Table of Contents
- Introduction
- Area of a Circle
- Circumference of a Circle
- Arc Length
- Inscribed Angle
- Inscribed Triangle
Introduction
A circle is a simple shape consisting of points on a plane which are all equal distance from a given point called the center. A circle contains 360 degrees.
The diameter is the length of a line segment that has endpoints on the edge of the circle and passes through the center of the circle. This line segment is the longest distance between any two points on the circle. The diameter of a circle is twice its radius.
A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle’s center, is the largest chord in a circle
An arc is formed from any two connected points on the circle’s edge.
Area of a Circle
The area of a circle is found with the following formula:
A = πr² , where is a constant, given as 3.14, and r is the radius
Example
Find the area of the circle below given the diameter is 8
If the diameter is 8, radius is = 4.
A = π4² = 16π
Circumference of a Circle
Circumference is the distance around a circle. It is found with the formula:
C = 2πr = πd
where r is the radius and d is the diameter
Example
Find the circumference of the circle below
C = 2π
Arc Length
The length of an arc can be found given the circumference of a circle and the central angle of the arc.
Example
What is the arclength of arc ab
Given the diameter, we calculate the circumference of the circle to be C = 3π . Since arc ab has endpoints on the diameter, we know that the central angle must be half the number of degrees in a circle or 180 degrees. 180/360 = 0.5. Multiply this by our circumference to find the arclength of ab. The arclength of ab is 1.5π
Inscribed Angle
The vast majority of calculations involving circles require knowing the central angle. However, the GMAT is deceptive and will often try to hide this information by providing an inscribed angle, or by only providing sufficient information to determine the inscribed angle. The inscribed angle is an angle that is formed from having three points on the circumference of the circle. Luckily, it’s easy to find the central angle, given the inscribed angle.
The central angle is equal to 2x its inscribed angle.
Inscribed Triangle
A triangle can be formed from the three points of an inscribed angle. Another rule that needs to be memorized: if a triangle is inscribed within a circle and one of the sides forms the diameter of the circle, then the triangle must be a right triangle. See example below:
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