ACARA v9 CONTENT DESCRIPTION “solve problems involving the circumference and area of a circle using formulas and appropriate units”
The parts of a circle
A circle is the set of points the same distance from a single point called the centre. That fixed distance from the centre out to the edge is the radius, written r. A straight line that runs all the way across the circle through the centre is the diameter, written d, and it is made of two radii laid end to end, so d = 2r. The distance all the way around the outside of the circle is the circumference. These four words, centre, radius, diameter, and circumference, are the vocabulary for everything that follows, so it helps to point each one out on a drawing before reaching for any formula.
The parts of a circle
Centre, radius r to the edge, diameter d across through the centre.
a circle has a centre, a radius r to the edge, and a diameter d across through the centre; the diameter is twice the radius, d = 2r.
Pi, the circle constant
If you measure the circumference of any circle and divide it by its diameter, you always get the same number, no matter how big or small the circle is. That constant ratio is called pi. Its value is about 3.14, and it is sometimes written as the fraction 22/7 for a quick estimate. Pi is an irrational number, which means its decimal goes on forever without ever settling into a repeating pattern, so in real calculations we round it, usually to 3.14. The key idea is that pi ties the distance around a circle directly to the distance across it: circumference divided by diameter is pi for every circle there is.
Pi is C divided by d
The circumference unrolls to about 3.14 diameters, every time.
pi is the circumference divided by the diameter, the same for every circle, about 3.14 (or 22/7).
The circumference of a circle
Because pi is the circumference divided by the diameter, multiplying both sides by the diameter gives the circumference formula, C = pi x d. Since the diameter is twice the radius, the same rule can be written as C = 2 x pi x r, which is handy when a problem gives you the radius instead of the diameter. Take a circle with radius r = 5 cm. Using pi about 3.14, the circumference is C = 2 x 3.14 x 5 = 31.4 cm. You would get the same answer through the diameter, since d = 10 and 3.14 x 10 = 31.4 cm. Circumference is a length, so it is measured in length units such as cm or m.
Circumference of a circle
C = pi x d = 2 x pi x r; for r = 5, C = 31.4 cm.
the circumference is C = pi x d = 2 x pi x r; for r = 5, C = 2 x 3.14 x 5 = 31.4 cm.
The area of a circle
The area of a circle is the amount of surface it encloses, and the formula is A = pi x r^2, which means pi times r times r. A short picture shows why this works. Imagine slicing the circle into many thin wedges, like the segments of an orange, and then laying the wedges side by side, pointing up and down in turn. They settle into a shape very close to a rectangle whose height is the radius r and whose width is about half the circumference, that is pi x r. Multiplying height by width gives r times pi x r, which is pi x r^2. For r = 5, the area is A = 3.14 x 5 x 5 = 78.5 cm^2. Area is always measured in square units.
Area of a circle
Rearranged wedges make a rectangle of height r and width about pi x r.
the area is A = pi x r^2; for r = 5, A = 3.14 x 5 x 5 = 78.5 cm^2.
Working a circle problem
When you meet a circle problem, first read whether you are given the radius or the diameter. If you are given the diameter, you can halve it to find the radius, or use C = pi x d directly for the circumference. Next decide which quantity is being asked for and pick the matching formula: C = 2 x pi x r for the distance around, or A = pi x r^2 for the surface inside. Substitute pi as about 3.14 and work through the arithmetic one step at a time. Keep the units straight, since a circumference comes out in cm while an area comes out in cm^2. Finally, glance at the answer and check it is a sensible size for the circle you started with.
Working a circle problem
From r = 5: circumference in cm, area in cm^2, units kept apart.
from r = 5: circumference 31.4 cm (a length) and area 78.5 cm^2 (a square measure) -- keep the units distinct.
Why this matters
Circles are everywhere once you start looking: wheels and gears, plates and lids, pipes and tanks, running tracks, and countless logos. The two formulas in this unit let you answer the two most common questions about any of them, namely how far it is around the outside, the circumference, and how much surface sits inside, the area. The two mistakes to watch for are mixing up the radius and the diameter, and confusing the length unit of a circumference with the square unit of an area. Keep those two apart and the rest is steady arithmetic. This unit stays with the flat, two dimensional circle; cylinders and spheres, which build on these ideas, come in a later unit.
Quick self-check
1. A circle has radius 5 cm. What is its diameter?
2. What does pi represent?
3. Using pi about 3.14, the circumference of a circle with radius 5 cm is about...
4. Using pi about 3.14, the area of a circle with radius 5 cm is about...