Area Guide — Formulas for 12 Common 2D Shapes

What area measures

Area is the amount of flat 2D space a shape covers — always expressed in square units (m², ft², cm², etc.).

Think of area as the number of unit-sized squares you could tile inside the shape without overlap. A 3×4 rectangle fits exactly 12 unit squares, so its area is 12.

Perimeter is a different measurement — the distance around the edge. Area is inside, perimeter is around. A square with side 10 has area 100 and perimeter 40; they're never the same number except at side length 4 (area 16, perimeter 16), which is a fun coincidence, not a theorem.

Different shapes use different formulas because the efficient way to cover space depends on the shape. A circle uses π; rectangles use straight multiplication; triangles halve the product; rings subtract one disc from another.

The full formula reference

Twelve shapes, twelve formulas. Once you spot the patterns (base × height, half × diagonal product, π × radius²) almost everything is variation.

Circle: A = π·r². Double the radius → quadruple the area (r² grows as the square).

Square: A = side². Same logic — doubling the side quadruples area.

Rectangle: A = width × height. Direct product of the two perpendicular sides.

Triangle (base-height): A = ½ · base · height. Any triangle is half a parallelogram with the same base and height.

Triangle (three sides, Heron): A = √(s·(s−a)·(s−b)·(s−c)) with s = (a+b+c)/2. Use this when you know the three sides but not the height.

Parallelogram: A = base · height. No ½ — it's a full rectangle's worth, just skewed.

Trapezoid: A = ½ · (a + b) · h. Average the two parallel sides, then multiply by the perpendicular height.

Ellipse: A = π·a·b. Generalisation of the circle — if a = b, you get πr².

Regular polygon (pentagon, hexagon, octagon, …): A = (n · s²) / (4 · tan(π/n)). n = number of sides, s = side length. Perimeter = n · s.

Annulus (ring): A = π · (R² − r²). Outer disc minus inner disc.

Sector of a circle: A = ½ · r² · θ (radians) or (θ°/360) · π · r². Arc length = r · θ.

Rhombus: A = ½ · d₁ · d₂. d₁ and d₂ are the two perpendicular diagonals.

Kite: A = ½ · d₁ · d₂. Same formula structure as rhombus.

Why each formula works (the visual proofs)

Each formula has a one-sentence geometric story. Once you know the story, the formula sticks without memorisation.

Circle: slice the disc into many thin wedges. Lay them alternately point-up / point-down in a row. The result approximates a rectangle of width r (the wedge "height") and length πr (half the circumference). Area = r · πr = πr².

Triangle: take any triangle and put a copy of it upside-down next to itself. The two triangles fit together as a parallelogram of base × height. So one triangle is exactly half: ½ · base · height.

Parallelogram: slice off the slanted left triangle and slide it to the right side. The figure becomes a rectangle of the same base × height. So the slant doesn't matter.

Trapezoid: imagine averaging the two parallel sides into one of length (a+b)/2 — the trapezoid becomes a rectangle of that average width and the same height.

Ellipse: an ellipse is a circle stretched independently along two perpendicular axes by factors a and b. The unit circle has area π; stretching by a and b scales the area by a·b → π·a·b.

Regular n-gon: split the polygon into n identical triangles by drawing lines from the centre to each vertex. Each triangle has base s (one polygon side) and height equal to the apothem. Add up n of them and the formula falls out.

Annulus: a ring is the outer disc minus the inner disc. πR² − πr² = π(R²−r²). The formula is exactly the visual.

Sector: a sector is a fraction θ/(2π) of the full disc. Multiply by the disc's area πr² and you get ½r²θ.

Rhombus / kite: the two perpendicular diagonals form a bounding rectangle whose width = d₁ and height = d₂. The shape is exactly half of that rectangle.

Heron's formula — triangle area from three sides

When you know all three sides of a triangle but not any height, Heron gives the area directly. Always works; doesn't depend on the triangle being right-angled.

Heron's formula: A = √(s · (s − a) · (s − b) · (s − c)), where s is the semiperimeter (a + b + c) / 2.

Works for any triangle — right, acute, obtuse — as long as the three side lengths satisfy the triangle inequality (each pair sums to more than the third).

Why it works: the formula comes from the law of cosines combined with the standard area formula. The derivation is algebraically ugly, but the result is clean.

Sectors of a circle and arc length

A sector is a "pie slice" of a disc — a wedge bounded by two radii and an arc. The area is a fraction of the full disc, and the arc length is a fraction of the circumference.

Sector area: A = ½ · r² · θ when θ is in radians, or (θ°/360) · π·r² in degrees. The fraction θ/(2π) tells you what proportion of the full disc you've cut out.

Arc length: L = r · θ (radians). It's the same fraction of the circumference 2πr.

Sector perimeter (the boundary you'd walk along): two radii + the arc, so 2r + r·θ.

When θ = 2π (or 360°), the sector is the full disc; the formulas collapse to πr² and 2πr respectively. When θ = π (180°), it's a semicircle.

Regular polygons — pentagon, hexagon, octagon, n-gon

A regular polygon has n equal sides and n equal angles. Knowing just n and one side length s gives you the area and the perimeter directly.

Area: A = (n · s²) / (4 · tan(π/n)). Perimeter: P = n · s. Derivation: split the polygon into n identical triangles from the centre — each has base s and height equal to the apothem (the perpendicular distance from the centre to a side). Sum of n triangles = the polygon.

The apothem is s / (2 · tan(π/n)). Multiply by half the perimeter (n·s/2) and you get the same area formula.

Interior angle of a regular n-gon: ((n − 2) · 180°) / n. Hexagons (n=6) have 120° interior angles, which is why they tile the plane with no gaps (3 × 120° = 360° at every vertex).

Common values worth memorising: triangle (n=3, s²·√3/4), square (n=4, s²), pentagon (n=5, s²·1.720), hexagon (n=6, s²·3√3/2 ≈ 2.598·s²), octagon (n=8, s²·2(1+√2) ≈ 4.828·s²).

Annulus — area of a ring

An annulus is the area between two concentric circles — the donut, the washer, the pipe cross-section.

Area: A = π · (R² − r²), where R is the outer radius and r is the inner radius. It's literally "outer disc minus inner disc": πR² − πr².

The visible boundary length (outer + inner circumferences) is 2π · (R + r). This is the perimeter of the ring, not of the disc.

Useful identity: π(R² − r²) = π(R + r)(R − r). The "(R + r)" half is twice the average circumference; the "(R − r)" half is the ring width. So area ≈ average circumference × ring width — the same intuition as the rectangle area.

Rhombus, kite — the diagonal-product shapes

Both a rhombus and a kite have two perpendicular diagonals. Both use the same area formula: half the product of the diagonals.

Rhombus: a quadrilateral with all four sides equal. The two diagonals are perpendicular AND bisect each other. Area = ½ · d₁ · d₂.

Kite: a quadrilateral with two pairs of adjacent equal sides. The diagonals are perpendicular but only one of them bisects the other. Area = ½ · d₁ · d₂ (same formula).

Why the same formula? In both cases the perpendicular diagonals form a bounding rectangle of width d₁ and height d₂. The shape inside is exactly half that rectangle, so area = ½ · d₁ · d₂.

Rhombus side length: each side is the hypotenuse of a right triangle with legs d₁/2 and d₂/2 (because the diagonals bisect each other). Side = √((d₁/2)² + (d₂/2)²). Perimeter = 4 sides.

Kite side length: depends on where the diagonals cross. Without that crossing point, you can compute the area but not a unique perimeter — different crossings give different kites with the same diagonal product.

The perpendicular-height trap

Triangles, parallelograms, and trapezoids all use "height" in their formulas — but it's always the perpendicular distance, never the slanted side.

The “height” in A = ½·base·height is the perpendicular distance from the base to the opposite vertex (or, for trapezoids, the opposite parallel side). It's the altitude, not the slanted side.

For a right triangle, the two legs are perpendicular to each other, so the leg is the height when the other leg is the base. For acute or obtuse triangles, you may need to drop a perpendicular from the vertex to find the height — or use Heron's formula instead.

For a parallelogram, the slant side is longer than the perpendicular height. Using the slant side will give a number bigger than the true area.

Common area mistakes

Most area errors are unit mistakes, height mix-ups, or π / 2π confusion.

1. Forgetting to square units. Area of a 3cm × 4cm rectangle is 12 cm², not 12 cm. Always report square units.

2. Using slant as height. Perpendicular height only.

3. Mixing π and 2π. Circle area is π·r². Circumference is 2·π·r. Double-check you're using the right one.

4. Heron with invalid sides. If a + b ≤ c (triangle inequality fails), the three sides can't form a triangle and Heron gives NaN or a negative under the root.

5. Trapezoid height as slant side. The height is the perpendicular distance between the two parallel sides, not a slant side.

6. Sector angle in degrees inside the radian formula. A = ½·r²·θ ONLY if θ is in radians. For degrees, use (θ°/360)·πr² — or convert first (θ_rad = θ° · π/180).

7. Annulus with inner ≥ outer. The hole can't be bigger than the disc; the calculator errors out, but if you do it by hand you'll get a negative area.

8. Regular polygon with non-integer or fewer-than-3 sides. n must be a whole number ≥ 3 (you can't have a 2-sided polygon).