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Area of Square inscribed in circle

Ratio of Areas

A circle is inscribed in an equilateral triangle and the square is inscribed in the circle. Find the ratio of the area of triangle to the area of square

A diagram of the geometric figure used in the problem statement is shown below:

$A square is inscribed in a circle which is inscribed in a equilateral triangle$

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Apr 16, 2013

Ratio of Areas
by: Staff

Answer

Part I

Calculate the radius of the inscribed circle:

All three interior angles of the equilateral triangle = 60°

$Interior Angles of an Equilateral Triangle$

Draw a straight line from the vertex of each interior angle so that each straight line bisects the angle.

The three straight lines will cross at the exact center of the triangle, circle, and square.

$Draw a straight line from the vertex of each interior angle so that each straight line bisects the angle. <br> <br>The three straight lines will cross at the exact center of the triangle, circle, and square.$

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Apr 16, 2013

Ratio of Areas
by: Staff

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Part II

The angle bisectors form six right triangles within the equilateral triangle.

Each right triangle is a 30°-60°-90° triangle.

$The angle bisectors form six right triangles within the equilateral triangle. Each right triangle is a 30°-60°-90° triangle.$

Since the triangle is a 30°-60°-90° triangle, the lengths of each side of the triangle are known.

If the radius of the circle is r (the short side of the triangle), the hypotenuse is 2r (twice the short side), and the length of the remaining side of the triangle is r√3.

$Since the triangle is a 30°-60°-90° triangle, the lengths of each side of the triangle are known.$

Since half of one of the diagonals of the square = r, the length of one of the diagonals = 2r.

since 1/2 of a side of the equilateral triangle is r√3, the length of each side is 2r√3.

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Apr 16, 2013

Ratio of Areas
by: Staff

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Part III

$Since half of one of the diagonals of the square = r, the length of one of the diagonals = 2r. <br> <br>Since 1/2 of a side of the equilateral triangle is r√3, the length of each side is 2r√3.$

We now have all the information needed to compute the ratio asked for in the problem statement

The area of the square can be calculated when the length of a diagonal is known.

$Calculate the area of the square using the length of a diagonal.$

The area of the equilateral triangle can be calculated when the length of a side is known.

$Calculate the area of the equilateral triangle using the length of one side.$

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Apr 16, 2013

Ratio of Areas
by: Staff

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Part IV

The ratio of the Area of the equilateral triangle to the Area of the square is:

$The ratio of the Area of the equilateral triangle to the Area of the square$

Thanks for writing.

Staff
www.solving-math-problems.com

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Area of Square inscribed in circle

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