Area of a circle - derivation

 

Circle

A line forming a closed loop, every point on which is a fixed distance from a center point.

Try this Drag an orange dot. The circle can be moved by dragging the center point and resized by dragging the point on the circle.

A circle is a type of line. Imagine a straight line segment that is bent around until its ends join. Then arrange that loop until it is exactly circular - that is, all points along that line are the same distance from a center point.

There is a difference between a circle and a disk. A circle is a line, and so, for example, has no area - just as a line has no area. A disk however is a round portion of a plane which has a circular outline. If you draw a circle on paper and cut it out, the round piece is a disk.

Properties of a circle

Center	A point inside the circle. All points on the circle are equidistant (same distance) from the center point.
Radius	The radius is the distance from the center to any point on the circle. It is half the diameter. See Radius of a circle.
Diameter	The distance across the circle. The length of any chord passing through the center. It is twice the radius. See Diameter of a circle.
Circumference	The circumference is the distance around the circle. See Circumference of a Circle.
Area	Strictly speaking a circle is a line, and so has no area. What is usually meant is the area of the region enclosed by the circle. See Area enclosed by a circle .
Chord	A line segment linking any two points on a circle. See Chord definition
Tangent	A line passing a circle and touching it at just one point. See Tangent definition
Secant	A line that intersects a circle at two points. See Secant definition

Calculator

ENTER ANY ONE VALUE
Radius		clear
Diameter		clear
Area		clear
Circumference		clear

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.

Pi

In any circle, if you divide the circumference (distance around the circle) by its diameter (distance across the circle), you always get the same number. This number is called Pi and is approximately 3.142. See Definition of pi.

Relation to ellipse

A circle is actually a special case of an ellipse. In an ellipse, if you make the major and minor axis the same length, the result is a circle, with both foci at the center. See Ellipse definition

Circle as a conic section

You can define a circle as the shape created when a plane cuts through a cone at right angles to the cone's axis. For more on this see Conic sections - circle.

Circle as a locus

A circle is the locus of all points a fixed distance from a given (center) point. This definition assumes the plane is composed of an infinite number of points and we select only those that are a fixed distance from the center. (See locus definition.)

Equations of a circle

In coordinate geometry, a circle can be described using sets of equations.
For more on this see Equations of circles and ellipses.

Other circle topics

General

• Circle definition
• Radius of a circle
• Diameter of a circle
• Circumference of a circle
• Parts of a circle (diagram)
• Semicircle definition
• Tangent
• Secant
• Chord
• Intersecting chords theorem
• Intersecting secant lengths theorem
• Intersecting secant angles theorem
• Area of a circle
• Concentric circles
• Annulus
• Area of an annulus
• Sector of a circle
• Area of a circle sector
• Segment of a circle
• Area of a circle segment (given central angle)
• Area of a circle segment (given segment height)

Equations of a circle

• Basic Equation of a Circle (Center at origin)
• General Equation of a Circle (Center anywhere)
• Parametric Equation of a Circle

Angles in a circle

• Inscribed angle
• Central angle
• Central angle theorem

Arcs

• Arc
• Arc length
• Arc angle measure
• Adjacent arcs
• Major/minor arcs
• Intercepted Arc
• Sector of a circle
• Radius of an arc or segment, given height/width
• Sagitta - height of an arc or segment

Area of a circle - derivation

This page describes how to derive the formula for the area of a circle. we start with a regular polygon and show that as the number of sides gets very large, the figure becomes a circle. By finding the area of the polygon we derive the equation for the area of a circle.

Try this. In the applet below we have a six-sided regular polygon. Keep clicking on 'more' and note that as the number of sides gets larger, the polygon approaches being a circle. As the number of sides becomes infinitely large, it is, in fact, a circle. Click 'reset' afterwards.

The polygon can be broken down into n isosceles triangles (where n is the number of sides), such as the one shown on the right.

In this triangle
s   is the side length of the polygon
r   is the radius of the polygon and the circle
h   is the height of the triangle.

The area of the triangle is half the base times height or

triangle area

=

1

2

s

h

There are n triangles in the polygon so

polygon area

=

n

 

1

2

s

h

 

This can be rearranged to be

polygon area

=

h

2

(

n

s

)

The term ns is the perimeter of the polygon (length of a side, times the number of sides). As the polygon gets to look more and more like a circle, this value approaches the circle circumference, which is 2πr. So, substituting 2πr for ns:

polygon area

=

h

2

(

2

π

r

)

Also, as the number of sides increases, the triangle gets narrower and narrower, and so when s approaches zero, h and r become the same length. So substituting r for h:

polygon area

=

r

2

(

2

π

r

)

Rearranging this, we get

area

=

π

r

2

If you know the diameter

The radius r of a circle is half the diameter d

r

=

d

2

Substituting r into the area formula

area

=

π

 

d

2

 

2

Which simplifies to

area

=

π

 

d

2

2

2

 

 

 

 

or

 

 

 

π

d

2

4

If you know the circumference

The circumference c of a circle radius r is given by

c

=

2

π

r

Dividing both sides by 2π

r

=

c

2

π

Substitute this into the area formula for r

area

=

π

 

c

2

π

 

2

Which simplifies to

area

=

c

2

4

π