How do you find the area of a sector of a circle with diameter?

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How do you find the area of a sector of a circle with diameter?

How do you find the area of a sector of a circle with diameter?

Slices

There are two main "slices" of a circle:

  • The "pizza" slice is called a Sector.
  • And the Segment, which is cut from the circle by a "chord" (a line between two points on the circle).

Try Them!

Sector Segment

images/circle-prop.js?mode=sector

images/circle-prop.js?mode=segment

Common Sectors

The Quadrant and Semicircle are two special types of Sector:

How do you find the area of a sector of a circle with diameter?
Half a circle is

a Semicircle.

How do you find the area of a sector of a circle with diameter?
Quarter of a circle is

a Quadrant.

You can work out the Area of a Sector by comparing its angle to the angle of a full circle.

Note: we are using radians for the angles.

How do you find the area of a sector of a circle with diameter?

This is the reasoning:

A circle has an angle of 2π and an Area of:πr2

A Sector has an angle of θ instead of 2π so its Area is : θ2π × πr2

Which can be simplified to:θ2 × r2

Area of Sector = θ 2 × r2   (when θ is in radians)

Area of Sector = θ × π 360 × r2   (when θ is in degrees)

How do you find the area of a sector of a circle with diameter?

Area of Segment

The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here).

There is a lengthy reason, but the result is a slight modification of the Sector formula:

Area of Segment = θ − sin(θ) 2 × r2   (when θ is in radians)

Area of Segment = ( θ × π 360 sin(θ)2 ) × r2   (when θ is in degrees)

How do you find the area of a sector of a circle with diameter?

Arc Length

The arc length (of a Sector or Segment) is:

L = θ × r   (when θ is in radians)

L = θ × π180 × r   (when θ is in degrees)

images/circle-prop.js?mode=arc

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Copyright © 2022 Rod Pierce

The area of sector of a circle is the amount of space enclosed within the boundary of the sector. A sector always originates from the center of the circle. The sector of a circle is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semi-circle is the most common sector of a circle, which represents half a circle. Let us learn more about the area of sector, its formula, and how to calculate the area of a sector using radians and degrees.

What is Area of Sector of a Circle?

The space enclosed by the sector of a circle is called the area of the sector. For example, a pizza slice is an example of a sector that represents a fraction of a pizza. There are two types of sectors: minor and major sectors. A minor sector is a sector that is less than a semi-circle, whereas, a major sector is a sector greater than a semi-circle.

The figure given below represents the sectors in a circle. The shaded region shows the area of the sector OAPB. Here, ∠AOB is the angle of the sector. It should be noted that the unshaded region is also a sector of the circle. So, the shaded region is the area of the minor sector and the unshaded region is the area of the major sector.

How do you find the area of a sector of a circle with diameter?

Now, let us learn about the area of a sector formula and its derivation.

Sector Definition

A sector is considered as a portion of a circle with two radii and an arc. A circle is divided into two sectors namely, the minor sector and the major sector where the minor is the smaller portion in the circle and the larger portion is the major portion.

Area of Sector Formula

In order to find the total space enclosed by the sector, we use the area of a sector formula. The area of a sector can be calculated using the following formulas,

  • Area of a Sector of Circle = (θ/360º) × πr2, where, θ is the sector angle subtended by the arc at the center, in degrees, and 'r' is the radius of the circle.
  • Area of a Sector of Circle = 1/2 × r2θ, where, θ is the sector angle subtended by the arc at the center, in radians, and 'r' is the radius of the circle.

Area of Sector Formula Derivation

Let us apply the unitary method to derive the formula for the area of the sector of a circle. We know that a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is given by πr2, where 'r' is the radius of the circle.

If the angle at the center of the circle is 1º, the area of the sector is πr2/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of Circle = (θ/360º) × πr2, where,

  • θ is the angle subtended at the center, given in degrees.
  • r is the radius of the circle.

In other words, πr2 represents the area of a full circle and θ/360º tells us how much of the circle is covered by the sector.

How do you find the area of a sector of a circle with diameter?

If the angle at the center is θ in radians, area of the sector of a circle = (1/2) × r2θ, where,

  • θ is the angle subtended at the center, given in radians.
  • r is the radius of the circle.

It should be noted that semi-circles and quadrants are special types of sectors of a circle with angles of 180° and 90° respectively.

Area of Sector Using Degrees

Let us use these formulas and learn how to calculate the area of the sector of a circle when the subtended angle is given in degrees with the help of an example.

Example: A circle is divided into 3 sectors and the central angles made by the radius are 160°, 100°, and 100° respectively. Find the area of all the three sectors.

Solution:

The angle made by the first sector is θ = 160°. Therefore, the area of the first sector = (θ/360°) × πr2 = (160°/360°) × (22/7) × 62 = 4/9 × 22/7 × 36 = 352/7 = 50.28 square units.

The angle made by the second sector is θ = 100°. Therefore, the area of the second sector is = (θ/360°) × πr2 = (100°/360°) × (22/7) × 62 = 5/18 × 22/7 × 36 = 220/7 = 31.43 square units.

The angle made by the third sector is the same as that of the second sector (θ = 100°). Thus, the area of the second sector is equal to the area of the third sector. Therefore, the area of the third sector = 31.43 square units.

Area of Sector in Radians

If we need to find the area of sector when the angle is given in radians, we use the formula, Area of sector = (1/2) × r2θ; where θ is the angle subtended at the center, given in radians, and 'r' is the radius of the circle. So, let us understand where the formula comes from. We know that the formula for the area of a sector (in degrees) = (θ/360º) × πr2 because it is a fraction of a circle. The same concept is applied to the formula when we want to express it in radians, but we just need to replace 360° with 2π because 2π (in radians) = 360°. This means, Area of sector in radians = (θ/2π) × πr2. On further simplifying the sector area formula, we get, area of sector = (θ/2) × r2 or (1/2) × r2θ. Let us understand how to find the area of a sector in radians with an example.

Example: Find the area of a sector if the radius of the circle is 6 units, and the angle subtended at the center = 2π/3

Solution: Given, radius = 6 units; Angle measure (θ)= 2π/3

The area of the given sector can be calculated with the formula, Area of sector (in radians) = (θ/2) × r2. On substituting the values in the formula, we get Area of sector (in radians) = [2π/(3×2)] × 62 = (π/3) × 36 = 12π.

Therefore, the area of the given sector in radians is expressed as 12π square units.

Real-Life Example of Area of Sector of Circle

One of the most common real-life examples of the area of a sector is the slice of a pizza. The shape of the slices of a circular pizza is like a sector. Observe the figure given below that shows a pizza that is sectioned into 6 equal slices, where each slice is a sector, and the radius of the pizza is 7 inches. Now, let us find the area of the sector formed by each slice by using the sector area formula. It should be noted that since the pizza is divided into 6 equal slices, the angle of sector is 60°. Area of Pizza slice = (θ/360°) × πr2 = (60°/360°) × (22/7) × 72 = 1/6 × 22 × 7 = 77/3 = 25.67 square units.

How do you find the area of a sector of a circle with diameter?

Tips on Area of Sector

Here is a list of a few important points that are helpful in solving the area of sector problems.

  • The area of a sector of a circle is the fractional area of the circle.
  • The area of a sector of a circle with radius 'r' is calculated with the formula, Area of a sector = (θ/360º) × π r2
  • The arc length of the sector of radius r can be calculated with the formula, Arc Length of a Sector = r × θ

☛ Related Articles

  • Area of a Circle
  • Arcs and Subtended Angles
  • Segment of a Circle
  • What is pi?

  1. Example 1: If the angle of a sector of a circle is 60°, and the radius of the circle is 7 inches, what is the area of the sector of this circle?

    Solution:

    The radius of the circle is 7 inches and the angle is 60°. So, let us use the area of sector formula. The area of sector = (θ/360°) × π r2 = (60°/360°) × (22/7) × 72 = 77/3 = 25.67 square units. Therefore, the area of the minor sector is 25.67 square units.

  2. Example 2: An umbrella has equally spaced 8 ribs. If viewed as a flat circle of radius 7 units, what would be the area between two consecutive ribs of the umbrella? (Hint: The area between two consecutive ribs would form a sector of a circle)

    Solution:

    The radius of the flat umbrella = 7 units. There are 8 ribs in the umbrella. Since a complete angle of a circle = 360°, the angle of each sector of the umbrella = 360/8 = 45° because the circle is divided into 8 equal sectors. Thus, the area of sector = (θ/360°) × π r2 = (45°/360°) × 22/7 × 72 = 77/4 = 19.25 square units. Therefore, the area between two consecutive ribs of the umbrella is 19.25 square units.

  3. Example 3: A circle with a diameter of 2 units is divided into 10 equal sectors. Can you find the area of each sector of the circle?

    Solution:

    The diameter of the circle is 2 units, therefore, the radius of the circle is 1 unit. Since a complete angle of a circle = 360°, the angle of each sector of the circle is 360/10 = 36° because the complete angle is divided into 10 equal parts. Area of Sector = (θ/360°) × πr2 = 36°/360° × 22/7 × 1 = 11/35 = 0.314 square units. Therefore, the area of each sector of the circle is 0.314 square units.​

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FAQs on Area of Sector of Circle

The space enclosed by the sector of a circle is called the area of the sector of a circle. The part of the circle that is enclosed by two radii and the corresponding arc is called the sector of the circle.

What is the Formula for Area of Sector of Circle?

The two main formulas that are used to find the area of a sector are:

  • Area of a Sector of Circle = (θ/360º) × πr2, where, θ is the angle subtended at the center, given in degrees, and 'r' is the radius of the circle.
  • Area of a Sector of Circle = 1/2 × r2θ, where, θ is the angle subtended at the center, given in radians, and 'r' is the radius of the circle.

How to Calculate the Area of a Sector using Degrees?

When the angle subtended at the center is given in degrees, the area of a sector can be calculated using the following formula, area of a sector of circle = (θ/360º) × πr2, where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.

What do you Mean by Sector of a Circle?

A sector is defined as the portion of a circle that is enclosed between its two radii and the arc adjoining them. The semicircle is the most common sector of a circle, which represents half of a circle.

What do you Mean by the Arc of a Circle?

A part of a curve or a part of a circumference of a circle is called the arc. Many objects have a curve in their shape. The curved portion of these objects is mathematically referred to as an arc.

How is the Area of Sector of Circle Formula Derived?

The area of the sector shows the area of a part of the circle's area. We know that the area of a circle is calculated with the formula, πr2. The formula for the area of a sector of a circle is derived in the following way:

  • Apply the unitary method to derive the formula of the area of a sector of circle.
  • We know, a complete circle measures 360º. The area of a circle with an angle measuring 360º at the center is given by πr2, where r is the radius of the circle.
  • If the angle at the center of the circle is 1º, the area of the sector is πr2/360º. So, if the angle at the center is θ, the area of the sector is, Area of a Sector of a Circle = (θ/360º) × πr2, where, θ is the angle subtended at the center, given in degrees, and r is the radius of the circle.
  • In other words, πr2 represents the area of a full circle and θ/360º tells us how much of the circle is covered by the sector.

How to Find the Area of Sector with Arc Length and Radius?

The area of a sector can be calculated if the arc length and radius is given. We first calculate the angle (θ) subtended by the arc with the formula, Length of Arc = (θ/360) × 2πr. Now, we already know the radius, and once the angle is known, the area of the sector can be calculated with the formula, Area of a Sector of a Circle = (θ/360º) × πr2

How to Find the Radius from Area of Sector?

If the area of a sector is known, and the angle (θ) subtended by the arc is known, the radius can be calculated by substituting the given values in the formula, Area of a Sector of a Circle = (θ/360º) × πr2. For example, let us find the radius if the area of a sector is 36π, and the sector angle is given as 90°. We will substitute the given values in the formula, Area of a Sector of a Circle = (θ/360º) × πr2, that is, 36π = (90/360) × πr2. So, the value of r2 = 144, which means r = 12 units.

How to Find the Area of Sector in Terms of Pi?

The area of sector can also be expressed in terms of pi (π). For example, if the radius of a circle is given as 4 units, and the angle subtended by the arc for the sector is 90°, let us find the area of the sector in terms of pi. Area of sector = (θ/360º) × πr2. Substituting the values in the formula, Area of sector = (90/360) × π × 42. After solving this, we get, the area as 4π.

How to Find the Area of a Sector in Radians?

In order to find the area of a sector with the central angle in radians, we use the formula, Area of sector = (θ/2) × r2; where θ is the angle subtended at the center, given in radians, and 'r' is the radius of the circle. For example, if the radius of the circle is 12 units, and the sector angle subtended by the arc at the center = 4π/3, let us find the area of the sector. Area of sector (in radians) = (θ/2) × r2. On substituting the values in the formula, we get Area of sector (in radians) = [4π/(3×2)] × 122 = (2π/3) × 144 = 96π. Therefore, the area of the sector in radians is expressed as 96π square units.

How to Find the Area of a Sector Without Angle?

If the sector angle is not given, but we know the arc length and the radius, the area of a sector can be calculated. We first find the sector angle by substituting the given values of the arc length and radius in the formula, Length of Arc = (θ/360) × 2πr. After calculating the angle, we can easily find the area of the sector with the formula, Area of a Sector of a Circle = (θ/360º) × πr2.

How to Find the Arc Length of a Sector?

Arc length is the distance along the part of the circumference of a circle. The arc length of a circle can be calculated using the following formulas:

  • Arc Length = θ × r; where θ = Central angle subtended by the arc, and r = radius of the circle. This formula is used when θ is in radian.
  • Arc Length = θ × (π/180) × r; where θ = Central angle subtended by the arc, and r = radius of the circle. This formula is used when θ is in degrees.