A circle is a basic closed curve in which all points on the curve are at a constant distance from a fixed point, called the ‘center’. As there is only one center to the circle, you can name the circle by naming its center. For example, if the center of the circle is O, the circle can be named as ‘circle O’.
The radius of a circle, r, is the distance between the circle’s center and any point on the circumference of the circle. Geometrically, all circles are similar.
Perimeter of Circle
The perimeter of a circle is nothing but the length of the circumference of a circle. In most of the articles, you will find the word circumference used instead of the perimeter of a circle.
The circumference of a circle is its boundary line. The formula of circumference of the circle,
C is C = 2π radius = 2πr units .
Since Radius = 2 times the diameter of the circle, C = πd units, where d=diameter of the circle.
Circumference of circle
The ratio of the circumference to the diameter is always equal to π. This was proved by a great mathematician Archimedes.
Example 1: Find the circumference of the circle given below. Use the approximate value of π = 3.14.
Radius of circle
Solution: Circumference, C = 2πr units = 2 × 3.14 × 7 = 43.96 in
Example 2: Find the circumference of the circle given below. Use π = 3.14.
Diameter of circle
Solution: Circumference, C = πd = 3.14 × 20 = 62.8 m
Perimeter of Semicircle:
A perimeter of a semicircle is the sum of the lengths of the diameter and the boundary of the semicircle(half of the circumference).
Perimeter of semicircle
Hence the perimeter of a semicircle in terms of radius = πr + d = πr + 2r = r(π + 2).
This implies that the perimeter of the semicircle in terms of the diameter = πr + d = πδ/2 + d = (d/2)(π + 2).
Example 3: Find the perimeter of the given semicircle. Use π = 3.14.
Diameter of semicircle
Solution: Diameter = 16 ft ⇒ radius = 8 ft
Perimeter of a semicircle = r(π + 2) = 8(3.14 + 2) = 41.12 ft
Example 4: If the perimeter of a semicircle is 61.68 m, find the radius of the circle.
Solution: Perimeter of a semicircle = r(π + 2)
61.68 = r(π + 2) ⇒ r = 61.68/(π + 2)
r = 61.68 / 5.14 ⇒ r = 12 m
Thus, the radius of the given semicircle is 12 m.
Area of a Circle
The region bounded by the circumference of a circle on a plane is called an area of the circle. The area of the circle in terms of the radius is given by the formula, A = πr2 sq units.
How do you arrive at the formula of the area of a circle?
A circle is divided into small sectors and arranged in a rectangular shape. Refer to the image below. More the number of sector portions considered the finer rectangular shape obtained.
Circle arranged into rectangular shape
Now, the area of the circle is the same as the area of the rectangle.
Circumference of the circle = 2πr and radius of the circle = r
Area of the rectangle = Base × Height
The base of the rectangle = 12(2πr); the height of the rectangle = r
Finding the area of a circle-turned-rectangle
Area of the circle = πr ×r= πr2 Sq. units
Example 5: Find the area of a circle whose circumference is 56.52 units.
Solution: Circumference, C = 2πr = 56.52 units
r = 56.52/(2π)
r = 9 units
Area of a circle, A = πr2 = 3.14 × 92 = 254.34 square units
Circle cut to find it’s area
Example 6: Find the circumference and area of a circle whose diameter is 22 m.
Solution: Diameter = 22 m ⇒ Radius = 22/2 = 11 m
Circumference, C = 2πr = 2 × 3.14 × 11 = 69.08 m
Area, A = πr2 = 3.14 × 112 = 379.94 sq. m
Area of a Semicircle = (1/2) Area of a Circle = (1/2) πr 2 Sq. units
Example 7: A cardboard is cut in the shape of a semicircle whose diameter is 32 cm. Find the area of the cardboard.
Solution: Diameter = 32 cm ⇒ Radius = 16 cm
Area of a semicircle = (1/2) πr2 = (1/2) × 3.14 × 162 = 401.92 sq. cm
Area of a sector of a circle
A portion of a circle enclosed by an intercepted arc and two radii is called a sector. The area of a circle is determined in radian or degrees depending on the angle of a sector. Refer to the image below.
Area of a circle in radian or degrees
Example 8: Calculate the area of a sector with a radius of 12 cm and an angle of 75°.
Solution: Area of the sector = 360r2 = 75360 × 3.14 × 122 = 94.2 sq. cm
Example 9: Find the central angle of a sector whose radius is 15 ft and the area is 157 sq. ft.
Solution: Area of the sector, A = 157 sq. ft, Radius, r = 15 ft
Area of the sector, A = 360r2
⟹ θ = A × 360r2 = 157 × 3603.14 × 152 = 80°
Thus, the central angle is 80°.
Example 10: Find the area of the major and minor sectors for the given circle whose central angle is 2/3. Solution:
Major and minor sector of a circle
Central angle, θ = 2π3
Area of the sector = 12r2 sq.in
Area of a minor sector = 12r2θ = 12922π3 = 84.78 sq. in
Area of a major sector = 12r2θ = 12922π – 2π3
A = 12924π3 = 169.56 sq. in
Know more about other concepts of Geometry on 3D Shapes, Herons Formula, Area of Triangle, What is Perimeter and What is Triangle.
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