Last updated: 2026-05-21

Triangle area

A triangle is a polygon with three sides and three angles, the simplest closed shape in two-dimensional geometry. Its area is the amount of flat surface it encloses, measured in square units. The area can be computed by four methods, depending on which measurements are known: base and height, Heron's formula (three sides), SAS (two sides and the included angle), and ASA (one side and two adjacent angles).

Base and Height Method

The most direct formula is:

$A = \frac{1}{2} \times b \times h$

where $b$ is the length of any chosen side (the base) and $h$ is the perpendicular height — the distance measured at a right angle from that base to the opposite vertex.

This formula works for every triangle type:

• Acute triangles: the foot of the altitude lies inside the base.
• Right triangles: one leg is the base and the other is the height.
• Obtuse triangles: the altitude foot falls outside the base, but the formula still applies.

The factor of one half

A triangle is exactly half of a parallelogram with the same base and height. Duplicating a triangle, rotating the copy 180°, and attaching it along one side produces a parallelogram with area $b \times h$. The triangle therefore has area $\frac{b \times h}{2}$.

Heron's Formula (Three Sides)

When only the three side lengths are known — no height, no angles — Heron's formula gives the area directly:

$s = \frac{a + b + c}{2}$

$A = \sqrt{s(s-a)(s-b)(s-c)}$

Here $s$ is the semi-perimeter (half the perimeter). The formula dates back to Heron of Alexandria (c. 60 CE), though there is evidence it was known even earlier.

Worked Example: 3-4-5 Right Triangle

$s = \frac{3 + 4 + 5}{2} = 6$

$A = \sqrt{6 \times (6-3) \times (6-4) \times (6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6$

Cross-check with the base-height method: legs 3 and 4 are perpendicular, so $A = \frac{1}{2} \times 3 \times 4 = 6$. Both methods agree.

SAS: Two Sides and the Included Angle

When two sides $p$ and $q$ and the angle $C$ between them are known, the area is:

$A = \frac{1}{2} \, p \, q \, \sin C$

This follows from the base-height formula: taking $p$ as the base, the height equals $q \sin C$ (the component of $q$ perpendicular to $p$).

Worked Example

With $p = 5$, $q = 7$, and $C = 60°$:

$A = \frac{1}{2} \times 5 \times 7 \times \sin 60° = \frac{35\sqrt{3}}{4} \approx 15.16$

Note: when $C = 90°$, $\sin 90° = 1$ and the formula reduces to the familiar right-triangle area $\frac{1}{2} p q$.

ASA: One Side and Two Adjacent Angles

When side $c$ and the angles $A$ and $B$ at its two endpoints are known, the area is:

$A = \frac{c^2 \sin A \sin B}{2 \sin(A + B)}$

This is derived by applying the Law of Sines to express the other two sides in terms of $c$, then substituting into the SAS formula. The third angle is $C = 180° - A - B$.

Worked Example

With $c = 8$, $A = 50°$, $B = 60°$ (so $C = 70°$):

$A = \frac{64 \times \sin 50° \times \sin 60°}{2 \times \sin 110°} = \frac{64 \times 0.766 \times 0.866}{2 \times 0.940} \approx 22.6$

Triangle Inequality

Not every set of three positive numbers defines a real triangle. Three lengths $a$, $b$, $c$ form a valid triangle if and only if:

$a + b > c, \quad b + c > a, \quad a + c > b$

If any of these three conditions fails, no triangle can be constructed. In Heron's formula, a violation causes the expression inside the square root to become negative — an immediate indicator that the input is geometrically impossible.

Real-World Applications

Architecture and construction. Triangular trusses are the backbone of roofs, bridges, and cranes because a triangle is the only polygon that is rigid under load — it cannot deform without changing a side length.

Surveying. Land surveyors measure the three sides of a plot using tape or laser and apply Heron's formula to calculate area without needing to establish a perpendicular.

Computer graphics. Every 3D surface is approximated by a mesh of triangles. The GPU computes triangle areas to determine lighting, shadows, and collision boundaries.

Navigation. Triangulation uses three known reference points to pinpoint an unknown location — the intersection of arcs whose radii are the measured distances to each reference.

If the triangle is a right triangle, the Pythagorean Theorem Calculator can find the missing side before computing the area. For circles, see the Circle Area & Circumference Calculator and Regular Polygon Calculator calculators for related polygon and curved-shape area calculations.