   Chapter 8.5, Problem 28E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

A circle can be inscribed in an equilateral triangle, each of whose sides has length 10 cm. Find the exact area of that circle.

To determine

To find:

The area of circle inscribed in an equilateral triangle.

Explanation

Formula:

Area of a triangle with an inscribed circle:

If P is the perimeter of the triangle and r is the length of radius of its inscribed circle, then the area A of the triangle is given by

A=12rP

Area of triangle:

If a, b and c are lengths of sides of triangle, then area of triangle is given by the formula:

A=s(s-a)(s-b)(s-c)

Where s is the semi perimeter which is given by s=12(a+b+c)

Area of circle:

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

Acircle=πr2

Calculation:

To find the area of inscribed circle, we have to find the radius of the circle. In order to find the radius of inscribed circle, we will use the following formula:

A=12rP,

Where A is the area of triangle, P is the perimeter of triangle and r is the radius of inscribed circle.

The given triangle is an equilateral triangle with length of side 10 cm.

The perimeter P of triangle is the sum of lengths of all sides. For an equilateral triangle, all sides are of equal length. That is, a=b=c=10

Hence, perimeter P=10+10+10=30

To find the area of triangle, let’s find the semi perimeter which is half of the perimeter of triangle. Thus, s=P2=302=15 cm.

Now, let’s find the area A of triangle.

A=15(15-10)(15-10)(15-10)

A=15×5×5×5

A=253 cm2

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