Area Guide — Formulas for All 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.

The full formula reference

Seven shapes, seven formulas. Learn the base-height pattern and most of these become the same idea.

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. Generalization of the circle — if a = b, you get πr².

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.

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.

Five 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. Assuming the trapezoid height is the slant side. The height is the perpendicular distance between the two parallel sides, not a slant side.