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A square and a circle are drawn such that one side of square is tangent to the circle, two vertices touching circle internally. Find ratio of area of circle to the area of square.
Give this question a try then check your answer with solution.
(Shortcut Image Solution with Calculations is gives below in this post)
Step 1:
Give names to all vertices,
Side of square = a;
Let, radius of circle = r;
Draw a radius to circle such that it will meet square at one of it's vertices located on the circle.
Step 2:
Draw a horizontal line passing through center of circle and tangential point on the side of square.
Step 3:
Here PO = OD = r;
⇒ QO = a - r;
Also, DQ = CQ = a/2;
Observe,
△DOQ is a right angled triangle,
⇒ OQ² + DQ² = OD²;
⇒ (a - r)² + (a/2)² = r²;
⇒ r = ⅝ × a;
Circle area = ²⁵⁄₆₄ × πa²;
Square area = a²
⇒ Circle area : Square area = 25π : 64
Image Solution:
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