5 Ways to Calculate the Area of a Circle

In this article, we’ll delve into five methods for calculating the area of a circle: using the radius, diameter, circumference, sector, and segment. Along the way, we’ll cover the fundamentals of circles and their formulas for area. Additionally, we’ll work through some examples related to circle area.

Area of a circle is defined as the region enclosed by the circumference of the circle. It is measured in square units.

Ways to calculate the area of a circle are:

• Area with Radius of Circle
• Area with Diameter of Circle
• Area with Circumference of Circle
• Area with Segments of Circle
• Area with Sectors of Circle

Area of circle formula with radius of circle is given by:

Area of Circle = πr²

where,

• r is the radius of circle

Definition of Radius

The distance between the center of circle and a point in the boundary of circle is called the radius of circle.

Derivation of Formula

To derive the area of circle formula we take a circle filled with concentric circle with radius r and cut it along its radius to form a triangle then, we find the area of the circle by using area of triangle formula.

As we know that the circumference of the circle is 2πr hence the base of the triangle is 2πr and also, we have cut the circle along its radius, so the height of triangle is r.

Area of Circle = (1/2) × B × H

Area of Circle = (1/2) × 2πr × r

Area of Circle = πr²

The area of circle formula with diameter of circle is given by:

Area of circle = π(D² / 4)

where,

• D is the diameter of circle

Definition of Diameter

The longest chord that passes through the center of the circle is called as the diameter of the circle. The diameter is sum of two radii. D = 2r.

Derivation of Formula

To derive the area with diameter we use the formula:

D = 2r

r = D / 2

Putting value of r in the area of circle formula

A = πr²

A = π(D / 2)²

A = π(D² / 4)

Hence the area of the circle with diameter is π(D² / 4).

The area of circle formula with the circumference of circle is given by:

Area of Circle = C² / 4π

where,

• C is the circumference of the circle

Definition of Circumference

Circumference is defined as the length of the boundary of the circle. Circumference of the circle = 2πr.

Derivation of Formula

To derive the formula for the area with circumference C we use formula:

C = 2πr

r = C / 2π

Putting the value of r in area of circle formula i.e., A = πr²

A = π(C / 2π)²

A = πC² / 4π²

A = C² / 4π

Hence, the area with circumference of circle A = C² / 4π.

The area of circle formula with the segments of circle is given by:

Area of Circle = Area of major segment + Area of minor segment

Definition of Segments

The region formed by the chord and the arc joining both the end of the chord is called segment of circle. The smaller segment is called the minor segment, and the larger segment is called major segment.

Derivation of Formula

To derive the formula for the area with segments we will use the following diagram.

In the above figure we can clearly see that the sum of areas of the minor segment and major segments gives the area of the circle.

We know that,

Area of minor segment = (1/2) × r² [(π/180) θ – sinθ]

Area of major segment = πr² – (1/2) × r² [(π/180) θ – sinθ]

Adding both areas we get,

Area of minor segment + Area of major segment = (1/2) × r² [(π/180) θ – sinθ] + πr² – (1/2) × r² [(π/180) θ – sinθ]

Area of minor segment + Area of major segment = πr²

We know that area of circle = πr²

Area of circle = Area of minor segment + Area of major segment

The area of circle formula with the sectors of circle is given by:

Area of Circle = Area of major sector + Area of minor sector

Definition of Sector

The region formed by two radii and the arc joining the two radii is called the sector of the circle. The smaller sector is called minor sector, and the larger sector is called the major sector.

Derivation of Formula

To derive the formula for the area with sectors we will use the following diagram.

In the above figure we can clearly see that the sum of areas of the minor sector and major sector gives the area of thee circle.

We know that,

Area of minor sector = (θ/360°) × πr²

Area of major sector = πr² – (θ/360°) × πr²

Adding both areas we get,

Area of minor sector + Area of major sector = (θ/360°) × πr² + πr² – (θ/360°) × πr²

Area of minor sector + Area of major sector = πr²

We know that area of circle = πr²

Area of circle = Area of minor sector + Area of major sector

Example 1: Find the area of circle given the radius of circle is 6 units.

Solution:

The area of the circle with radius of circle is given by:

Area of Circle = πr²

Area of Circle = π × 6²

Area of Circle = 36π

Area of Circle = 113.1 square units

Example 2: If the diameter of the circle is 10 units, then find the area of the circle.

Solution:

The area of the circle with diameter is given by:

Area of circle = π(D² /4)

Area of circle = π(10² /4)

Area of circle = π(100 /4)

Area of circle = 25π

Area of circle = 78.54 square units

Example 3: The circumference of the circle is 12 units then find the area of the circle.

Solution:

The area of the circle with circumference is given by:

Area of Circle = C² / 4π

Area of Circle = 12² / 4π

Area of Circle = 144 / 4π

Area of Circle = 36 / π

Area of Circle = 11.46 square units

Example 4: If the area of major segment and minor segment of a circle is 15 square units and 5 square units then find the area of circle.

Solution:

The area of the circle with segments of circle is given by:

Area of Circle = Area of major segment + Area of minor segment

Area of Circle = 15 + 5

Area of Circle = 20 square units

Example 5: If the area of major sector and minor sector of a circle is 18 square units and 10 square units then find the area of circle.

Solution:

The area of the circle with sectors of circle is given by:

Area of Circle = Area of major sector + Area of minor sector

Area of Circle = 18 + 10

Area of Circle = 28 square units

Q1. Find the area of circle given the radius of circle is 21 units.

Q2. If the diameter of the circle is 32 units, then find the area of the circle.

Q3. The circumference of the circle is 28 units then find the area of the circle.

Q4. If the area of major segment and minor segment of a circle is 20 square units and 11 square units then find the area of circle.

Q5. If the area of major sector and minor sector of a circle is 15 square units and 7 square units then find the area of circle.

Related Articles:

• Circle Formulas For Diameter, Area and Circumference
• Area Of A Circle Calculator – Free Online Calculator
• Areas of Sector and Segment of a Circle

What is the Area of Circle?

The region bounded by the circle is called as the area of the circle.

How is the area of a circle different from its circumference?

The area of a circle represents the total space enclosed by its circumference, while the circumference is the distance around the outer edge of the circle.

Can the area of a circle be negative?

No, the area of a circle cannot be negative. It is always a positive value or zero.

How does changing the radius of a circle affect its area?

Increasing the radius of a circle increases its area proportionally, following the formula A = πr². Conversely, decreasing the radius reduces the area accordingly.

What are some practical applications of knowing the area of a circle in real-life scenarios?

Real-life applications include calculating areas of fields, ponds, or circular plots of land, determining the amount of material needed for circular constructions like roads or pipes, and designing circular objects such as wheels or discs.