# What is the circumference of a circle?

 A circle is easy to make: Draw a curve that is “radius” away from a central point. And so: All points are the same distance from the center. ## You Can Draw It Yourself

Put a pin in a board, put a loop of string around it, and insert a pencil into the loop. Keep the string stretched and draw the circle!

Try dragging the point to see how the radius and circumference change.

(See if you can keep a constant radius!)

1. The Radius is the distance from the center outwards.
2. The Diameter goes straight across the circle, through the center.
3. The Circumference is the distance once around the circle.
4. And here is the really cool thing:

When we divide the circumference by the diameter we get 3.141592654…
which is the number π (Pi)

 So when the diameter is 1, the circumference is 3.141592654… We can say:

Circumference = π × Diameter Distance walked = Circumference = π × 100m

= 314m (to the nearest m)

• Also note that the Diameter is twice the Radius:
• Diameter = 2 × Radius
• And so this is also true:
• Circumference = 2 × π × Radius
• In Summary:

### Remembering

• Radius is the shortest word and shortest measure
• Diameter is longer
• Circumference is the longest

### Definition

 The circle is a plane shape (two dimensional), so:

### Area

1. The area of a circle is π times the radius squared, which is written:
2. A = π r2
3. Where
• A is the Area

To help you remember think “Pie Are Squared” (even though pies are usually round):

Area= πr2

= π × 1.22

= 3.14159… × (1.2 × 1.2)

= 4.52 (to 2 decimals)

Or, using the Diameter:

A = (π/4) × D2

### Area Compared to a Square

A circle has about 80% of the area of a similar-width square.
The actual value is (π/4) = 0.785398… = 78.5398…%

And something interesting for you:

See Circle Area by Lines

### Names

• Because people have studied circles for thousands of years special names have come about.
• Nobody wants to say “that line that starts at one side of the circle, goes through the center and ends on the other side” when they can just say “Diameter”.
• So here are the most common special names:

### Lines

1. A line that “just touches” the circle as it passes by is called a Tangent.
2. A line that cuts the circle at two points is called a Secant.
3. A line segment that goes from one point to another on the circle's circumference is called a Chord.

4. If it passes through the center it is called a Diameter.
5. And a part of the circumference is called an Arc.
• There are two main “slices” of a circle.
• The “pizza” slice is called a Sector.
• And the slice made by a chord is called a Segment.
1. The Quadrant and Semicircle are two special types of Sector:
2. Quarter of a circle is called a Quadrant.
3. Half a circle is called a Semicircle.

A circle has an inside and an outside (of course!).

But it also has an “on”, because we could be right on the circle.

Example: “A” is outside the circle, “B” is inside the circle and “C” is on the circle.

### Ellipse

A circle is a “special case” of an ellipse.

Activity: Approximate Value For Pi

## Circumference of a Circle

A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If you measure the distance around a circle and divide it by the distance across the circle through the center, you will always come close to a particular value, depending upon the accuracy of your measurement. This value is approximately 3.14159265358979323846… We use the Greek letter (pronounced Pi) to represent this value. The number  goes on forever. However, using computers,  has been calculated to over 1 trillion digits past the decimal point. The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter.  is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to . This relationship is expressed in the following formula:

where  is circumference and  is diameter. You can test this formula at home with a round dinner plate. If you measure the circumference and the diameter of the plate and then divide  by , your quotient should come close to . Another way to write this formula is:  where · means multiply. This second formula for finding the circumference of a circle is commonly used in problems where the diameter is given and the circumference is not known (see the examples below).

The radius of a circle is the distance from the center of a circle to any point on the circle.

If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius.

See also:  Why is a nasa spacecraft currently orbiting an asteroid? | everyday einstein

This relationship is expressed in the following circumference of a circle formula: , where  is the diameter and  is the radius.

Circumference, diameter and radii are measured in linear units, such as inches and centimeters. A circle has many different radii and many different diameters, each passing through the center. A real-life example of a radius is the spoke of a bicycle wheel.

A 9-inch pizza is an example of a diameter: when one makes the first cut to slice a round pizza pie in half, this cut is the diameter of the pizza. So a 9-inch pizza has a 9-inch diameter. Let's look at some examples of finding the circumference of a circle.

In these examples, we will use  = 3.14 to simplify our calculations.

Example 1: The radius of a circle is 2 inches. What is the diameter?

• Solution:
•  = 2 · (2 in)
•  = 4 in

Example 2: The diameter of a circle is 3 centimeters. What is the circumference?

1. Solution:
2.  =  *
3.  = 3.14 · (3 cm)
4.  = 9.42 cm

Example 3: The radius of a circle is 2 inches. What is the circumference?

• Solution:
•  = 2 · (2 in)
•  = 4 in
•  =  *
•  = 3.14 · (4 in)
•  = 12.56 in

Example 4: The circumference of a circle is 15.7 centimeters. What is the diameter?

Solution:

=  *

15.7 cm = 3.14 ·

15.7 cm ÷ 3.14 =

= 15.7 cm ÷ 3.14

= 5 cm

Summary: The number  is the ratio of the circumference of a circle to its diameter. The value of  is approximately 3.14159265358979323846…The diameter of a circle is twice the radius.

Given the diameter or radius of a circle, we can find the circumference. We can also find the diameter (and radius) of a circle given the circumference. The formulas for diameter and circumference of a circle are listed below.

We round  to 3.14 in order to simplify our calculations.

=  *

### Exercises

Circumference of a circle questions: Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Use  = 3.14 to calculate your answers.

 1 The diameter of a nickel is 2 centimeters. What is the circumference?
 2 The circumference of a bicycle wheel is 50.24 inches. What is the diameter?
 3 The radius of a circular rug is 4 feet. What is the circumference?
 4 The circumference of a compact disc is 28.26 centimeters. What is the radius?
 5 The diameter of your bicycle wheel is 25 inches. How far will you move in one turn of your wheel?

## Circumference

Circumference (C in black) of a circle with diameter (D in cyan), radius (R in red), and centre (O in magenta). Circumference = π × diameter = 2 × π × radius.

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In geometry, the circumference (from Latin circumferens, meaning “carrying around”) is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure.
Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk.

### Circle

The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition.

Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound.

 The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

When a circle's diameter is 1, its circumference is π.
When a circle's radius is 1—called a unit circle—its circumference is 2π.

### Relationship with π

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter π. The first few decimal digits of the numerical value of π are 3.141592653589793 … Pi is defined as the ratio of a circle's circumference C to its diameter d:

π
=

C
d

.

{displaystyle pi ={frac {C}{d}}.}

Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:

C

=
π

d

=
2
π

r

.

{displaystyle {C}=pi cdot {d}=2pi cdot {r}.!}

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.

In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio (C/d, since he did not use the name π) was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides. This method for approximating π was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.

### Ellipse

Main article: Ellipse § Circumference

Circumference is used by some authors to denote the perimeter of an ellipse.

There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions.

However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse,

x

2

a

2

+

y

2

b

2

=
1
,

{displaystyle {frac {x^{2}}{a^{2}}}+{frac {y^{2}}{b^{2}}}=1,}

is

C

e
l
l
i
p
s
e

π

2
(

a

2

+

b

2

)

.

{displaystyle C_{
m {ellipse}}sim pi {sqrt {2(a^{2}+b^{2})}}.}

Some lower and upper bounds on the circumference of the canonical ellipse with

a

b

{displaystyle ageq b}

are

2
π
b

C

2
π
a
,

{displaystyle 2pi bleq Cleq 2pi a,}

π
(
a
+
b
)

C

4
(
a
+
b
)
,

{displaystyle pi (a+b)leq Cleq 4(a+b),}

4

a

2

+

b

2

C

π

2
(

a

2

+

b

2

)

.

{displaystyle 4{sqrt {a^{2}+b^{2}}}leq Cleq pi {sqrt {2(a^{2}+b^{2})}}.}

Here the upper bound

2
π
a

{displaystyle 2pi a}

is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound

4

a

2

+

b

2

{displaystyle 4{sqrt {a^{2}+b^{2}}}}

is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.

The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind. More precisely, we have

C

e
l
l
i
p
s
e

=
4
a

0

π

/

2

1

e

2

sin

2

θ

d
θ
,

{displaystyle C_{
m {ellipse}}=4aint _{0}^{pi /2}{sqrt {1-e^{2}sin ^{2} heta }} d heta ,}

where again

a

{displaystyle a}

is the length of the semi-major axis and

e

{displaystyle e}

is the eccentricity

1

b

2

/

a

2

.

{displaystyle {sqrt {1-b^{2}/a^{2}}}.}

### Graph

In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph.

• Arc length
• Area
• Isoperimetric inequality

### References

1. ^ San Diego State University (2004). “Perimeter, Area and Circumference” (PDF). Addison-Wesley. Archived from the original (PDF) on 6 October 2014.
2. ^ Bennett, Jeffrey; Briggs, William (2005), Using and Understanding Mathematics / A Quantitative Reasoning Approach (3rd ed.

), Addison-Wesley, p. 580, ISBN 978-0-321-22773-7

3. ^ Jacobs, Harold R. (1974), Geometry, W. H. Freeman and Co., p. 565, ISBN 0-7167-0456-0
4. ^ Sloane, N. J. A. (ed.). “Sequence A000796”. The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. ^ Katz, Victor J.

(1998), A History of Mathematics / An Introduction (2nd ed.), Addison-Wesley Longman, p. 109, ISBN 978-0-321-01618-8

6. ^ Jameson, G.J.O. (2014). “Inequalities for the perimeter of an ellipse”. Mathematical Gazette. 98 (499): 227–234. doi:10.2307/3621497. JSTOR 3621497.

7. ^ Almkvist, Gert; Berndt, Bruce (1988), “Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary”, American Mathematical Monthly, 95 (7): 585–608, doi:10.

2307/2323302, JSTOR 2323302, MR 0966232, S2CID 119810884

8. ^ Harary, Frank (1969), Graph Theory, Addison-Wesley, p. 13, ISBN 0-201-02787-9

 The Wikibook Geometry has a page on the topic of: Arcs
 Look up circumference in Wiktionary, the free dictionary.
• Numericana – Circumference of an ellipse

## Circumference of a Circle (Definition, Formula & Examples)

Circumference of the circle or perimeter of the circle is the measurement of the boundary across any two-dimensional circular shape including circle.  Whereas the area of circle, defines the region occupied by it.  If we open a circle and make a straight line out of it, then its length is the circumference. It is usually measured in unit cm or unit m.

When we use the formula to calculate the circumference of the circle, then the radius of the circle is taken into account. Hence, we need to know the value of radius or the diameter to evaluate the perimeter of circle.

### Circumference of a Circle Formula

• The Circumference (or) perimeter of a circle = 2πR
• where,
• R is the radius of the circle
• π is the mathematical constant with an approximate (up to two decimal points) value of 3.14
• Again,
• Pi (π) is a special mathematical constant, it is the ratio of circumference to diameter of any circle.
• where C = π D
• C is the circumference of the circle
• D is the diameter of the circle
• For example: If radius of the circle is 4cm then find its circumference.
• Circumference = 2πr
• = 2 x 3.14 x 4
• = 25.12 cm
• Also, check:
 Secant Of A Circle Sector Of A Circle Circles Class 9 Circles For Class 10

### Area of a Circle Formula

Area of any circle is the region enclosed by the circle itself or the space covered by the circle. The formula to find the area of the circle is;

A = πr2

Where r is the radius of the circle. This formula is applicable to all the circles with different radii.

### Perimeter of Semi-Circle

The semi-circle is formed when we divide the circle into two equal parts. Therefore, the perimeter of the semi-circle also becomes half.

Hence, Perimeter = 2πr/2 = πr

### Area of Semi-Circle

Area of the semi-circle is the region occupied by a semi-circle in a 2D plane. The area of the semi-circle is equal to half of the area of a circle, whose radii are equal.

Therefore, Area = πr2/2

Summary

 Circumference of Circle 2πr Area of circle πr2 Perimeter of semi-circle πr Area of semi-circle πr2/2

## What Is the Circumference of a Circle?

Scientific American presents Math Dude by Quick & Dirty Tips. Scientific American and Quick & Dirty Tips are both Macmillan companies.

Throughout history, circles have symbolized many things: unity, protection, the Sun, infinity, and the Olympic Games, to name a few. Of course, philosophers and symbologists aren’t the only people to have taken an interest in circles. Mathematicians have spent millennia studying them, too.

Which is precisely why today’s article is all about circles.

In particular, after a quick refresher of circle basics, we’re going to figure out why the equation for the circumference of a circle that we all learned in school works, and we’re also going to learn how this equation is used to set up lots of Olympic track and field events.

What Is a Circle? My favorite way to define a circle is in terms of how to draw one. In the episode What Is Pi? we learned how to draw a circle arts-and-crafts style. To do this, start by cutting a 3-inch piece of string to serve as the radius of the circle. What’s the radius? It’s half the diameter.

Okay, but what’s the diameter? As you’ll soon be able to test with your finished drawing, it’s the greatest distance between any two points on the circle. Now, pin one end of the string down with your finger near the center of a normal sheet of binder paper, and then hold the loose end of the string up against the lead of your pencil.

Finally, pull the pencil so the string is taut and trace out your circle.

What does this all mean? Believe it or not, it means we’ve found a very good way to define a circle. Namely, a circle is the set of all points (that’s the curve you drew with your pencil) that are all the same distance from some common point (that’s the spot where you pinned the string down with your finger).

## Circumference (Perimeter) of a circle

The distance around the edge of a circle. Also 'periphery' , 'perimeter'.

Try this Drag the orange dots to move and resize the circle. The circumference is shown in blue. Note the radius changes and the circumference is calculated for that radius.

You sometimes see the word 'circumference' to mean the curved line that goes around the circle. Other times it means the length of that line, as in “the circumference is 2.11cm”.

The word 'perimeter' is also sometimes used, although this usually refers to the distance around polygons, figures made up of straight line segments.

### If you know the radius

Given the radius of a circle, the circumference can be calculated using the formula where:
R  is the radius of the circle
π  is Pi, approximately 3.142

If you know the diameter of a circle, the circumference can be found using the formula where:

D  is the diameter of the circle

π  is Pi, approximately 3.142

### If you know the area

If you know the area of a circle, the circumference can be found using the formula where:

A  is the area of the circle

π  is Pi, approximately 3.142

### Calculator

Use the calculator above to calculate the properties of a circle.

Enter any single value and the other three will be calculated.
For example: enter the radius and press 'Calculate'. The area, diameter and circumference will be calculated.

Similarly, if you enter the area, the radius needed to get that area will be calculated, along with the diameter and circumference.

### Related measures

• Radius The radius is the distance from the center of the circle to any point on the perimeter. See radius of a circle.
• Diameter The distance across the circle. See Diameter of a Circle for more.
• Inscribed angle
• Central angle
• Central angle theorem

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