In these lessons, 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

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.

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.

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.

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.

An arc is a part of a circle.

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

An arc can be measured in degrees. In the circle above, the measure of arc BC is equal to its central angle BOC, which is 45°.

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.

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.

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.

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 = *2*πr*

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:

Step 1: Write down the formula: | C = πd |

Step 2: Plug in the value: | C = 8π |

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

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

Solution:

Step 1: Write down the formula: | C = 2πr |

Step 2: Plug in the value: | C = 10π |

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

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

It is given by the formula:

A = πr^{2}(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 = πr^{2} |

Step 2: Change diameter to radius: | |

Step 3: Plug in the value: | A = π5^{2} = 25π |

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

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

Solution:

Step 1: Write down the formula: | A = πr^{2} |

Step 2: Plug in the value: | A = π10^{2} = 100π |

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

**How to relate the radius and diameter of a circle with its area?**

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

It explains how to find the area of a sector.

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