Triangle Area Calculator

How to Use the Triangle Area Calculator

This triangle area calculator supports four methods — Base & Height, Heron's Formula (three side lengths), SAS (two sides and the included angle), and Coordinates (three vertex points) — so you can find the area no matter what information you have. All methods return the area; when all three sides are known, the calculator also shows the perimeter and triangle type (equilateral, isosceles, or scalene; acute, right, or obtuse).

For geometry problems involving the Pythagorean theorem, our Pythagorean theorem calculator solves for any side of a right triangle given the other two.

The Four Triangle Area Formulas

Depending on what information you have, different formulas apply:

• Base & Height: A = ½ × b × h — the simplest formula. Requires the perpendicular height (not a slant side).
• Heron's Formula: s = (a+b+c)/2; A = √(s(s−a)(s−b)(s−c)) — uses all three side lengths. No angle or height needed.
• SAS: A = ½ × a × b × sin(C) — uses two sides and the angle between them. Requires a calculator for the sine value.
• Coordinates: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| — the shoelace formula. Works directly from vertex coordinates.

All four formulas give the same result for the same triangle — they are different routes to the same answer.

Heron's Formula Step by Step

Heron's formula is the most commonly used method when only the three side lengths are known. Here is how to apply it:

1. Verify the triangle inequality: the sum of any two sides must be greater than the third side.
2. Compute the semi-perimeter: s = (a + b + c) / 2
3. Compute the product: s(s−a)(s−b)(s−c)
4. Take the square root: A = √(s(s−a)(s−b)(s−c))

Example: sides a = 5, b = 6, c = 7. Semi-perimeter s = 9. Product = 9 × 4 × 3 × 2 = 216. Area = √216 ≈ 14.70 square units.

Heron's formula is named after Hero of Alexandria (c. 10–70 AD), but evidence suggests it may have been known to Archimedes centuries earlier. For distance problems involving points in a coordinate system, our distance calculator can find the side lengths you need.

Triangle Classification

Triangles are classified in two ways — by their sides and by their angles:

• By sides:
  • Equilateral — all three sides equal; all angles are 60°
  • Isosceles — two sides equal; the two base angles are equal
  • Scalene — all sides different; all angles different
• By angles:
  • Acute — all angles less than 90°
  • Right — one angle exactly 90° (satisfies the Pythagorean theorem)
  • Obtuse — one angle greater than 90°

Every triangle fits one category from each group — for example, a right isosceles triangle has two equal sides and a 90° angle. The calculator determines the type automatically using the law of cosines to test the largest angle.

Real-World Applications

Triangle area calculations appear in many practical fields:

• Construction — calculating the area of a triangular gable end for siding or insulation; estimating the area of a triangular plot of land
• Surveying — using coordinates from GPS or total station measurements to compute land parcel areas
• Engineering — computing cross-sectional areas of triangular structural members
• Computer graphics — all 3D surfaces are rendered as meshes of triangles; triangle area is fundamental to shading and physics calculations
• Navigation — triangular plotting of position fixes in maritime navigation

The Triangle Inequality

For any triangle to exist, the three side lengths must satisfy the triangle inequality: the sum of any two sides must be strictly greater than the third side. This must hold for all three combinations:

• a + b > c
• a + c > b
• b + c > a

If any one of these fails, the sides cannot form a closed triangle. For example, sides 2, 3, 10 do not work because 2 + 3 = 5 < 10. Geometrically, the shorter two sides are too short to reach each other after being connected to the ends of the longest side. When using Heron's formula, a violated triangle inequality causes the product s(s−a)(s−b)(s−c) to become negative, making the square root undefined — which is how the formula signals an invalid triangle.