The area of a disc

A disc is a circle together with everything inside it.

• The circle is a line, so it has a length – the circumference.
• The disc is a flat region, so it has an area – the area of the disc.
• A disc’s centre, radius and diameter are those of the circle around it.

We 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: the arcs of the sectors of one colour are at the bottom, those of the other colour at the top.
• Half of the circumference $\displaystyle 2\pi r$ is $\displaystyle \pi r$.
• The more and 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 $\displaystyle {\pi r\cdot r}={\pi r^2}$.

• $\displaystyle r^2$ is $\displaystyle r\cdot r$ – the radius times itself, not $\displaystyle 2r$.
• If the diameter is given, divide it by 2 first.
• Area is measured in square units: cm², m².

A pizza is 30 cm across.
What is its area?

• The radius is $\displaystyle {30:2}={15}$ cm.
• $\displaystyle {A}={3.14\times15^2}={3.14\times225}={706.5}$ cm².

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

$\displaystyle \pi$ can also be kept in the answer, like a letter, instead of being replaced by 3.14.

• A disc of radius 3 cm has an area of $\displaystyle {\pi\cdot3^2}={9\pi}$ cm², and a circumference of $\displaystyle {2\cdot\pi\cdot3}={6\pi}$ cm.
• $\displaystyle 9\pi$ is the exact answer. $\displaystyle {9\times3.14}={28.26}$ is an approximation of it.

The area of a half disc is half the area of the disc.
For example, a window in the shape of a half disc 80 cm across:

• The radius is 40 cm.
• The whole disc: $\displaystyle {3.14\times40^2}={3.14\times1600}={5024}$ cm².
• The window: $\displaystyle {5024:2}={2512}$ cm².

A ring is the shape between two circles with the same centre.

Its area is the area of the big disc minus the area of the small disc.
For example, $\displaystyle {R}={5}$ cm and $\displaystyle {r}={3}$ cm:

• The big disc: $\displaystyle {3.14\times5^2}={78.5}$ cm².
• The small disc: $\displaystyle {3.14\times3^2}={28.26}$ cm².
• The ring: $\displaystyle {78.5-28.26}={50.24}$ cm².

A CD’s outer radius is 10 cm.
In the middle of the CD there is a round hole of radius 6 cm.
What is the CD’s area? ($\displaystyle \pi\approx3.14$)

The largest disc that fits in a square of side 10 cm is cut out of it.
How many cm² are left of the square? ($\displaystyle \pi\approx3.14$)

At a pizzeria you can buy one pizza 40 cm across, or two pizzas 20 cm across each.
Which gives more pizza? How many times as much?

A disc’s area is $\displaystyle 49\pi$ cm².
What is its radius?
What is its circumference? Write the answer with $\displaystyle \pi$.