Calculate the area of a triangle using base and height, three sides (Heron's formula), or two sides and an angle.
Last reviewed: May 2026
Calculating the area of a triangle is one of the most fundamental operations in geometry, with applications spanning architecture, engineering, surveying, computer graphics, and physics. Unlike rectangles where area is simply length times width, triangles require different formulas depending on what measurements are known.1
The most commonly taught formula is A = ½ × base × height, where the base is any side of the triangle and the height is the perpendicular distance from that base to the opposite vertex. This formula works for every triangle — equilateral, isosceles, scalene, right, acute, and obtuse — but requires knowing the perpendicular height, which is not always directly measurable.
When height is unknown or difficult to measure, alternative formulas become essential. Heron's formula uses only the three side lengths. The trigonometric formula uses two sides and their included angle. The coordinate formula uses vertex positions on a Cartesian plane. Each approach has specific advantages depending on the available measurements and the context of the problem.2
The standard formula A = ½ × base × height is the simplest and most intuitive. The base can be any side of the triangle — whichever side you choose as the base determines which perpendicular distance serves as the height. Importantly, the height must be perpendicular to the chosen base, not simply the length of another side.
For right triangles, the two legs serve as base and height (they are already perpendicular), making calculation trivial. For acute triangles, the perpendicular height falls inside the triangle. For obtuse triangles, the height from the obtuse-angle vertex falls outside the triangle, requiring extension of the base line — a common source of confusion for students. This calculator handles all three cases automatically.1
When you know all three side lengths but not the height, Heron's formula provides an elegant solution. First compute the semi-perimeter s = (a + b + c) / 2, then apply A = √[s(s − a)(s − b)(s − c)]. Named after Hero of Alexandria (c. 10–70 AD), this formula has been used for over 2,000 years and remains one of the most practical tools in surveying and land measurement.
Heron's formula is particularly useful in real-world applications where you can measure distances between three points but cannot easily measure perpendicular heights — for example, determining the area of an irregularly shaped plot of land by dividing it into triangles and measuring the side lengths with a tape measure or GPS device.3
The trigonometric formula A = ½ × a × b × sin(C) calculates area using two side lengths and the angle between them. This is often the most practical formula in navigation, physics, and engineering where angles are easily measured with protractors, theodolites, or accelerometers.
Note that C must be the included angle — the angle formed between sides a and b at their shared vertex. Using a non-included angle produces an incorrect result. When the included angle is 90°, sin(90°) = 1, and the formula simplifies to A = ½ × a × b, which matches the base-height formula with the two legs as base and height.2
| Known Values | Formula | Best For |
|---|---|---|
| Base and height | A = ½ × b × h | Right triangles, known perpendicular height |
| Three sides (a, b, c) | A = √[s(s−a)(s−b)(s−c)] | Surveying, land measurement, GPS coordinates |
| Two sides + included angle | A = ½ × a × b × sin(C) | Navigation, physics, engineering |
| Coordinates (x₁,y₁), (x₂,y₂), (x₃,y₃) | A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| | Computer graphics, mapping, CAD |
| Equilateral (side s) | A = (s² × √3) / 4 | Regular polygons, structural engineering |
Right triangle: The two legs are perpendicular by definition, so A = ½ × leg₁ × leg₂. No need to calculate a separate height. The hypotenuse is never used as the "height" in this formula.
Equilateral triangle: All sides equal (s), so the height = (s × √3) / 2 and the area = (s² × √3) / 4 ≈ 0.433 × s². An equilateral triangle has the largest possible area for any triangle with a given perimeter — a fact used in structural optimization.
Isosceles triangle: With two equal sides of length a and base b, the height = √(a² − b²/4), and the area = (b/4) × √(4a² − b²). This formula is derived by splitting the isosceles triangle into two right triangles at its axis of symmetry.
Obtuse triangle: The standard ½bh formula still applies, but the height from the vertex at the obtuse angle falls outside the triangle. Use Heron's formula or the trigonometric formula to avoid confusion with external heights.3
When triangle vertices are given as coordinates on a plane, the shoelace formula (also called the surveyor's formula) computes area directly: A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. The absolute value ensures a positive area regardless of vertex ordering direction.
This formula is fundamental in computer graphics for calculating the area of triangulated surfaces, in GIS for measuring land parcels from GPS coordinates, and in computational geometry for determining point-in-triangle containment. It extends to any polygon — for n vertices, the shoelace formula sums cross products of consecutive vertex pairs.2
| Application | Preferred Formula | Why |
|---|---|---|
| Land surveying | Heron's formula | Side lengths measured directly with surveying equipment |
| Architecture/Construction | ½ × base × height | Right angles common; heights measured from blueprints |
| Navigation | ½ab sin(C) | Bearings give angles; distances from charts or radar |
| Computer graphics | Shoelace formula | Vertices stored as coordinates; batch computation |
| Physics (force vectors) | ½|A × B| (cross product) | Vectors define two sides; magnitude gives area |
Area is always expressed in squared units — if the side lengths are in meters, the area is in square meters (m²). If measurements are in feet, the area is in square feet (ft²). Mixing units within a single calculation is one of the most common sources of error. Before applying any formula, convert all measurements to the same unit system. If one side is given in inches and another in feet, convert both to the same unit before calculating.
For unit conversions involving area: 1 ft² = 144 in², 1 m² = 10.764 ft², 1 acre = 43,560 ft², 1 hectare = 10,000 m². When converting a triangle's area from one unit to another, convert the linear measurements first, then compute the area — or compute the area in the original units and multiply by the square of the conversion factor. For example, to convert from ft² to m², multiply by (0.3048)² = 0.0929.
When triangle vertices are given as 3D coordinates (x, y, z) rather than 2D, the shoelace formula does not directly apply. Instead, use the cross product method: given vertices A, B, C, form vectors AB = B − A and AC = C − A, compute their cross product AB × AC, and take half its magnitude. The area equals ½|AB × AC|. This method is essential in computer graphics, 3D modeling, and finite element analysis where triangulated surfaces exist in three-dimensional space.
The cross product method generalizes naturally to computing surface areas of complex 3D objects. A mesh of thousands of triangles approximates any curved surface — the total surface area equals the sum of all individual triangle areas computed via cross products. This is the standard technique in CAD software, video game engines, and scientific visualization for computing surface properties of arbitrary geometric models.2
Using slant height instead of perpendicular height: The height in the ½bh formula must be perpendicular to the base, not the length of a slanted side. In an obtuse triangle, the perpendicular height from the longest side often extends outside the triangle — if this is confusing, use Heron's formula instead.
Angle in wrong units: The sin(C) formula requires the angle in the same units your calculator expects. Most scientific calculators default to degrees, but programming languages use radians. Converting incorrectly produces wildly wrong results — 30° and 30 radians give completely different sine values.
Forgetting absolute value in coordinate formula: The shoelace formula can produce negative results depending on vertex ordering. Always take the absolute value of the final result. The sign indicates orientation (clockwise vs. counterclockwise), not a negative area.
→ Choose the formula that matches your known values. If you have three sides, use Heron's formula directly — do not try to calculate the height first, as that introduces unnecessary complexity and potential rounding error.
→ Verify with a second method. For critical applications (construction, surveying), calculate the area using two different formulas and confirm they agree. Discrepancies indicate a measurement or calculation error.
→ Remember the triangle inequality. The sum of any two sides must exceed the third side. If a + b ≤ c, no triangle exists and Heron's formula produces an imaginary number (the expression under the square root becomes negative).
→ Use Heron's formula for irregular land. Divide any polygon into triangles by connecting vertices, measure all side lengths, and sum the triangle areas. This method works for any shape and requires only distance measurements.
See also: Triangle Calculator · Area Calculator · Pythagorean Theorem · Area of Circle · Perimeter Calculator