Calculating the length of a radius

Help Questions

GMAT Quantitative › Calculating the length of a radius

Questions 1 - 10
1

The arc of a circle measures . The chord of the arc, , has length . Give the length of the radius of the circle.

Explanation

A circle can be divided into three congruent arcs that measure

.

If the three (congruent) chords are constructed, the figure will be an equilateral triangle. The figure is below, along with the altitudes of the triangle:

Circle and triangle

Since , it follows by way of the 30-60-90 Triangle Theorem that

and

The three altitudes of an equilateral triangle split each other into segments that have ratio 2:1. Therefore,

2

A arc of a circle measures . Give the radius of this circle.

Explanation

A arc of a circle is of the circle. Since the length of this arc is , the circumference is this, or

The radius of a circle is its circumference divided by ; therefore, the radius is

3

The arc of a circle measures . The chord of the arc, , has length . Give the length of the radius of the circle.

Explanation

A circle can be divided into congruent arcs that measure

.

If the (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord , or .

4

The points and form a line which passes through the center of circle Q. Both points are on circle Q.

To the nearest hundreth, what is the length of the radius of circle Q?

Explanation

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

Plug in our points and simplify:

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

5

Two circles in the same plane have the same center. The larger circle has radius 10; the area of the region between the circles is . What is the radius of the smaller circle?

Explanation

The area of a circle with radius is .

Let be the radius of the smaller circle. Its area is . The area of the larger circle is . Since the area of the region between the circles is , and is the difference of these areas, we have

The smaller circle has radius .

6

Two circles in the same plane have the same center. The smaller circle has radius 10; the area of the region between the circles is . What is the radius of the larger circle?

Explanation

The area of a circle with radius is .

Let be the radius of the larger circle. Its area is . The area of the smaller circle is . Since the area of the region between the circles is , and is the difference of these areas, we have

The smaller circle has radius .

7

If the circumference of a circle is , what is its radius?

Explanation

Using the formula for the circumference of a circle, we can solve for its radius. Plugging in the given value for the circumference, we have:

8

If the area of a circle is , what is its radius?

Explanation

Using the formula for the area of a circle, we can solve for its radius. Plugging in the given value for the area of the circle, we have:

9

A square and a circle have the same area. What is the ratio of the length of one side of the square to the radius of the circle?

Explanation

Let be the sidelength of the square is the square and be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

We find the ratio of to - that is, - as follows:

The correct ratio is .

10

Given that the area of a circle is , determine the radius.

Explanation

To solve, use the formula for the area of a circle, , and solve for .

Page 1 of 2
Return to subject