What is the area of the circle?

Geometry Level 2

ABCD is a square of length 1 unit. A circle is tangent to 2 sides of ABCD and passes through exactly one of its vertices. Then, the area of the circle is:

SOURCE: Arpit Shukla

(3-4√ 2)π unit² (4-6√ 2)π unit² (6-4√ 2)π unit² (4+6√ 2)π unit²

Prakkash Manohar by Prakkash Manohar , 22, India

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2 solutions

Prakkash Manohar
Apr 6, 2014

Please refer to this image.

AB = BC = CD = DA = 1unit

So, BD = √2 units

But, BD = r + r + x = 2r + x

So, 2r + x = √2

x = √2 - 2r -------------(1)

In right triangle OFB, Applying Pythagoras theorem, we get,

OB² = OF² + BF²

(r + x)² = r² + r²

r² + x² + 2rx = 2r²

r² - x² - 2rx = 0

r² - (√2 - 2r)² - 2r(√2 - 2r) = 0 [From ------(1)]

Solving, we get, r² + 2√2r - 2 = 0

Solving for x, we get, r = -√2 + 2 or -√2 - 2

But r cannot be negative. So, r = 2 - √2

So, area of the circle = πr² = π(2 - √2)² = (6 - 4√2)π unit²

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Vince Policarpio
Apr 12, 2014
  1. If side of the square is 1, and r is radius of the circle, then r + [r / sqrt(2)] = 1. Then r = 2 - sqrt(2).
  2. Area of the circle is A = pi (r^2). Then A = pi [2 - sqrt(2)]^2 = pi [6 - 4 * sqrt(2)].
1 Helpful 0 Interesting 0 Brilliant 0 Confused

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