# Find the Area between Two Semicircles

Find the area between two semicircles, when a small semicircle is inscribed in a larger semicircle as shown in the figure. (find the area of blue shaded region). The diameter of the smaller semicircle is the chord of the larger semicircle

## Solution

## Area of the blue region = Area of the larger semicircle *−* Area of the smaller semicircle

We can redraw the figure as shown below

Here OA = OB

∠BOQ = 90°

Let the radius of the larger semicircle =* r*, then

OR = *r −* 2 → OR is the radius of the smaller semicircle

OR = OA = OB = *r* *−* 2 {*radius of the semicircle*}

### Apply Pythagorean theorem in triangle BOQ

OQ² + OB² = BQ²

⇒ (*r* *−* 1)² + (*r* *−* 2)² = *r*²

⇒ *r*² *−* 2*r* +1 + *r*² *−* 4*r* + 4 = *r*²

Now, *r*² *−* 6*r* + 5 = 0

Factorise *r*² *−* 6*r* + 5

Then, *r*² *−* 6*r* + 5 = *r*² *−* *r* *−* 5*r* + 5 = *r*(*r* *−* 1) *−* 5(*r* *−* 1) = (*r* *−* 1)(*r* *−* 5) = 0

so *r* = 1 cm*or* *r* = 5 cm

The radius of the larger semicircle is greater than **2 cm** so ** r = 5 cm** and

*r**−*2 = 3 cmThat is radius of larger semicircle = 5 cm and radius of the smaller semicircle = 3 cm

### Now we can find the area of semicircles

**Area of the larger semicircle = **½ × *π *×* *5² =** 25 π/2 cm²**

**Area of the smaller semicircle = **½ × *π *×* *3² =** 9 π/2 cm²**

### Difference between area of semicircles gives the blue area, so

Blue area = 25*π*/2 * −* 9

*π*/2 = 16

*π*/2

⇒ **Blue area = 8 π cm² **