What is the formula for area of sector?

Anytime you cut a slice out of a pumpkin pie, a round birthday cake, or a circular pizza, you are removing a sector. A sector is created by the central angle formed with two radii, and it includes the area inside the circle from that center point to the circle itself. The portion of the circle's circumference bounded by the radii, the arc, is part of the sector.

Arcs of a Circle

Acute central angles will always produce minor arcs and small sectors. When the central angle formed by the two radii is 90°, the sector is called a quadrant (because the total circle comprises four quadrants, or fourths). When the two radii form a 180°, or half the circle, the sector is called a semicircle and has a major arc.

Unlike triangles, the boundaries of sectors are not established by line segments. True, you have two radii forming the central angle, but the portion of the circumference that makes up the third "side" is curved, so finding the area of the sector is a bit trickier than finding area of a triangle. The distance along that curved "side" is the arc length.

How to Find Area of a Sector

You cannot find the area of a sector if you do not know the radius of the circle. Be careful, though; you may be able to find the radius if you have either the diameter or the circumference. You may have to do a little preliminary mathematics to get to the radius.

Find the Radius of a Circle

Given the diameter, d, of a circle, the radius, r, is:

r=d2

Given the circumference, C of a circle, the radius, r, is:

r=C(2π)

Once you know the radius, you have the lengths of two of the parts of the sector. You only need to know arc length or the central angle, in degrees or radians.

Area of a Sector Formula

The central angle lets you know what portion or percentage of the entire circle your sector is. A quadrant has a 90° central angle and is one-fourth of the whole circle. A 45° central angle is one-eighth of a circle.

Those are easy fractions, but what if your central angle of a 9-inch pumpkin pie is, say, 31°?

[insert drawing of pumpkin pie with sector cut at +/- 31°]

This formula helps you find the area, A, of the sector if you know the central angle in degrees, n°, and the radius, r, of the circle:

For your pumpkin pie, plug in 31° and 9 inches:

A = (31360) × π × 92

A = 0.086111 × π × 81

A = 21.9126 in2

Area of Sector Radians

If, instead of a central angle in degrees, you are given the radians, you use an even easier formula.

To find Area, A, of a sector with a central angle θ radians and a radius, r:

Our beloved π seems to have disappeared! It hasn't, really. Radians are based on π (a circle is 2π radians), so what you really did was replace n°360° with θ2π. When θ2π is used in our original formula, it simplifies to the elegant (θ2) × r2.

Area of a Sector of a Circle Examples

You have a personal pan pizza with a diameter of 30 cm. You have it cut into six equal slices, so each piece has a central angle of 60°. What is the area, in square centimeters, of each slice?

A = (n°360°) × π × r2

Try it yourself first, before you look ahead!

A = (60°360°) × π × 152

A = (16) × π × 225

A = 117.8097 cm2

Did you remember to take half the diameter to find the radius?

Area of a Sector Example Using Radians

Suppose you have a sector with a central angle of 0.8 radians and a radius of 1.3 meters. Your formula is:

Try it yourself before you look ahead!

A = (0.82) × 1.32

A = 0.676 m2

Arc Length and Sector Area

You can also find the area of a sector from its radius and its arc length. The formula for area, A, of a circle with radius, r, and arc length, L, is:

Here is a three-tier birthday cake 6 inches tall with a diameter of 10 inches.

[insert cartoon drawing, or animate a birthday cake and show it getting cut up]

You cut it into 16 even slices; ignoring the volume of the cake for now, how many square inches of the top of the cake does each person get?

Each slice has a given arc length of 1.963 inches. The radius is 5 inches, so:

A = (5 × 1.963)2

A = 4.9075 in2

Since the cake has volume, you might as well calculate that, too: V = (π × 52 × 6 × 22.5°)360° = 29.452 in3, or cubic inches.

Next Lesson:

Equations Of A Circle

Related Pages
Circles, Sectors, Segments
Area Of Circles
More Geometry Lessons

The following table gives the formulas for the area of sector and area of segment for angles in degrees or radians. Scroll down the page for more explanations, examples and worksheets for the area of sectors and segments.

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. This area is proportional to the central angle. In other words, the bigger the central angle, the larger is the area of the sector.

The following diagrams give the formulas for the area of circle and the area of sector. Scroll down the page for more examples and solutions.

We will now look at the formula for the area of a sector where the central angle is measured in degrees.

Recall that the angle of a full circle is 360˚ and that the formula for the area of a circle is πr2.

Comparing the area of sector and area of circle, we derive the formula for the area of sector when the central angle is given in degrees.

This formula allows us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of a sector (degrees)

Calculate The Area Of A Sector (Using Formula In Degrees)

We can calculate the area of the sector, given the central angle and radius of circle.

Example:
Given that the radius of the circle is 5 cm, calculate the area of the shaded sector. (Take π = 3.142).

Solution:

Area of sector = 60°/360° × 25π
    = 13.09 cm2

Calculate Central Angle Of A Sector

We can calculate the central angle subtended by a sector, given the area of the sector and area of circle.

Example:
The area of a sector with a radius of 6 cm is 35.4 cm2. Calculate the angle of the sector. (Take π = 3.142).

Solution:

How To Derive The Formula To Calculate The Area Of A Sector In A Circle?

It explains how to find the area of a sector of a circle. The formula for the area of a circle is given and the formula for the area of a sector of a circle is derived.

Example:
Janice needs to find the area of the red section of the circular table top in order to buy the right amount of paint. What is the area of the red section of the circular table top?

Solution:
Step 1: Find the area of the entire circle using the area formula A = πr2.
Step 2: Find the fraction of the circle by putting the angle measurement of the sector over 360°, the total number of degrees in a circle.
Step 3: Multiply the fraction by the area of the circle. Leave your answer in terms of π.

• Show Video Lesson

How To Calculate The Area Of A Sector Using The Formula In Degrees And The Missing Radius Given The Sector Area And The Size Of The Central Angle?

Example 1: Find the area of the shaded region.

Example 2: Find the radius of the circle if the area of the shaded region is 50π

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Formula For Area Of Sector (In Radians)

Next, we will look at the formula for the area of a sector where the central angle is measured in radians. Recall that the angle of a full circle in radians is 2π.

Comparing the area of sector and area of circle, we get the formula for the area of sector when the central angle is given in radians.

This formula allows us to calculate any one of the values given the other two values.

Worksheet to calculate arc length and area of sector (radians)

The following video shows how we can calculate the area of a sector using the formula in radians.

Example:
A lawn sprinkler located at the corner of a yard rotates through 90° and sprays water 30ft. What is the area of the sector watered?

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How To Determine The Area Of A Sector?

The formula is given in radians.
How to determine the area of a segment? (the area bounded by a chord and an arc).

Example 1: Find the area of the sector of a circle with radius 8 feet formed by a central angle of 110°

Example 2: Find the area of the shaded region in the circle with radius 12cm and a central angle of 80°.

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Area Of Segment (Angle In Degrees)

The segment of a circle is a region bounded by the arc of the circle and a chord.

The area of segment in a circle is equal to the area of sector minus the area of the triangle.

How To Derive The Area Of A Segment Formula?

How do you find the area of a segment of a circle?

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How To Calculate The Area Of Segments Of Circles?

It uses half the product of the base and the height to calculate the area of the triangle.

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How To Calculate The Area Of Sector And The Area Of Segment?

It uses the sine rule to calculate the area of triangle.

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Area Of Segment (Angle In Radians)

Finding the area of a segment (angle given in radians)

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