4 Sides Area Calculator

It is natural to expect an area from four side lengths and nothing else. That cannot be done, and it is worth seeing why before you trust a number. Build a square from four rods of 10 ft, hinged at the corners. It covers 100 ft². Now push on one corner until it leans into a diamond with a 30° angle. The rods have not changed, yet the area has fallen to 50 ft². Keep pushing and it approaches zero.

So you need one more measurement to pin the shape down. The three options above are listed from most to least useful.

Formula

A = √( s₁(s₁ − a)(s₁ − b)(s₁ − p) ) + √( s₂(s₂ − c)(s₂ − d)(s₂ − p) )

a, b, c, d

the four sides, in order around the shape

p

the diagonal, which separates sides a and b from sides c and d

s₁

half of a + b + p

s₂

half of c + d + p

4 sides and a diagonal (exact)

A diagonal cuts the shape into two triangles, and a triangle cannot flex once its sides are known. Heron's formula gives each triangle's area, and adding them gives the total. This method is usually the most practical on real ground, because it takes only a tape.

Sides of 80, 60, 70 and 50 with a diagonal of 90 give triangles of about 2,352.5 and 1,741.2, for a total of about 4,093.7. Keep the sides in order around the shape: a and b on one side of the diagonal, c and d on the other. If a corner points inward, measure the diagonal that stays inside the shape.

4 sides and 2 angles (exact)

If you can measure angles, take a pair of opposite corners: A between sides d and a, and C between sides b and c. Bretschneider's formula then gives the area. Sides of 80, 30, 80 and 50 with both angles at 60° give s = 120 and √(40 × 90 × 40 × 70 − 80 × 30 × 80 × 50 × cos² 60°) = √(10,080,000 − 2,400,000) ≈ 2,771.28.

The sides and one angle already fix the shape, so the second angle acts as a cross-check. Each angle implies a length for the diagonal between the other two corners. If those lengths differ by more than about 2%, no shape can have all your measurements, and the calculator says so rather than printing a number that belongs to nothing.

• A = √( (s − a)(s − b)(s − c)(s − d) − a × b × c × d × cos²(½(A + C)) )
• s = (a + b + c + d) ÷ 2

4 sides only (upper bound)

With only the sides, the one thing that can be said for certain is the most they could enclose. That maximum occurs when all four corners sit on a single circle, which makes opposite angles add up to 180°. Brahmagupta's formula gives it, and the result is labelled Largest possible area because your shape can be smaller, and often is.

A rectangle is the friendly exception. Its corners do lie on a circle, so sides of 40, 30, 40 and 30 give 1,200, which is also 40 × 30. For anything less regular, treat the figure as a ceiling. Those 80, 30, 80 and 50 sides reach about 3,174.90 at most, while the real shape above covers about 2,771.28.

• A ≤ √( (s − a)(s − b)(s − c)(s − d) )

Same sides, different areas

The table holds the sides at 80, 60, 70 and 50 and changes only the diagonal. The spread shows how much is at stake when a calculator quietly assumes one shape for you.

Which sides can close at all

Whichever method you use, the longest side must be shorter than the other three put together. Sides of 100, 10, 10 and 10 cannot meet, because the three short ones fall 70 short of spanning the long one, and the calculator reports that. With a diagonal, the same rule applies to each triangle: every one of its three lengths must be shorter than the other two combined.

Frequently asked questions