing the ratio of the two areas as a percentage, we have

$\frac{35.49}{42.42} \times 100\approx 83.7 %$ %