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.
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
Area with Diameter of Circle
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).
Area with Circumference of Circle
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.
Area with Segments of Circle
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
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
Solved Examples on Area of Circle
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
Practice Questions on Area of Circle
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.
