Question

26. The ratio of the areas of the incircle and circumcircle of a square is
(a) 1 : 2
(b) 1 : 3
(c) 1 : 4
(d) 1 : √2

Answer 1

Answer:

a

Step-by-step explanation:

Let side of square = x units

∴ Diagonal of the square = √2 x units

Diameter of the incircle = x units

Diameter of the circumcircle = √2 x units

$\frac{Area of incircle}{Area of circumcircle}\\ =\frac{\pi (\frac{x}{2}) ^{2} }{\pi (\frac{\sqrt{2} x}{2} )^{2} } \\ =\frac{1}{2}$

Answer 2

1:2

Step-by-step explanation:

Let the side of a square be denoted as "a"

Radius of incircle is half the length of side of a square

$= \frac{a}{2}$

Area of incircle A1

$= \pi ({ \frac{a}{2} })^{2} = \frac{\pi {a}^{2} }{4}$

Radius of circumcircle is half the length of the diagonal of square

$= \frac{a}{ \sqrt{2} }$

Area of circumcircle A2

$= \pi( { \frac{a}{ \sqrt{2} } })^{2} = \frac{\pi {a}^{2} }{2}$

$∴ \frac{A1}{A2} = \frac{ \frac{\pi {a}^{2} }{4} }{ \frac{\pi {a}^{2} }{2} }$

$= \frac{1}{2}$

$1:2$