Circumference and Area of Circles, Pi, Perimeter and Area of Triangles/Rectangles
What is a Circle?
A circle is a shape where the circumference (or the perimeter of the circle) consists of points that are equally far away from a fixed point (the center).
The radius is the distance from the center of the circle to any point on the circle itself (must be from the center)
The diameter is the distance from one point on the circle to the other side (must go through the center)
The diameter is always twice the length of the radius.
What is the Circumference a Circle?
The circumference is the distance around a circle (this is a special term for the perimeter of circles).
To find the circumference of a circle, you use the equation:
C = dπ, which is more commonly written as C = 2πr
C = dπ = 2πr
Ex.1: What is the circumference of a circle in terms of pi with a radius of 12?
C = 2πr = 2π(12) = 24π
Answer: 24π
What is the Area a Circle?
The area of a circle is the amount of space that the circle covers (total amount of space inside the circle)
To find the area of a circle, you use the equation:
Area of circle=π × radius2 , or
A = πr2
Ex. 1: In terms of pi, what is the area of a circle with a radius of 5?
A = πr2 = π × (5)2 = 25π
Answer: 25π = ~78.54
Ex. 2: What is the area of a circle in terms of pi with a diameter of 16?
Diameter = 2 × (Radius)
16 = 2 (Radius)
Radius = 8
Area = π × radius2 = π × (8)2 = 64π
Answer: 64π = ~201.16
What is Pi?
We have been using it a lot in equations involving circles lately, so what exactly is it? Pi is the ratio of any circle's circumference to its diameter (this is true for all circles!). If you have a circle with a diameter of 1, for example, then the circumference of the circle is 3.141592653…or pi.
Pi is written as the Greek symbol π.
From the drawing, the circumference divided by the diameter of the circle is π (this is where our equation for circumference comes from).
Finding the Circumference using the Diameter
Ex. 1: You walk around a circle which has a diameter of 100 meters, how far have you walked?
Distance walked = circumference
Circumference = πd
= π(100) = 314 m (don't forget units!)
Finding the Diameter using the Circumference
Ex. 1: Sam measured 80 mm around the outside of a pipe, what is its diameter?
Diameter = Circumference / π
= 80 mm / π
= 25.464
Answer: 25 mm (closest mm
Pi is approximately equal to:
3.14159265358979323846264338327950…
A quick and easy approximation for π using fractions is 22/7
● 22/7 = 3.1428571…
○ Keep in mind that 22/7 is not exactly equal to π
○ π is not exactly equal to the ratio of any two whole numbers because π is an irrational number
● Better approximation: 355/113
○ 335/113 = 3.1415929…
When using π in calculations, usually 3.14 or 3.14159 is fine. Some calculators have a built in button for π that uses a very good approximation for the value (you can use this button instead if you want).
What is a Triangle?
A triangle is a shape that has three sides and three angles.
1) The three angles always add up to 180 degrees.
2) There are 3 different types of triangles, with three different angle classifications:
You can also combine names together to create triangles. The triangle on the right is called an isosceles right triangle because it has a right angle and two equal angles.
What is the Perimeter of a Triangle?
The perimeter is the total distance around the edge of the triangle.To find the perimeter of a triangle, you simply add up all the side lengths (P = a + b + c).
Ex. 1: What is the perimeter of the triangle on the right?
Perimeter = 11cm + 13cm + 8cm = 32cm
What is the Area of a Triangle?
The area of a triangle is the total amount of space inside the triangle. Area = ½ x b x h , where b is the distance along the base and h is the height (measured at a right angle to the base).
A = ½ bh
Ex.1: What is the area of the triangle to the right?
Area = ½ x b x h
= ½ x 30 x 21
= 315
Answer: Area = 315 units2
What is a Rectangle?
A rectangle is a shape with four sides, where every angle is a right angle (90°). The opposite sides of a rectangle are equal to each other (the dashes indicate that they are equal)
What is the Perimeter of a Rectangle?
The perimeter of a rectangle is the distance around the edges. To find the perimeter, you can add up all the sides if they are given, or more commonly, use the equation: Perimeter = 2(w+h), where w is the width and h is the height of the rectangle. (you can also use length and width instead).
P = 2(w + h)
Ex. 1: A rectangle has a width of 10 cm, and a height of 6 cm, what is its perimeter?
Perimeter = 2 x (10cm + 6 cm)
= 2 x (16cm)
= 32 cm
Answer: Perimeter = 32 cm
What is the Area of a Rectangle?
The area of a rectangle is the amount of space it takes up, and is found using the equation: Area = w x h , where w is the width and h is the height.
A = wh
Ex. 1: A rectangle is 8m wide and 5m high, what is its area?
Area = w x h
= 8m x 5m
= 40 m2
Answer: Area = 40 m2
