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!

So when the diameter is 1, the circumference is 3.141592654...

Distance walked = Circumference = π × 100m

= 314m (to the nearest m)

The circle is a plane shape (two dimensional):

And the definition of a circle is:

Lines

A line that goes from one point to another on the circle's circumference is called a Chord.

If that line passes through the center it is called aDiameter.

A line that "just touches" the circle as it passes by is called a Tangent.

And a part of the circumference is called an Arc.

Slices

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.

Quarter of a circle is called a Quadrant.

Half a circle is called a Semicircle.

A circle has an inside and an outside (of course!). But it also has an "on", because you could be right on the circle.

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

Draw a circle with a radius of 1.

The distance half way around the edge of the circle
will be 3.14159265... a number known as Pi

Or you could draw a circle with a diameter of 1.

Then the circumference (the distance all the way
around the edge of the circle) will be Pi

Pi (the symbol is the Greek letter π) is:

The ratio of the Circumference 
to the Diameter 
of a Circle.

In other words, if you measure the circumference, and then divide by the diameter of the circle you get the number π

It is approximately equal to:

3.14159265358979323846…

The digits go on and on with no pattern. In fact, π has been calculated to over two quadrillion decimal places and still there is no pattern.

Distance walked = Circumference = π × 100m = 314.159...m

= 314m (to the nearest m)

3.14159265358979323846264338327950288419716939937510 58209749445923078164062862089986280348253421170679...

Slices

There are two main "slices" of a circle:

• The "pizza" slice is called a Sector.
• And the slice made by a chord is called aSegment.

Quarter of a circle is called a Quadrant.

Half a circle is called a Semicircle.

Area of a Sector

You can work out the Area of a Sector by comparing its angle to the angle of a full circle.

Note: I am using radians for the angles.

This is the reasoning:

• A circle has an angle of 2π and an Area of: πr²
• So a Sector with an angle of θ (instead of 2π) must have an area of: (θ/2π) × πr²
• Which can be simplified to: (θ/2) × r²

Area of Sector = ½ × θ × r²   (when θ is in radians)

Area of Sector = ½ × (θ × π/180) × r²   (when θ is in degrees)

Arc Length

By the same reasoning, the arc length (of a Sector or Segment) is:

L = θ × r   (when θ is in radians)

L = (θ × π/180) × r   (when θ is in degrees)

Area of Segment

The Area of a Segment is the area of a sector minus the triangular piece (shown in light blue here).

There is a lengthy reason, but the result is a slight modification of the Sector formula:

Area of Segment = ½ × (θ - sin θ) × r²   (when θ is in radians)

Area of Segment = ½ × ( (θ × π/180) - sin θ) × r²   (when θ is in degrees)

Area of a Circle by Cutting into Sectors

Here is a way to find the formula for the area of a circle:

Cut a circle into equal sectors (12 in this example)

Divide just one of the sectors into two equal parts. You now have thirteen sectors – number them 1 to 13:

Rearrange the 13 sectors like this:

Which resembles a rectangle:

What are the (approximate) height and width of the rectangle?

The height is the circle's radius: just look at sectors 1 and 13 above. When they were in the circle they were "radius" high.

The width (actually one "bumpy" edge) is half of the curved parts along the circle's edge ... in other words it is about half the circumference of the circle.

We know that:

Circumference = 2 × π × radius

And so the width is about:

Half the Circumference = π × radius

And so we have (approximately):

Now we just multply the width by the height to find the area of the rectangle:

Area = (π × radius) × (radius)

= π × radius²

Note: The rectangle and the "bumpy edged shape" made by the sectors are not an exact match.

But we could get a better result if we divided the circle into 25 sectors (23 with an angle of 15° and 2 with an angle of 7.5°).

And the more we divided the circle up, the closer we would get to being exactly right.

Conclusion

Area of Circle = π r²

Annulus

Area

Because it is a circle with a circular hole, you can calculate the area by subtracting the area of the "hole" from the big circle's area:

Area = πR² - πr² 
= π( R² - r² )

Example: a steel pipe has an outside diameter (OD) of 100mm and an inside diameter (ID) of 80mm, what is the area of the cross section?

Convert diameter to radius for both outside and inside circles:

• R = 100 mm / 2 = 50 mm
• r = 80 mm / 2 = 40 mm

Now calculate area:

Area = π( R² - r² )

Area = 3.14159... × ( 50² - 40² )

Area = 3.14159... × ( 2500 - 1600 )

Area = 3.14159... × 900

Area = 2827 mm² (to nearest mm²)

Circle Theorems

Some interesting things about angles and circles.

Inscribed Angle

First off, a definition:

Inscribed Angle: an angle made from points sitting on the circle's circumference.

 
A and C are "end points"
B is the "apex point"

Inscribed Angle Theorems

An inscibed angle a° is half of the central angle 2a°

(Called the Angle at the Center Theorem)

And (keeping the endpoints fixed) ...

... the angle a° is always the same, no matter where it is on the circumference:

Angle a° is the same.
(Called the Angles Subtended by Same Arc Theorem)

Example: What is the size of Angle POQ? (O is circle's center)

Example: What is the size of Angle CBX?

Angle in a Semicircle

An angle inscribed in a semicircle is always a right angle:

(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)

So there we go! No matter where that angle is 
on the circumference, it is always 90°

Example: What is the size of Angle BAC?

Cyclic Quadrilateral

Example: What is the size of Angle WXY?