Area of Triangle Calculator

How the Area of a Triangle Works

The area of a triangle is the amount of two-dimensional space enclosed by its three sides. Triangles are the simplest polygon and the fundamental building block of all complex shapes in geometry — every polygon can be decomposed into triangles. According to Wolfram MathWorld, over 400 triangle-related theorems and formulas exist, but for practical area calculations, two formulas cover nearly all situations.

This calculator supports two methods: the base-height formula (when you know a base and its perpendicular height) and Heron's formula (when you know all three sides). Surveyors, architects, and engineers use these formulas daily for tasks ranging from land measurement to structural design. If your shape is a rectangle, its area is simply double the triangle formed by its diagonal.

The Triangle Area Formulas

Method 1 — Base and Height:

A = ½ × base × height

The height must be perpendicular to the chosen base. Any side can be the base as long as you use the corresponding perpendicular height.

Method 2 — Heron's Formula (three sides):

A = √[s(s - a)(s - b)(s - c)], where s = (a + b + c) / 2

First documented by Heron of Alexandria around 60 CE, this formula uses only the three side lengths. The variable s is the semi-perimeter (half the perimeter).

Worked example (base-height): A triangular garden bed has a base of 8 ft and a height of 5 ft. Area = ½ × 8 × 5 = 20 square feet.

Worked example (Heron's): A land plot has sides of 30 m, 40 m, and 50 m. Semi-perimeter s = 60 m. Area = √(60 × 30 × 20 × 10) = √360,000 = 600 square meters. (Note: this is a right triangle since 30² + 40² = 50².)

Key Terms You Should Know

• Base — any side of the triangle chosen as the reference for the height measurement.
• Height (altitude) — the perpendicular distance from the base to the opposite vertex (corner).
• Semi-perimeter (s) — half the sum of all three sides: s = (a + b + c) / 2. Used in Heron's formula.
• Triangle inequality — the rule that the sum of any two sides must exceed the third. If a + b ≤ c, the sides cannot form a triangle.
• Hypotenuse — in a right triangle, the side opposite the 90° angle. Always the longest side.

Triangle Types and Their Area Formulas

Practical Examples

Example 1 — Roof pitch calculation: A roof rises 6 ft over a horizontal run of 12 ft. The triangular cross-section has area = ½ × 12 × 6 = 36 sq ft. For a 30-ft wide house, each gable end has the same cross-section. The roof slope (pitch) is 6/12 or a 6:12 pitch, one of the most common roof angles in the United States.

Example 2 — Irregular land survey: A triangular lot has sides measured at 45 m, 60 m, and 75 m. Using Heron's formula: s = 90, A = √(90 × 45 × 30 × 15) = √1,822,500 = 1,350 sq m (0.135 hectares). According to FAO data, the average farm plot in many developing countries is 1-2 hectares, making this about one-tenth of a typical plot.

Example 3 — Sail area for a sailboat: A triangular mainsail has a luff (height) of 35 ft and a foot (base) of 12 ft. Sail area = ½ × 12 × 35 = 210 sq ft. Racing rules often limit total sail area, so accurate calculation matters. Use the trapezoid calculator if the sail is cut with a roach (curved leech).

Tips for Accurate Triangle Area Calculations

• Height is always perpendicular: The most common mistake is using a slanted side instead of the perpendicular height. For obtuse triangles, the height may fall outside the triangle when extended from certain bases.
• Use Heron's for surveying: When measuring land, it is easier to measure three sides with a tape than to determine a perpendicular height. Heron's formula is more practical in field conditions.
• Check the triangle inequality first: Before calculating, verify that a + b > c for all side combinations. If any pair sums to less than or equal to the third, the triangle is impossible.
• Right triangles have a shortcut: If one angle is 90°, the two shorter sides are the base and height. No need for Heron's formula — just use ½ × leg₁ × leg₂. Use the hypotenuse calculator if you only know two sides.
• Double-check with both methods: If you know all three sides AND the height, calculate area using both formulas. If the results match, your measurements are consistent.