GRE Exam: A Simple Guide To Solving Geometry Problems - Circles

A circle is a group of points that are all equidistant from a central point, known as the center. Any line segment that connects the center to a point on the circle is called the radius. Any circle has infinite radii, and all radii for a given circle are equal.

A chord is any line that connects two points on a circle. The diameter of a circle is any chord that passes through the center of the circle. The diameter is the longest possible chord.

Note that for any given circle, the diameter is twice the radius. Thus when you are provided with the diameter of a circle, you can determine its radius and vice versa. A diameter is one type of chord.

Circumference and Area
Let’s now look at calculations that involve the diameter and radius. The circumference of a circle is the perimeter of the circle. Think of it in the following way: If you flattened the circle to be a straight line, the length of that straight line would be the circle’s circumference.

There is a constant relationship between circumference and diameter:

From this, you can derive the formula for circumference:

Pi (π) might appear intimidating, but it’s simply a constant. Its value is roughly 3.14.

For the GRE, though, almost any question that involves π will expect you to represent your answer using π instead of a decimal.

For example, if you are asked to solve for the circumference of a circle with a diameter of 8, the answer would be 8π instead of 8 × 3.14 = 25.12.
 

What is the radius of a circle whose circumference is 24π?
6
12
18
24
30

SOLUTION: Use the circumference formula:

Since the radius is half of the diameter, the radius of the circle is 12. The correct answer is B.

The next element of a circle that you can calculate using radius and diameter is the area. The area of a circle represents the amount of space within that circle.

The formula for area of a circle is πr2.

If a circle has a circumference of 16π, what is its area?
4π
8π
16π
64π
256π

SOLUTION: To solve for the circle’s area, you need to determine the radius. Use the formula for circumference to determine the radius:

Since d = 2r, r = 8. Substitute 8 for r in the area formula: area = π82 = 64π. The answer is D.

Sector Area and Arc Length
Sector area represents the area of a piece of a circle. Arc length represents the length of a piece of a circle’s circumference.

Keep in mind the following:
- To determine sector area, you will need to know the area of the circle and what fraction the given piece is of the entire circle’s area.
- To determine arc length, you will need to know the circumference of the circle and what fraction the arc length is of the entire circumference.

If you are told what fraction the arc length or sector area is of a given circle, the question is pretty straightforward. Let’s say a circle has an area of 16π. If you are asked to determine the area of of this circle, just divide the area by 4 and arrive at 4π.

Likewise, let’s say a circle has a circumference of 8π. If you are asked to calculate of the circumference, just divide the circumference by 4, and arrive at 2π.

Unfortunately, a real GRE question will not make things so simple. Instead of telling you what fraction the sector is of the entire circle, a typical question will expect you to infer this information from the central angle of the sector or arc.

The central angle is simply the angle formed by two intersecting radii. For example, when you formed a quarter circle in the preceding example, the central angle was 90 degrees:

From the preceding example, you can infer the following two relationships:

Why do these relationships hold?

Because the central angle is simply a fraction of the entire circle’s angle measurement.

Since the angles around a circle’s center measure 360, a central angle of 60 degrees means the sector area is of the circle’s area.

Similarly, if the central angle is 40, then the arc length is simply of the circumference.

So how will you use the central angle to determine sector area?

Let’s look at an example.

If a circle has an area of 90π, what is the area of a sector with a central angle of 60 degrees?
1. Determine what fraction the sector is of the entire circle. Use the formula:

In this case, . Thus the sector area is of the circle’s entire area.
2. Multiply the area of the circle by .
In the example, you used the central angle to determine the sector area, but you can also use sector area or arc length to determine the central angle.

A sector has an arc length of 9π and an area of 100π. What is the central angle of the sector?
SOLUTION: Use the formula central angle/360 = arc length/circumference. You know the arc length, so to determine the central angle, you need to determine the circumference, and then solve the proportion.

You can use the area to determine the radius: 100π= πr2.

Thus r = 10. If r = 10, then the diameter = 20, and the circumference = 20π.

Now you can plug these values into the original formula: substitute 20π for the circumference and 9π for the arc-length:


Triangle in a Semicircle
If a triangle is inscribed in a semicircle, two properties follow:
1. The triangle is a right triangle.
2. The diameter of the circle = the triangle’s hypotenuse.