Question

Pls help 15. Find the exact area of the shaded region of the circle shown in the diagram.

Answer 1

• The shaded region is 9.83 cm²

Explanation:

Refer to attached diagram with added details.

Given

Circle O with:

• OA = OB = OD - radius
• OC = OD = 2 cm

To find

• The area of segment ADB.

Solution

Since r = OC + CD, the radius is 4 cm.

Consider right triangles OAC or OBC:

• They have one leg of 2 cm and hypotenuse of 4 cm, so the hypotenuse is twice the short leg.

Recall the property of 30°x60°x90° triangle:

• a : b : c = 1 : √3 : 2, where a- short leg, b- long leg, c- hypotenuse.

It means OC: OA = 1 : 2, so angles AOC and BOC are both 60° as adjacent to short legs.

In order to find the shaded area we need to find the area of sector OADB and subtract the area of triangle OAB.

Area of sector:

• A = π(θ/360)r², where θ- central angle,
• A = π*((mAOC + mBOC)/360)*r²,
• A = π*((60 + 60)/360))(4²) = 16.76 cm².

Area of triangle AOB:

• A = (1/2)*OC*(AC + BC), AC = BC = OC√3 according to the property of 30x60x90 triangle.

• A = (1/2)(2*2√3)*2 = 4√3 = 6.93 cm²

The shaded area is:

• A = 16.76 - 6.93 = 9.83 cm²