How to find the ratio of diameter and circumference

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GRE Quantitative Reasoning › How to find the ratio of diameter and circumference

Questions 1 - 2

1

A rectangle is inscribed inside of a circle such that every corner touches the edge of the circle. If the area of the rectangle is and the perimeter of the rectangle is , what is the area of the circle in inches squared?

$420.25\pi$

$1681\pi$

$600.25\pi$

$2401\pi$

The answer cannot be determined.

Explanation

To find the area of the circle, it is important to know either its diameter or radius. For the geometry described in this problem, this is the same as the diagonal of the rectangle.

Gre circle rectangle

However, to find the diagonal of the rectangle, the sides must first be known. They can be found, since the perimeter and area are given:

This system of equation can be solved by substitution:

Followed by:

Note that this gives two possible values for , or , though the one selected is irrelevant; the other value will be the value for .

Knowing these two values, the diagonal can be found; it is the hypotenuse of a right triangle formed by these two lengths:

Since the diagonal is also the diameter, the area of the circle is given by:

$\pi \left(\frac{d}{2}\right)^2=\pi \frac{1681^}{4}=420.25\pi$

2

A circle with an area of $30\pi$ is divided into sectors with areas in a ratio of . What is the area of the largest sector?

$15\pi$

$10\pi$

$18\pi$

$9\pi$

Explanation

From the ratio given , it may be easier to write it such that the terms sum up to . This can be taken by dividing each term by the sum of the terms:

$\frac{1}{6}:\frac{2}{6}:\frac{3}{6}$

or

$\frac{1}{6}:\frac{1}{3}:\frac{1}{2}$

The largest sector thus has an area equal to

$\frac{1}{2}30\pi=15\pi$

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