The area of a disc

Let us compare the area of a disc with the area of a square whose side is the radius, $\displaystyle r$ – call it a radius square (the coloured square in the drawing).

• The disc is inside a big square made of 4 radius squares, so its area is less than 4 radius squares.
• The four corners outside the disc are together smaller than one radius square, so the disc is more than 3 radius squares.
• Noa drew on squared paper a disc whose radius is 10 squares, and counted the squares inside it: about 314. A radius square holds $\displaystyle {10\times10}={100}$ squares, so the disc is about 3.14 radius squares.

The area of a disc is $\displaystyle \pi$ radius squares – the same $\displaystyle \pi$ as in the circumference.

Here is why: cut a disc into 16 equal sectors, like slices of a pizza, and lay them in a row – one with its point up, the next with its point down:

The shape we get is almost a rectangle:

• Its height is a radius, $\displaystyle r$.
• Its length is half the circumference: half of the arcs are at the bottom, half at the top. Half of $\displaystyle 2\pi r$ is $\displaystyle \pi r$.
• The thinner the sectors, the closer the shape is to a rectangle.

A rectangle’s area is its length times its width, so the area of the disc is $\displaystyle \pi r\times r$.

• $\displaystyle r^2$ (read “$\displaystyle r$ squared”) is $\displaystyle r\times r$ – the radius times itself, not $\displaystyle 2\times r$.
• To calculate: $\displaystyle 3.14\times r\times r$.
• If the diameter is given, halve it first.
• Area is measured in square units: cm², m².

A pizza’s diameter is 30 cm.
What is its area?

• The radius is $\displaystyle {30:2}={15}$ cm.
• The area: $\displaystyle {3.14\times15\times15}={706.5}$ cm².

Find the area of the disc, in cm² ($\displaystyle \pi\approx3.14$).

When the radius is doubled, the area is 4 times as big:

• Radius 1 cm: $\displaystyle {3.14\times1\times1}={3.14}$ cm².
• Radius 2 cm: $\displaystyle {3.14\times2\times2}={12.56}$ cm² – 4 times as much.
• The radius appears twice in the calculation, and $\displaystyle {2\times2}={4}$. The radius square, too, becomes 4 times as big.
• The circumference only doubles: the radius appears in it once.

A pizzeria sells one pizza 40 cm across, or two pizzas 20 cm across each.
Which gives more pizza?
Explain.

The area of a part of a disc is found from the area of the whole disc:

• A half disc – half the disc’s area.
• A ring – the shape between two circles with the same centre. Its area is the big disc’s area minus the small disc’s.

For example, a ring whose outer radius ($\displaystyle R$) is 5 cm and inner radius ($\displaystyle r$) 3 cm:

• The big disc: $\displaystyle {3.14\times5\times5}={78.5}$ cm².
• The small disc: $\displaystyle {3.14\times3\times3}={28.26}$ cm².
• The ring: $\displaystyle {78.5-28.26}={50.24}$ cm².

A window is shaped like half a disc.
Its diameter is 80 cm.
What is the area of its glass, in cm²? ($\displaystyle \pi\approx3.14$)

A CD’s radius is 6 cm.
In its middle there is a round hole whose radius is 1 cm.
What is the CD’s area, in cm²? ($\displaystyle \pi\approx3.14$)

A gate is made of a square whose side is 2 m, with a half disc on top of it.
What is the gate’s area, in m²? ($\displaystyle \pi\approx3.14$)

The biggest disc that fits was cut out of a square of cardboard whose side is 10 cm.
How many cm² of cardboard are left? ($\displaystyle \pi\approx3.14$)