# GMAT Math : Circles

## Example Questions

### Example Question #11 : Calculating The Area Of A Circle

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The circumscribed circle has circumference . Give the area of the inscribed circle.

Explanation:

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the inscribed circle has half the radius of the circumscribed circle.

The circumscribed circle has circumference , so its radius is

The inscribed circle has radius half this, or 5, so its area is

### Example Question #13 : Calculating The Area Of A Circle

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The inscribed circle has circumference . Give the area of the circumscribed circle.

Explanation:

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the circumscribed circle has twice the radius of the inscribed circle.

The inscribed circle has circumference , so its radius is

The circumscribed circle has radius twice this, or 20, so its area is

### Example Question #11 : Calculating The Area Of A Circle

Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.

The inscribed circle has circumference . Give the area of the circumscribed circle.

Explanation:

Examine the diagram below, which shows the hexagon, segments from its center to a vertex and the midpoint of a side, and the two circles.

Note that the segment from the center of the hexagon to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the hexagon can be proved to form a 30-60-90 triangle.

The inscribed circle has circumference , so its radius - and the length of the longer leg of the right triangle - is

By the 30-60-90 Theorem, the length of the shorter leg is this length divided by , or ; the length of the hypotenuse, which is the radius of the circumscribed circle, is twice this, or

The area of the circumscribed circle can now be calculated:

### Example Question #281 : Gmat Quantitative Reasoning

Two circles are constructed; one is inscribed inside a given square, and the other is circumscribed about the same square.

The inscribed circle has circumference . Give the area of the circumscribed circle.

Explanation:

Examine the diagram below, which shows the square, segments from its center to a vertex and the midpoint of  a side, and the two circles.

Note that the segment from the center of the square to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the square form a 45-45-90 triangle, so by the 45-45-90 Theorem, the radius of the circumscribed circle is  times that of the inscribed circle.

The inscribed circle has circumference , so its radius is

The circumscribed circle has radius  times this, or , so its area is

### Example Question #12 : Calculating The Area Of A Circle

Two circles are constructed; one is inscribed inside a given square, and the other is circumscribed about the same square.

The circumscribed circle has circumference . Give the area of the inscribed circle.

The correct answer is not among the other responses.

Explanation:

Examine the diagram below, which shows the square, segments from its center to a vertex and the midpoint of  a side, and the two circles.

Note that the segment from the center of the square to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two radii and half a side of the square form a 45-45-90 Triangle, so by the 45-45-90 Theorem, the radius of the inscribed circle is equal to that of the circumscribed circle divided by .

The inscribed circle has circumference , so its radius is

Divide this by  to get the radius of the circumscribed circle:

The circumscribed circle has area

### Example Question #12 : Calculating The Area Of A Circle

Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.

The circumscribed circle has circumference . Give the area of the inscribed circle.

Explanation:

Examine the diagram below, which shows the hexagon, segments from its center to a vertex and the midpoint of a side, and the two circles.

Note that the segment from the center of the hexagon to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the hexagon can be proved to form a 30-60-90 triangle.

The circumscribed circle has circumference , so its radius - and the length of the hypotenuse of the right triangle -

By the 30-60-90 Theorem, the length of the shorter leg is half this, or 5. The length of the longer leg, which is the radius of the inscribed circle, is  times this, or .

The area of the inscribed circle can now be calculated:

### Example Question #11 : Calculating The Area Of A Circle

For Data Sufficiency questions, select either A, B, C, D, E based on the following rubric:

• A) statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
• (B) statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
• (C) both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
• (D) each statement alone is sufficient
• (E) statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Is the area of circle X less than the area of square Y?

1.  The diameter of X is 2 less than the square of Y's diagonal.

2.  The perimeter of Y is less than the circumference of X

C

B

A

D

E

D

Explanation:

In order to compare the areas of the square and circle, we need to find an equivalence between the radius of the circle and the side-length of the square.

Let's take the first statement.  Tranlating it into math, we have:

Where r= the radius of the circle, and d= the length of the side of the square.  The important point is that we DO NOT HAVE TO SOLVE THIS.  All we need to recognize is that the equation makes a comparison between r and d.  That is what we are looking for.

Because the first statement makes a direct, numerical comparison between r and d, it is sufficient by itself to answer the question.

Let's look at the second statement.

Translating that into math, we get:

,

where d is the length of the square's side and r is the radius of the circle.

This also sets an equivalence (technically, an inequality) between d and r, and we could use that information to show whether the area of the circle is less than the area of the square.

Statement 2 is ALSO sufficient to answer the question.

So, the answer choice is D.  Both statements are sufficient on their own to answer the question.

### Example Question #19 : Calculating The Area Of A Circle

In the figure shown below, line segment  passes through the center of the circle and has a length of . Points , , and  are on the circle. Sector  covers  of the total area of the circle. Answer the following questions regarding this shape.

Find the area of the given shape.

Explanation:

Finding the area here is straightforward if we can use what we're given wisely. First, the area of a circle can be found using the following:

where  is area and  is our radius. Now, although we are not given our radius, we are given the diameter, just not by name.

Segment  goes from one point on the outside of the circle through the center to another point on the circle; that means its length is the circle's diameter.

If our diameter is , then our radius must be . Plug this into the area equation to find the area of our circle:

### Example Question #41 : Geometry

Consider the Circle :

(Figure not drawn to scale.)

If Circle  represents the bottom of a silo, what is the area of the base of the silo in square meters?

Explanation:

The area of circle is found by the following equation:

where  is our radius, which is .

So,

### Example Question #41 : Geometry

A circle has a radius of . Calculate the area of the circle.