Area of a Disc

How much paint do you need for a circular wall? The question reduces to: what is the area enclosed by a circle of a given radius? Archimedes answered it twenty-three centuries ago by squeezing polygons in and out of the circle until they met, a method-of-exhaustion argument that prefigured calculus by two millennia. The answer is shockingly simple: $A = \\pi r^2$.

First, define $\\pi$

The number $\\pi$ is defined as the ratio of a circle's circumference to its diameter:

This ratio is the same for every circle, large or small, which is one of the deep facts of Euclidean geometry. Numerically $\\pi \\approx 3.14159\\ldots$; it is irrational (Lambert, 1761) and transcendental (Lindemann, 1882).

The area formula

For a disc of radius $r$:

If you know the diameter $d = 2r$ instead, substitute: $A = \\pi (d/2)^2 = \\pi d^2 / 4$.

• Engineering: Pipe-flow rates depend on cross-sectional area $\\pi r^2$; doubling the pipe radius quadruples the flow capacity, which is why municipal water mains are large even though pressure stays modest.
• Astronomy: Telescope light-gathering power scales as the disc area $\\pi r^2$ of the primary mirror; a 10-metre mirror collects 4x the light of a 5-metre, which is why bigger telescopes resolve fainter objects.
• Statistics: The standard normal density has its mass concentrated within a disc-like region around the mean; computing "probability within radius $r$" in 2D Gaussian samples uses $\\pi r^2$ weighted by the density.

Slide the number of rings $n$ from low to high. The disc on the left is partitioned into $n$ concentric annuli; on the right those rings are unrolled into rectangles stacked into a near-triangle. As $n$ grows, the rectangles fill out the triangle of base $2\\pi r$ and height $r$, whose area is $\\tfrac{1}{2}(2\\pi r) \\cdot r = \\pi r^2$.

Where does $\\pi r^2$ come from?

Inscribe a regular $n$-gon inside the circle. Slice it into $n$ thin isosceles triangles meeting at the centre. Each triangle has base $b\_n$ (one polygon side) and height roughly $r$. The polygon area is $A\_n = \\tfrac{1}{2} (n b\_n) \\cdot \\text{height}$.

As $n \\to \\infty$, the polygon's perimeter $n b\_n$ approaches the circumference $2\\pi r$, and the height approaches $r$. So $A \\to \\tfrac{1}{2} (2\\pi r) \\cdot r = \\pi r^2$. Archimedes' beautiful idea: an inscribed polygon's area is less than the disc's, a circumscribed polygon's area is more, and both bounds converge to $\\pi r^2$, so the disc's area must equal that limit.

Sectors and slices of pizza

A sector is a "pie slice" of the disc bounded by two radii and an arc. If the central angle is $\\theta$ in degrees, the sector occupies the fraction $\\theta / 360$ of the disc:

A semicircle (\\theta = 180^\\circ) is $\\tfrac{1}{2}\\pi r^2$; a quarter circle is $\\tfrac{1}{4}\\pi r^2$.

Try it

• Predict first: what is the area of a unit disc? Set $r = 1$ and verify the readout shows $\\pi \\approx 3.14$, the simplest case worth memorising.
• Before adjusting: if you double $r$, by what factor should the area change? Set $r = 2$ and verify the area is $4\\pi \\approx 12.57$, quadruple, because $(2r)^2 = 4r^2$.
• For a pizza of radius $12$ inches, eat a 45^\\circ slice. Slice area $= (45/360) \\cdot \\pi \\cdot 144 = 18\\pi \\approx 56.5$ square inches.

A trap to watch for

The most common beginner mistake is confusing area $\\pi r^2$ with circumference $2 \\pi r$. Both contain $\\pi$ and $r$, but they measure different things and have different units. Area is two-dimensional (square units); circumference is one-dimensional (linear units). Sanity check: if $r = 1$, the area is $\\pi \\approx 3.14$ but the circumference is $2\\pi \\approx 6.28$, they differ. If you ever find yourself writing $\\pi r^2$ for the perimeter of a circular field, stop and re-derive.

What you now know

You can compute the area of any disc or sector, you understand why $\\pi$ shows up (it is the polygon-perimeter limit), and you know that area grows as the square of the radius. The next section turns the same constant $\\pi$ to a different question: the circumference, or perimeter, of the same circle.

Quick check

Mark section complete →

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 7, Section 1, the area of a disc and its sector formulas.
• Archimedes (c. 250 BCE). Measurement of a Circle (in T. L. Heath, The Works of Archimedes, Dover, 1953). The original polygon-exhaustion derivation of $A = \\pi r^2$.
• Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. Section 1.6 covers circles, sectors, and arc length.
• Berggren, L., Borwein, J., and Borwein, P. (2004). Pi: A Source Book, 3rd ed. Springer. Comprehensive history of $\\pi$, including the original Archimedes text.