A circle of radius 5 has a circumference of 31.42 and an area of 78.54. Double the radius and the circumference doubles while the area quadruples, because one is linear in r and the other goes with r². That single fact explains most of the situations where geometry feels counter-intuitive — pizza sizes, pipe capacity, and why a 65-inch television has 40 per cent more screen than a 55-inch one rather than 18.
The same scaling runs one dimension higher for volume, which is why a modest increase in a tank’s dimensions produces a much larger increase in what it holds.
What are the formulas worth knowing?
Five shapes cover the overwhelming majority of practical area problems.
| Shape | Area |
|---|---|
| Rectangle | length × width |
| Triangle | ½ × base × height |
| Circle | πr² |
| Trapezoid | ½ × (a + b) × height |
| Ellipse | π × a × b (semi-axes) |
The ellipse is the one with a sting in it. Its area is straightforward and its perimeter has no exact closed form at all — it requires an elliptic integral, and Ramanujan’s approximation is what practical work uses instead. It is a rare case where a shape everyone recognises has no elementary answer.
What can you do with three sides and no angles?
Get the area, via Heron’s formula. From the semi-perimeter s, the area is √(s(s−a)(s−b)(s−c)) — no angle, no height, and it has been known for two thousand years.
That is genuinely useful on site, where three distances are measurable and an angle is not. For a right triangle the hypotenuse is √(a² + b²), which is the same relationship in its most familiar form: 3 and 4 give 5.
For any triangle with two sides and the angle between them, the cosine rule generalises Pythagoras — and reduces to it exactly when that angle is 90 degrees, because the cosine of 90 is zero.
Why do gradians exist?
Because a right angle is exactly 100 of them and a full turn is 400, which makes decimal arithmetic on angles trivial. They were part of the same metric reform that gave us the metre, and they survive mainly in some surveying.
Degrees are the historical convention — 360 divides evenly by a great many numbers, which mattered when arithmetic was done by hand. Radians are the mathematical one: 180 degrees is π radians, and calculus only works cleanly in radians because the derivative of sin x is cos x only when x is measured that way.
Converting is one multiplication: degrees to radians is × π/180, and back is × 180/π. A 45-degree angle is 0.7854 radians.
What does the discriminant tell you?
Whether to bother solving. The quadratic formula gives x = (−b ± √(b² − 4ac)) ÷ 2a, and the expression under the root — the discriminant — settles the shape of the answer before you compute it: positive means two real roots, zero means one repeated root, negative means a complex conjugate pair.
For x² − 3x + 2 the discriminant is 1, so there are two real roots, and they are 2 and 1.
Completing the square is where the formula comes from, and it gives something the formula does not: the vertex. Writing ax² + bx + c in completed form puts the turning point directly in the expression, which is what you actually want when the question is about a maximum rather than a crossing.
What does a determinant mean?
The factor by which a matrix scales area in two dimensions and volume in three. A determinant of 2 doubles areas; a determinant of −1 preserves them and flips orientation; a determinant of 0 collapses the space onto a line or a point.
That last case is why a matrix is invertible only when its determinant is non-zero — a transformation that has flattened a dimension cannot be undone, because the information is gone.
Matrix multiplication has its own constraint worth remembering: the columns of the first must match the rows of the second, which is why AB and BA are frequently both defined and different, or one defined and the other not.
Questions people ask
Why is a large pizza such better value? Because area goes with the square of the diameter. A 16-inch pizza has 1.78 times the area of a 12-inch one, not 1.33 — which is usually a much larger ratio than the price difference.
Does doubling a pipe’s diameter double its flow? No — it roughly quadruples the cross-section, and flow rises faster still because friction falls with size. Pipe sizing is one of the places the square law is most consequential.
Is π ever needed to more than a few places? Almost never. Forty digits of π computes the circumference of the observable universe to within the width of a hydrogen atom; fifteen is more than any engineering problem requires.
Why does a right angle appear in so many formulas? Because it makes the cross term vanish. Pythagoras, the dot product and the cosine rule all simplify at 90 degrees for the same underlying reason.
Linear, square, cube — three scaling laws behind most of the surprises. The area calculator covers the standard shapes, the circle and triangle calculators go from any known value to the rest, the angle calculator converts between the three unit systems, and the quadratic solver and matrix calculator handle the algebra that sits behind them.