Roll a coin along a ruler and mark where it lands after one full turn. The distance is always a little more than three times the coin's width — never four, never two, always the same stubborn three-and-a-bit. That number has a name, π, and it is the secret tucked inside every circle ever drawn. This lesson is the story of that number, and of the two formulas it unlocks.
The parts of a circle
A circle is the set of all points the same distance from a single fixed point. That fixed point is the centre. The fixed distance — from the centre out to the edge — is the radius, written \(r\).
If you walk in a straight line from one edge, through the centre, to the opposite edge, you cross two radii back to back. That longest possible chord is the diameter, \(d\). So the diameter is simply twice the radius:
\[ d = 2r \]And the distance once around the outside — the length of the curve itself — is the circumference, \(C\). Think of it as the perimeter of the circle, the answer to "how far if I walk all the way around?"
The discovery that became π
Here is the remarkable thing — the fact that took humanity centuries to pin down. Pick any circle at all: a coin, a wheel, a dinner plate, the rim of a galaxy. Measure its circumference and divide by its diameter. You always get the same number.
\[ \frac{C}{d} = \pi \approx 3.14159\ldots \]Bigger circles have a bigger circumference, but they have a proportionally bigger diameter too, and the ratio between them never budges. That constant ratio is what we call pi, written with the Greek letter \(\pi\). Its digits run on forever without repeating, so we round it: \(\pi \approx 3.14\) is plenty for most work.
In words π is not a length and not an area — it is a comparison. It says how many diameters it takes to wrap around the outside: a little over three, for every circle in the universe.
Circumference: C = πd = 2πr
Once you accept that \(C \div d\) is always \(\pi\), rearranging gives the formula directly. Multiply both sides of \(\frac{C}{d} = \pi\) by \(d\):
\[ C = \pi d \]And because \(d = 2r\), you can write the same thing in terms of the radius — which is usually what a problem hands you:
\[ C = 2\pi r \]The two formulas say exactly the same thing; pick whichever matches the number you were given. Drag the radius slider below and watch the circumference grow in perfect step with it — and notice the ratio \(C \div d\) holding fast at π no matter how large the circle gets.
- Choose the radius form, since we know \(r\): \(\;C = 2\pi r\).
- Substitute \(r = 7\): \(\;C = 2 \times \pi \times 7 = 14\pi\) cm. That is the exact answer.
- For a decimal, use \(\pi \approx 3.14\): \(\;C \approx 2 \times 3.14 \times 7 = 43.96\) cm.
The circumference is \(14\pi \approx 43.96\) cm.
Area: A = πr²
Circumference measures the curve. Area measures the flat space inside — how much paint to fill the disc. The formula is short:
\[ A = \pi r^2 \]But where does it come from? Here is the picture that makes it click. Slice the circle into many thin wedges, like a pizza, then unfold them and lay them side by side, points up and points down alternately.
The wiggly shape they form is almost a rectangle — and the thinner you slice, the more rectangular it gets. Its height is one radius, \(r\). Its width is made of the curved tops of half the wedges, which together are half the circumference, \(\frac{1}{2}C = \frac{1}{2}(2\pi r) = \pi r\). A rectangle's area is width times height:
\[ A = (\pi r)(r) = \pi r^2 \]So the circle's area is exactly \(\pi r^2\). Be careful that the radius is squared, not the whole thing — square the radius first, then multiply by π.
- Start from the area formula: \(\;A = \pi r^2\).
- Square the radius first: \(\;r^2 = 3^2 = 9\).
- Multiply by π for the exact answer: \(\;A = 9\pi\) m².
- For a decimal, use \(\pi \approx 3.14\): \(\;A \approx 9 \times 3.14 = 28.26\) m².
The area is \(9\pi \approx 28.26\) m².
Tip — keep π to the end. Leaving your answer "in terms of π" (like \(14\pi\) or \(9\pi\)) is exact and tidy. Only swap in \(3.14\) at the very last step, when a decimal is actually asked for. It avoids rounding errors and shows your reasoning clearly.
Slices of a circle: arcs and sectors
You do not always want the whole circle. A piece of the circumference is called an arc; the pizza-slice region between two radii is a sector. Both are just fractions of the whole, set by the angle at the centre out of the full \(360°\).
A quarter-circle uses \(90°\), which is \(\frac{90}{360} = \frac{1}{4}\) of the turn — so its arc is \(\frac{1}{4}\) of the circumference and its area is \(\frac{1}{4}\) of \(\pi r^2\). In general, scale by \(\dfrac{\theta}{360}\):
\[ \text{arc length} = \frac{\theta}{360}\,(2\pi r), \qquad \text{sector area} = \frac{\theta}{360}\,(\pi r^2) \]You will meet these in depth later. For now, the idea to hold on to is simple: a fraction of the angle gives the matching fraction of both the curve and the area.
Practice
Try each one yourself, then reveal the full solution.
1. A circle has radius \(r = 7\) cm. Find its circumference, using \(\pi \approx 3.14\).
Use the radius form of the circumference formula:
\[ C = 2\pi r = 2 \times 3.14 \times 7 \]Work left to right: \(2 \times 3.14 = 6.28\), then \(6.28 \times 7 = 43.96\).
The circumference is 43.96 cm (exactly \(14\pi\) cm).
2. A circle has radius \(r = 3\) cm. Find its area, giving the answer both exactly in terms of π and as a decimal.
Use the area formula and square the radius first:
\[ A = \pi r^2 = \pi \times 3^2 = 9\pi \]That \(9\pi\) is the exact answer. For a decimal, use \(\pi \approx 3.14\): \(\;9 \times 3.14 = 28.26\).
The area is \(9\pi\) cm² ≈ 28.26 cm².
3. A circle has circumference \(C = 31.4\) cm. Find its radius (use \(\pi \approx 3.14\)).
Start from \(C = 2\pi r\) and solve for \(r\) by dividing both sides by \(2\pi\):
\[ r = \frac{C}{2\pi} = \frac{31.4}{2 \times 3.14} = \frac{31.4}{6.28} \]And \(31.4 \div 6.28 = 5\).
The radius is 5 cm. (Quick check: \(2 \times 3.14 \times 5 = 31.4\) ✓.)