Geometry: Circles

In this lesson, we will learn

• about circles
• the properties of circles
  • diameter
  • chord
  • radius
  • arc
  • semicircle, minor arc, major arc
  • tangent
  • secant
  • circumference and formula
  • area and formula
  • sector and formula

Circle

In geometry, a circle is a closed curve formed by a set of points on a plane that are the same distance from its center O. That distance is known as the radius of the circle.

Diameter

The diameter of a circle is a line segment that passes through the center of the circle and has its endpoints on the circle. All the diameters of the same circle have the same length.

Chord

A chord is a line segment with both endpoints on the circle. The diameter is a special chord that passes through the center of the circle. The diameter would be the longest chord in the circle.

Radius

The radius of the circle is a line segment from the center of the circle to a point on the circle.

In the above diagram, O is the center of the circle and and  are radii of the circle. The radii of a circle are all the same length. The radius is half the length of the diameter.

Arc

An arc is a part of a circle.

In the diagram above, the part of the circle from B to C forms an arc.

An arc can be measured in degrees.

In the circle above, arc BC is equal to the BOC that is 45°.

Semicircle, Minor Arc and Major Arc

A semicircle is an arc that is half a circle. A minor arc is an arc that is smaller than a semicircle. A major arc is an arc that is larger than a semicircle.

The following video shows how to identify semicircle, minor arc and major arc.

Tangent

A tangent to a circle is a line that touches a circle at only one point. A tangent is perpendicular to the radius at the point of contact.

In the above diagram, the line containing the points B and C is a tangent to the circle.

It touches the circle at point B and is perpendicular to the radius

is perpendicular to  i.e.

Secant

A secant is a straight line that cuts the circle at two points. A chord is the portion of a secant that lies in the circle.

Circumference

The circumference of a circle is the distance around a circle.

Calculating the circumference of a circle involves a constant called pi with the symbol π. The value of π (pi) is approximately 3.14159265358979323846... but usually rounding to 3.142 should be sufficient. (see a mnemonic for π)

The formula for the circumference of a circle is

C = πd     (see a mnemonic for this formula)

or

C = 2πr

where C is the circumference, d is the diameter and r is the radius.

If you are given the diameter then use the formula C = πd

If you are given the radius then use the formula C = 2πr

Worksheet to calculate the circumference of a circle.

Worksheet to calculate the area and circumference of a circle.

Example 1: Find the circumference of the circle with a diameter of 8 inches.

Solution:

Answer: The circumference of the circle is 8π ≈ 25.163 inches.

Example 2: Find the circumference of the circle with a radius of 5 inches.

Solution:

Answer: The circumference of the circle is 10 π ≈ 31.42 inches.

This video will explore the relationship between the radius, diameter and circumference of a circle.

Area

The area of a circle is the region enclosed by the circle.

It is given by the formula:

A = πr²     (see a mnemonic for this formula)

where A is the area and r is the radius.

Since the formula is only given in terms of radius, remember to change from diameter to radius if necessary.

Worksheet to calculate the area of a circle.

Worksheet to calculate the area and circumference of a circle.

Worksheet to calculate problems that involve the radius, diameter, circumference and area of circle.

Example 1: Find the area the circle with a diameter of 10 inches.

Solution:

Step 1: Write down the formula:	A = πr2
Step 2: Change diameter to radius:
Step 3: Plug in the value:	A = π52 = 25π

Answer: The area of the circle is 25π ≈ 78.55 square inches.

Example 2: Find the area the circle with a radius of 10 inches.

Solution:

Answer: The area of the circle is 100π ≈ 314.2 square inches.

Sector

A sector is like a "pizza slice" of the circle. It consists of a region bounded by two radii and an arc lying between the radii. The area of a sector is a fraction of the area of the circle

The formula to calculate the area of a sector is

The following video shows how to derive the formula to calculate the area of a sector in a circle.

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