Circumference (Perimeter) of a Circle – Definition, Formula & Examples

The word ‘circumference’ refers to the path or the boundary that surrounds a shape or a figure. In other words, the circumference is the perimeter of a 2D shape. A person walking across a circular path covers a distance equal to the circumference of a circle in one round. The circumference of a circle depends on the diameter of a circle and is equal to the product of $\pi$ and the diameter.

The circumference and area are the two important concepts in circles and in this article you’ll learn about the circumference of a circle or the perimeter of circle.

Circle – A 2D Plane Figure

A circle is a two-dimensional figure formed by a set of points that are at a constant or at a fixed distance from a fixed point on the plane. The fixed point is called the centre of the circle and the fixed distance of the points from the centre is called the radius.

Some of the important terms related to the circle are

• Semicircle: A semicircle is one of the two halves when a circle is cut along the diameter. The two halves (semicircles) are of equal measure.
• Arc of a Circle: An arc of a circle is referred to as a curve, a part, or a portion of its circumference.
• Segment of a Circle: The area enclosed by the chord and the corresponding arc in a circle is called a segment. There are two types of segments – minor segment, and major segment.
• Sector of a Circle: The sector of a circle is defined as the area enclosed by two radii and the corresponding arc in a circle. There are two types of sectors – minor sector, and major sector.

What is the Circumference of a Circle?

The circumference of a circle is its boundary or the length of the complete arc of a circle. If you open a circle by cutting along its radius, the circumference can be stretched as a straight line, therefore, circumference is a linear measure and is measured in units like $mm$, $cm$, $m$, $ft$, $in$, etc.

Circumference of a Circle Formula

The circumference of a circle depends on its diameter or radius. The formula for the circumference of a circle is expressed in the form $C = \pi d$ or $C = 2 \pi r$, where $d$ and $r$ are the diameter and the radius of a circle respectively.

Note:

• Diameter is double the radius of a circle or radius is half of the diameter of a circle. $\left(d = 2r \text{ or } r = \frac {d}{2} \right)$.
• $\pi$ used to calculate the circumference and area of a circle is an irrational number. During calculation the approximate value of $\pi = \frac {22}{7}$ or $\pi = 3.14$ is considered.

Why Circumference is $2 \pi r$?

Let’s consider two small arcs $AB$ and $CD$ of two circles of radii $x$ and $y$ units. Both of which subtend equal angles $\left( \theta \right)$ at their respective centres $P$ and Q respectively.

For smaller values of $\theta$, the arc length $\widehat{AB}$ is almost equal to the line segment $\overline{AB}$.  Similarly, $\widehat{CD} = \overline{CD}$.

Since, the angle $\theta$ is same in both the triangles $APB$ and $CQD$, therefore, $\frac {AP}{PB} = \frac {CQ}{QD} = \frac {1}{1}$. (As $\triangle APB \sim \triangle CQD$ by $SAS$ rule of similarity.)

Therefore, it follows $AB : AP = CD : CQ$.

Thus, we can say that for all values of $\theta$, the ratios $\widehat{AB} : AP$  and  $\widehat{CD} : CQ$ are equal. It follows that the ratio $\text{Circumference} : \text{Radius}$ which is equal to the ratio $\text{Circumference} : \text{Diameter}$ is constant for any circle.

If this constant is denoted by a number $\pi$, then $\text {Circumference} = \pi \times \text{Diameter} = \pi \times 2 \times \text{Radius}$.

Examples

Ex 1: Find the circumference of a circle of radius $28 cm$.

Radius of a circle $r = 28 cm$.

Circumference of a circle C = $2 \pi r = 2 \times \frac {22}{7} \times 28 = 2 \times 22 \times 4 = 176 cm$

Therefore, the circumference of a circle of radius $28 cm$ is $176 cm$.

Ex 2: Find the circumference of a circle of diameter $10.5 mm$

Diameter of a circle $d = 10.5 mm$.

Circumference of a circle C = $\pi d = \frac {22}{7} \times 10.5 = 22 \times 1.5 = 33 mm$

Therefore, the circumference of a circle of diameter $10.5 mm$ is $33 mm$.

Ex 3: Find the radius of a circle whose circumference is $44 cm$.

Circumference of a circle $C = 44 cm$

Let the radius of the circle be $r$

Therefore, $C = 2 \pi r => 44 = 2 \times \frac {22}{7} \times r => r = \frac {44 \times 7}{2 \times 22} = 7 cm$.

Ex 4: Find the diameter of a circle whose circumference is $154 m$.

Circumference of a circle $C = 154 m$

Let the diameter of the circle be $d$

$C = \pi d => 154 = \frac {22}{7} \times d => d = \frac {154 \times 7}{22} = 49 m$.

Therefore, the diameter of a circle of circumference $154 m$ is $49 m$.

Circumference of a SemiCircle

A semicircle is a region formed by cutting a circle into two equal halves along a diameter of a circle. The circumference or perimeter is the boundary bounded by the semicircle and the diameter of a circle.

Therefore, the circumference of a semicircle is the sum of arc formed by the diameter and the diameter and is given by $\pi r + r + r = \pi r + 2r = \left( \pi + 2\right)r$.

Examples

Ex 1: Find the circumference of a semicircle whose radius is $3.5 ft$.

Radius of semicircle $r = 3.5 ft$.

Circumference of a semicircle = $\left( \pi + 2\right)r = \left( \frac {22}{7} + 2\right) \times 3.5 = \frac {36}{7} \times 3.5 = 18 ft$

The circumference of a semicircle of radius $3.5 ft$ is $18 ft$.

Ex 2: Find the radius of a semicircle whose circumference is $72 cm$.

Circumference of a semicircle $C = 72 cm$

Let the radius of the semicircle be $r$

Therefore, $C = \left( \pi + 2\right)r => 72 = \left( \frac {22}{7} + 2\right)r => 72 = \frac {22 + 14}{7} r => 72 = \frac {36}{7} r => r = \frac {72 \times 7}{36} = 14 cm$

Length of Arc of a Circle

The arc length is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the endpoints of the arc.

For example, in the circle shown below, $AB$ is the arc of the circle with centre $O$. The arc length of this arc $AB$ is given as $L$.

The arc length formula in radians can be expressed as, arc length = $\theta \times \left(\frac {\pi}{180} \right) \times r$, where

• $\theta$ = Central angle of the arc in degrees
• $r$ = Radius of the circle

Examples

Ex 1: Find the length of an arc of a circle of radius $21 cm$ subtending an angle of $45^{\circ}$ at the centre of the circle.

Radius of circle $r = 21 cm$

Central angle of the arc $\theta = 45^{\circ}$

Arc length = $\theta \times \left(\frac {\pi}{180} \right) \times r = 45 \times \left(\frac {\pi}{180} \right) \times 21$

$= \frac {\pi}{4} \times 21 = 16.5 cm$

Therefore, the length of an arc of a circle of radius $21 cm$ which subtends an angle of $45^{\circ}$ at the centre of the circle is $16.5 cm$.

Ex 2: If an arc of length $\frac {11}{3} cm$ has a central angle of $30^{\circ}$, then find the radius of a circle.

Length of arc $L = \frac {11}{3} cm$

Central angle $\theta = 30^{\circ}$

$L = \theta \times \left(\frac {\pi}{180} \right) \times r => \frac {11}{3} = 30 \times \left(\frac {\pi}{180} \right) \times r => \frac {11}{3} = \frac {\pi}{6} \times r$

$=> r = \frac {11}{3} \times \frac {6}{\pi} => r = \frac {22}{\pi} => r = 22 \times \frac {7}{22} = 7 cm$

Perimeter of Sector of a Circle

A sector of a circle is a pie-shaped part of a circle made of the arc along with its two radii. A portion of the circumference (also known as an arc) of the circle and $2$ radii of the circle meet at both endpoints of the arc forming a sector. The shape of a sector of a circle looks like a pizza slice or a pie.

For example, in the circle shown below, $AB$ is the arc of the circle with centre $O$. Then the boundary covered by this arc $AOB$ is the perimeter of a sector of a circle.

The perimeter of sector formula in radians can be expressed as, perimeter of sector = $\frac {\pi \theta r}{180} + r + r = \frac {\pi \theta r}{180} + 2r = \left(\frac {\pi \theta }{180} + 2\right)r$, where

• $\theta$ = Central angle of the arc in degrees
• $r$ = Radius of the circle

Examples

Ex 1: Find the perimeter of a sector of a circle of radius $3.5 cm$ subtending an angle of $90^{\circ}$ at the centre of the circle.

Radius $r = 3.5 cm$

Central angle $\theta = 90^{\circ}$

Perimeter of a sector of a circle = $\left(\frac {\pi \theta }{180} + 2\right)r = \left(\frac {\pi \times 90 }{180} + 2\right) \times 3.5 = \left(\frac {\pi}{2} + 2\right) \times 3.5 = \frac {\pi + 4}{4} \times 3.5 = 6.25 cm$

Perimeter of Segment of a Circle

A segment of a circle is the region that is bounded by an arc and a chord of the circle. Let us recall what is meant by an arc and a chord of the circle.

An arc is a portion of the circle’s circumference.

A chord is a line segment that joins any two points on the circle’s circumference.

There are two types of segments

• Minor segment: A minor segment is made by a minor arc of the circle
• Major segment: A major segment is made by a major arc of the circle

The perimeter of a segment of a circle is the sum of arc length and the chord of a circle. The formula to find the perimeter of a segment of a circle is given by $\theta \times \left(\frac {\pi}{180} \right) \times r + l$, where

• $\theta$ = Central angle of the arc in degrees
• $r$ = Radius of the circle
• $l$ = length of chord

Examples

Ex 1: Find the perimeter of a segment of a circle of radius $3.5 cm$ subtending an angle of $30^{\circ}$ at the centre of the circle and containing a chord of length $4 cm$.

Radius of a circle $r = 3.5 cm$

Central angle $\theta = 30^{\circ}$

Chord length $l = 4 cm$

The perimeter of a segment P = $\theta \times \left(\frac {\pi}{180} \right) \times r + l = 30 \times \left(\frac {\pi}{180} \right) \times 3.5 + 4 = \frac {\pi}{6} \times 3.5 \times 4 = 5.83 cm$

Conclusion

The circumference of a circle is the same as the perimeter of a circle which is the curved boundary around a circle. Numerically the circumference of a circle is equal to the product of the diameter of a circle and a constant $\pi$.

Practice Problems

1. Find the circumference of a circle whose diameter is
• $84 mm$
• $35 cm$
2. Find the diameter of a circle whose circumference is
• $66 m$
• $110 cm$
3. Find the circumference of a circle whose radius is
• $0.35 cm$
• $56 in$
4. Find the perimeter a semicircle of radius
• $56 cm$
• $77 mm$

FAQs

What is the circumference of a circle?

The circumference of a circle is the length of the boundary or the length of the complete arc of a circle. The circumference of the circle is the product of $\pi$ and the diameter of the circle.

What is $2\pi r$ in a circle?

$2 \pi r$ is the formula used to find the circumference of a circle.

How do you find the circumference of a circle?

The formula used to find the circumference of a circle of radius $r$ is $2 \pi r$.

How to find the diameter from the circumference of a circle?

The circumference of a circle is given by $C = \pi d$, which gives $d = \frac {C}{d}$.

Therefore, the diameter of a circle can be calculated by dividing the circumference of a circle by $\pi$.

What is the unit of the circumference of a circle?

The circumference is the perimeter of a circle. As the perimeter is the length of the boundary surrounding a plane figure, therefore, the unit of the circumference is $mm$, $cm$, $m$, $ft$, etc.