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
![The diameter of the smaller semicircle is the chord of the larger semicircle](https://aplusreach.com/wp-content/uploads/2021/11/Find-the-Area-between-Two-Semicircles-1-1024x576.jpg)
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](https://aplusreach.com/wp-content/uploads/2021/11/Find-the-Area-between-Two-Semicircles-2-1024x576.jpg)
Apply Pythagorean theorem in triangle BOQ
OQ² + OB² = BQ²
⇒ (r − 1)² + (r − 2)² = r²
⇒ r² − 2r +1 + r² − 4r + 4 = r²
Now, r² − 6r + 5 = 0
Factorise r² − 6r + 5
Then, r² − 6r + 5 = r² − r − 5r + 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 cm
That 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 = 25π/2 cm²
Area of the smaller semicircle = ½ × π × 3² = 9π/2 cm²
Difference between area of semicircles gives the blue area, so
Blue area = 25π/2 − 9π/2 = 16π/2
⇒ Blue area = 8π cm²