Base and height
A = b × h ÷ 2
The height is perpendicular to the base.
Area of a Triangle Calculator (Base, 3 Sides or Angle)
Calculate the area of any triangle from the data you have: base × height ÷ 2, three sides with Heron's formula, or two sides and the included angle (½ab sin C). Formulas explained, worked examples (gable wall, plot of land) and units of your choice.
Area
-
Was this tool useful?
How does this tool work?
This calculator gives the area of a triangle whatever information you have: a base and its matching height, the three sides, or two sides and the angle between them. It covers geometry class exercises as well as practical jobs: the area of a gable wall to render or side, a sail, a triangular plot measured with a tape, or a roof slope.
Base and height: A = b × h ÷ 2. The height is the perpendicular distance from the base to the opposite vertex, not the length of a slanted side. Every triangle is half of a parallelogram with the same base and height, hence the division by two.
Three sides: the tool applies Heron's formula. It computes the semi-perimeter s = (a + b + c) ÷ 2, then A = √(s(s − a)(s − b)(s − c)). This method is ideal in the field, where sides are easy to measure but a height rarely is. The three lengths must satisfy the triangle inequality: each side must be shorter than the sum of the other two, otherwise no triangle exists and no result is shown.
Two sides and the included angle (SAS): A = ½ × a × b × sin C, where C is the angle between sides a and b. The angle can be entered in degrees, radians or gradians and must be strictly between 0 and 180°. With a right angle, sin 90° = 1 and you get half the product of the two legs.
The result is expressed in the square of the first length field's unit. All three methods give the same area for the same triangle, which makes them a good way to double-check a calculation done by hand.
How to use this tool
Choose the method
Base + height, Three sides or Two sides + included angle, depending on the data you have.
Enter the measurements
Each length has its own unit; the angle can be entered in degrees, radians or gradians.
Read the area
It appears in square units as soon as the values form a valid triangle.
The three formulas for the area of a triangle
Each formula matches a set of data; they all give the same result for the same triangle.
Heron's formula
A = √(s(s − a)(s − b)(s − c)), with s = (a + b + c) ÷ 2
From the three sides, via the semi-perimeter s.
Two sides and included angle
A = ½ × a × b × sin C
C is the angle formed by sides a and b.
Equilateral triangle
A = (√3 ÷ 4) × a²
The special case where all three sides are equal.
- A
- area, in square units
- b, h
- base and the height relative to that base
- a, b, c
- lengths of the three sides
- s
- semi-perimeter
- C
- angle between sides a and b
Example: b = 10 m and h = 4 m, A = 10 × 4 ÷ 2 = 20 m².
Use cases
Middle and high school
Check an area found with the height, or practice Heron's formula and the trigonometric formula ½ab sin C.
Construction and roofing
Calculate the area of a gable, a hip end or a triangular roof face for rendering, insulation or roofing material.
Surveying and land
Get the area of a triangular plot from its three measured sides, or split an irregular lot into triangles.
Hobbies and fabrication
Estimate the area of a shade sail, a pennant, a piece of fabric or a sheet of plywood.
Examples
Base and height
8 × 5 ÷ 2 = 20 cm².
Input
b = 8 cm, h = 5 cmResult
A = 20 cm²3-4-5 triangle (Heron)
s = 6, A = √(6 × 3 × 2 × 1) = √36 = 6 cm²; it is a right triangle.
Input
a = 3, b = 4, c = 5 cmResult
A = 6 cm²Any three sides
s = 12, A = √(12 × 5 × 4 × 3) = √720 ≈ 26.83 m².
Input
a = 7, b = 8, c = 9 mResult
A ≈ 26.83 m²Two sides and the angle
½ × 6 × 8 × sin 30° = 24 × 0.5 = 12 m².
Input
a = 6 m, b = 8 m, C = 30°Result
A = 12 m²House gable
The triangular area to render or side above the rectangular part of the wall.
Input
base 8 m, height to the ridge 3 mResult
A = 12 m²Triangular plot
Heron: s = 60, A = √(60 × 30 × 20 × 10) = √360,000 = 600 m², or 0.06 hectare (about 0.15 acre).
Input
sides of 30, 40 and 50 mResult
A = 600 m²Tips and best practices
- The height must be perpendicular to the chosen base: in an obtuse triangle, it can fall outside the triangle.
- If you do not know the height, measure the three sides and use the Heron method.
- With two sides and an angle, the angle must be the one between those two sides, otherwise the result is wrong.
- Any polygon can be split into triangles: then add up their areas.
- If no result appears with three sides, check the triangle inequality: the longest side must be shorter than the sum of the other two.
Sources and references
- Weisstein E.W., Triangle Area (opens in a new tab), Wolfram MathWorldPage in English
- Weisstein E.W., Heron's Formula (opens in a new tab), Wolfram MathWorldPage in English
- Programme de mathématiques pour le cycle 4 (annexe 2) (opens in a new tab), Ministère de l’Éducation nationalePage in French
Frequently asked questions
What is the formula for the area of a triangle?
The basic formula is A = base × height ÷ 2. Without the height, use Heron's formula with the three sides, or A = ½ × a × b × sin C with two sides and the included angle.
How do you find the area of a triangle with only the three sides?
Compute the semi-perimeter s = (a + b + c) ÷ 2, then A = √(s(s − a)(s − b)(s − c)). That is Heron's formula; choose the Three sides tab to apply it automatically.
How do you find the area of a right triangle?
The two legs act as base and height: A = a × b ÷ 2. For legs of 6 cm and 8 cm, the area is 24 cm².
How do you find the height of a triangle from its area?
Rearrange the formula: h = 2 × A ÷ b. A triangle with an area of 30 cm² and a 12 cm base has a height of 5 cm.
Why does the tool show nothing with my three sides?
The lengths do not form a triangle: the longest side is greater than or equal to the sum of the other two. For example, 2, 3 and 6 cm cannot close up.
What is the area of an equilateral triangle?
For a side a, A = (√3 ÷ 4) × a² ≈ 0.433 × a². An equilateral triangle with 10 cm sides covers about 43.30 cm²; the Three sides method gives the same result.
Related tools
Keep going with these tools
Tools similar or complementary to this one.
Similar tools
Add up the three sides of a triangle, or find the third side with the law of cosines from two sides and the included angle.
Mathematics & calculationUse this tool
Find the hypotenuse or a missing leg of a right triangle, or check whether a triangle is right-angled with the converse, with a full worked solution.
Mathematics & calculationUse this tool
Choose a 2D shape or 3D solid, enter the dimensions you know and get the area, perimeter, diagonals, angles, surface area and volume.
Mathematics & calculationUse this tool
Read next
GuideConcepts and technologiesBeginner
Geometry calculations: formulas, methods and examples for 2D and 3D shapes
Learn how to calculate areas, perimeters, diagonals, angles, surfaces and volumes, and open the Bethemesh geometry calculator directly on the shape you need.
10 August 20266 min