Name: Area of Segment
Uploaded: 2019-07-19
Duration: 7 min 50 s
Description: Do you know the result for the area of a segment? Well, this video talks about the derivation of it. So, let's dive in and master the concept in just a few m...

Question

Semi-circles are described on the sides

A B and C D

of a rectangle

A B C D

as diameters, both inside the rectangle. If

A B = 14 cm

and

B C = 7 cm

, find the area enclosed by the-semi circles.

Accepted Answer

Given, $A B = C D = 14 cm$ and $B C = A D = 7 cm$

So, radius of semicircles $= \frac{1}{2} \times A B = \frac{1}{2} \times 14 cm = 7 cm$

Let radius of the semicircles, $G E = G F = F H = E H = 7 cm$

And $E F = 7 cm$

So, $∆ G E F and ∆ H E F$ are equilateral triangles and they are equal.

So, $∠ G E F = ∠ E G F = ∠ G F E = ∠ F E H = ∠ E H F = ∠ E F H = 60 °$

Now to find the common area in both semicircles, we need to find the area of triangles $∆ G E F and ∆ E H F$ and all the four minor segments $G E, H E, H F, F G$ .

So, area of triangle $∆ G E F = \frac{\sqrt{3}}{4} {πr}^{2} = \frac{\sqrt{3}}{4} \times 7^{2} = 21.19 {cm}^{2}$

Now, area of minor segment $G E =$ Area of sector $G F E -$ Area of triangle $∆ G E F$

$= \frac{θ}{360 °} {πr}^{2} - \frac{\sqrt{3}}{4} {(side)}^{2}$

$= \frac{60 °}{360 °} \times \frac{22}{7} \times 7^{2} - \frac{\sqrt{3}}{4} \times 7^{2}$

$= 25.67 - 21.19$

$= 4.48 {cm}^{2}$

So, area of common region in both the semicircles $= 4 \times$ Area of minor segment $+ 2 \times$ Area of equilateral triangle

$= 4 \times 4.48 + 2 \times 21.19$

$= 17.92 + 42.38$

$= 60.3 {cm}^{2}$

So, area enclosed by semicircles $= 2 \times$ Area of semicircle $-$ Area of common region in both the semicircles

$= 2 \times \frac{1}{2} {πr}^{2} - 60.3$

$= \frac{22}{7} \times 7^{2} - 60.3$

$= 154 - 60.3$

$= 93.7 {cm}^{2}$

Hence, area enclosed by the semicircle is $93.7 {cm}^{2}$ .

Semi-circles are described on the sides $A B and C D$ of a rectangle $A B C D$ as diameters, both inside the rectangle. If $A B = 14 cm$ and $B C = 7 cm$ , find the area enclosed by the-semi circles.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

Semi-circles are described on the sides AB and CD of a rectangle ABCD as diameters, both inside the rectangle. If AB=14 cm and BC=7 cm, find the area enclosed by the-semi circles.

Important Questions on Mensuration

Learn and Prepare for any exam you want !

Semi-circles are described on the sides $A B and C D$ of a rectangle $A B C D$ as diameters, both inside the rectangle. If $A B = 14 cm$ and $B C = 7 cm$ , find the area enclosed by the-semi circles.