Circumference & Area Of Circles
Hello there. This math post is on circumference (perimeter) and area of circles. This guide is ideal for middle school students (Grades 6 to 8).
The math text images were rendered with the use of QuickLaTeX.com. Circle images are screenshots from using the Math Is Fun website Drawing tool and the Geogebra online Geometry tool.
Topics
- Quick Terminology
- Circumference Of Circles
- Area Of Circles
- Dealing With Semi-Circles & Quarter Circles
- Practice Problems
- Solutions To Practice Problems
Quick Terminology
Here are some quick terminology terms for reference.
Radius
The distance from the centre of the circle to the boundary/outside of the circle.
Diameter
Twice the radius measure. This can also be viewed as the distance from side of the circle to the other side of the circle while passing through the centre of the circle.
Circumference
Another way of saying the perimeter of the circle. The distance around a circle.
Π (Pi)
A famous (irrational) number with never ending digits. To two decimals places the value for Π is 3.14. Pi is the ratio of the circle's circumference divided by the circle's diameter.
Circumference Of Circles
When it comes to circles, the circumference is a fancy way of saying the perimeter or distance around a circle. Given a radius, the circumference of a circle is twice the radius multiplied by Π.
Alternatively, this circumference is the circle's diameter multiplied by Pi. This is because the diameter is twice the radius.
Example
A circle with a radius of 5 units has a circumference of 10 pi units. This is from:
Area Of Circles
The area of a circle can be somewhat viewed as taking the area of a square. Given a radius length, the area of the circle is taking the radius multiplied by itself and then multiplying by pi.
Example
With a circle with a radius of 10 centimetres, the area of this circle is 100 Pi square centimetres. The calculation is as follows:
Dealing With Semi-Circles & Quarter Circles
So far we have dealt with whole circles. What would happen if we have to compute the perimeter and areas of semi-circles and quarter circles?
Semi-Circle Example
The main thing to remember with the semi-circle is that it is half of a regular circle. Refer to the image below where the radius length is five units.
The area of this semi-circle is half of 25 Π square units which 12.5 Π square units. Assuming that Π is 3.14, this approximates to 39.25 square units.
Taking the perimeter of the half circle does involve taking half of the circumference. However, you have the straight line diameter that is leftover. The straight line distance of 10 from the point (0, 0) to (0, 10) is added on top of the half circumference to obtain the true semi-circle perimeter.
Dealing With Quarter Circles
Working with a quarter circle is similar to working with half circles. The area is one quarter of the area of the full circle. With the perimeter of the quarter circle, it is one quarter of the full circle's circumference plus two the radii distances from the outside.
In the picture, the radius of the quarter circle is 3 units. The area is 9 pi square units divided by four which is 2.25 pi square units. This quarter circle's circumference is 6 pi divided by 4 plus 6 units. This is 1.5Π + 6 units.
Practice Problems
What is the circumference of a circle with 20 centimetres as its radius?
What is the area of a circle with 12 centimetres as its radius?
What is the area of a circle with 10 decimetres as its diameter?
What is the circumference of a circle with 8 centimetres as its diameter?
Given a diameter of pi units, what is the circumference of the circle?
Determine the area and circumference of the half circle with a radius of 2 metres.
Determine the area and circumference of the quarter circle with a radius of 4 metres.
Solutions To Practice Problems
40Π centimetres
144Π square centimetres
25Π square decimetres
8Π centimetres
Π squared square units
Area = 2Π square meters, Circumference = 2 Π + 4 metres
Area = 4Π square meters, Circumference = 2 Π + 8 metres
