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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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Prove that the area of a circle circumscribed about a square is twice the area of the circle inscribed within the square.

To determine

To prove:

The area of a circle circumscribed about a square is twice the area of the circle inscribed within the square.

Explanation

Formula:

The area of the circle A=πr2

Calculation:

The square circumscribed about the circle and inscribed the circle.

Consider the diagram as shown below:

Let a be the length of the sides of the square.

The small circle touches at midpoints of the sides of the square.

Therefore diameter of the small circle is equals to the length of the side of the square.

So radius of the small triangle is half of the length of the side of the triangle.

Radius of the small circle = a2

Area of the small circle AS=πr2

AS=π(a2)2=πa24

ABO, is the right angled triangle.

Hypotenuse of the triangle ABO is the radius of the larger circle.

The circumscribed circle has a radius that is equal to half the diagonal of the square, for it touches each vertex of the square. Using 45-45-90 triangles in the square, we know that this is the same as:

R=d2

R=a22

Area of the larger circle AL=πR2

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