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Elementary Geometry for College St...

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

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BuyFindarrow_forward

Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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Determine a formula for the area of the shaded region determined by the circle and its inscribed square.

Chapter 8.5, Problem 33E, Determine a formula for the area of the shaded region determined by the circle and its inscribed

To determine

To find:

A formula for area of shaded region.

Explanation

Formula:

Area of circle:

If r is the radius of circle, then area of circle is given by the formula:

Acircle=πr2.

Area of square:

If the length of side of a square is a, then the area of square is given by the formula:

Asquare=a2.

Pythagoras theorem:

In a right angled triangle, if the length of hypotenuse is c and the length of remaining two sides of triangle is a and b, then by Pythagoras theorem, c2=a2+b2.

Calculation:

The area of shaded region can be determined by subtracting the area of inscribed square from the area of circle.

Ashaded region=Acircle-Ainscribed square.

The length of side of square is given as s.

Thus, Asquare=s2

From the diagram, it is clear that the diameter of circle is equal to the diagonal of square.

Let the length of diagonal be d.

All the angles of a square are equal and they measure 900. Thus, the diagonal of square and a pair of adjacent sides form a right angled triangle.

The length of diagonal of the square can be found by using Pythagoras theorem.

The diagonal is the hypotenuse of right angled triangle.

d2=a2+b2

Here, the lengths of remaining two sides of triangle are equal to the side of square. Thus, a=b=s. Let’s substitute s in place of a and b.

d2=s2+s2

d2=2s2

Taking square root on both sides,

d=2×s

Hence, the length of diameter of circle =2×s

We know that, radius of a circle is half of the length of diameter

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