Show that the area of the circumference of any square is twice the area of its incircle - Maths - Areas Related to Circles

Question

Show that the area of the circumference of any square is twice
the area of its incircle

Ankush Jain · Accepted Answer

The correct query should be: Show that the area of the circumcircle of any square is twice the area of its incircle Solution:

Let ABCD be a square of side 'a' whose incircle and circum circle are drawn as shown in the above figure Clearly, the diameter of circum circle is AC and diameter of incircle is EF which is equal to side AB i e , 'a' In triangle ABC by pythagoras theorem, we get AC2 = AB2 + BC2 â‡’ AC2 = a2+ a2 = 2a2 â‡’ AC = âˆš2a So, radius of circum circle, R $= \frac{A C}{2} = \frac{\sqrt{2} a}{2} = \frac{a}{\sqrt{2}}$ Now, area of circum circle $= π R^{2} = π \times {(\frac{a}{\sqrt{2}})}^{2} = \frac{a^{2} π}{2}$ Similarly, diameter of incircle = EF = a So, radius of incircle = a/2 Therefore, area of incircle $= π r^{2} = π \times {(\frac{a}{2})}^{2} = \frac{a^{2} π}{4}$ $\Rightarrow \frac{A r e a o f c i r c u m c i r c l e}{A r e a o f i n c i r c l e} = \frac{\frac{a^{2} π}{2}}{\frac{a^{2} π}{4}} = 2 \Rightarrow A r e a o f c i r c u m c i r c l e = 2 \times a r e a o f i n c i r c l e$ [Hence Proved]

Show that the area of the circumference of any square is twice the area of its incircle.