Pentagon Area Calculator

Five sides do not fix the area of an irregular pentagon. Like a loop of hinged rods, a 5-sided outline can flex into many shapes while every side keeps its length, and each shape encloses a different area. Any calculator that accepts only five side lengths is assuming a shape you did not give it. This one asks for two diagonals as well, because they make the shape rigid and the answer exact.

Formula

A = T₁ + T₂ + T₃

T₁

the triangle made by sides a and b with diagonal p

T₂

the triangle made by diagonals p and q with side c

T₃

the triangle made by sides d and e with diagonal q

Each triangle's area comes from Heron's formula. A regular pentagon has a shorter route: A = ¼ × √(5(5 + 2√5)) × s².

Irregular pentagon: 5 sides and 2 diagonals

Choose one corner and label it A. Going around the shape from A, the sides are a, b, c, d and e. From A, two diagonals reach the two corners that are not next to it: p ends where b meets c, and q ends where c meets d. That divides the pentagon into three triangles, and a triangle cannot flex.

With sides of 30, 25, 20, 30 and 25 and diagonals of 45 and 40, the three triangles come to about 353.55, 399.95 and 374.53, for a total of about 1,128.04. Both diagonals must start from the same corner. If the calculator reports that some lengths cannot form a triangle, a diagonal has usually been measured from the wrong corner or the sides have been entered out of order.

Why 5 sides are not enough

Compare two shapes with every side equal to 10. The regular pentagon covers about 172.05. A house outline, made of a 10 by 10 square topped with an equilateral triangle, covers 100 + 43.30 = 143.30. Its diagonals from a bottom corner are about 14.14 and 19.32, and entering those gives the same 143.30. Same five sides, about 17% less area. Any single formula based on sides alone would be wrong for at least one of these.

Regular pentagon: the side alone

When all five sides and all five angles are equal, a single length settles everything. Multiply the side squared by ¼√(5(5 + 2√5)), which is about 1.7205. A regular pentagon with sides of 10 has an area of about 172.05, a perimeter of 50 and diagonals of about 16.18, since the diagonal is the side times the golden ratio, about 1.618.

Equal sides alone do not make a pentagon regular, as the house outline shows. The angles must match too, each one 108°. If you are unsure, measure a diagonal or two: on a regular pentagon they are all the same length, and if they are not, use the irregular method.

• A = ¼ × √(5(5 + 2√5)) × s²
• A ≈ 1.7205 × s²

Using it on real shapes

Five-sided outlines are common in lots with one clipped corner, rooms with a bay wall, and gable-end walls, which are the house outline at a larger scale. A tape is all you need, since no angles are involved. Pick a corner from which both diagonals run inside the shape, and if one has to pass outside because a corner points inward, try another corner. For a shape with six or more corners, or if a diagonal is blocked by a building, listing corner coordinates in the irregular shape calculator is the better route.

Frequently asked questions