How to use the triangle area calculator
- Under What do you know?, pick Base and height, Three sides or Two sides and angle.
- Optionally pick a Unit of length. With No unit you get plain numbers and the words "square units".
- Type your numbers: Base (b) and Height (h); or Side a, Side b and Side c; or Side a, Side b and Angle C between them in degrees.
- Read the Area and the working. Three sides, and two sides with an angle, also show the perimeter, angles and triangle type.
The formulas
- Base and height: A = ½ × b × h.
- Three sides (Heron's formula): s = (a + b + c) ÷ 2, then A = √(s(s − a)(s − b)(s − c)), evaluated in Kahan's stable order so thin triangles keep their accuracy.
- Two sides and the angle between them: A = ½ × a × b × sin(C), with the third side from the law of cosines.
Worked example
The calculator opens on Base and height with b = 8 and h = 5, so A = ½ × 8 × 5 = 20 square units. Three sides starts at 3, 4 and 5: s = 6 and A = √(6 × 3 × 2 × 1) = √36 = 6, a right scalene triangle with angles 36.8699°, 53.1301° and 90°. Two sides and angle starts at 5, 7 and 60°: A = ½ × 5 × 7 × 0.866025 = 15.1554.
Limits and edge cases
- Every number must be greater than 0 and at most 1,000,000,000,000. Thousands commas are fine; use a period for decimals.
- Three sides must satisfy the triangle inequality. If the two shorter sides add up to the longest (within about one part in a trillion), the triangle is flat and the page says so instead of showing 0.
- The angle must be between 0 and 180 degrees and sit between the two sides you typed. For radians, convert first with the angle converter.
- The unit menu only labels your numbers; it does not convert them.
- For a right triangle the legs are the base and height. Find a missing leg with the Pythagorean theorem calculator, or compare with the rectangle area calculator.
Triangle area FAQ
How do I find the area of a triangle with only three sides?
Use Heron's formula: halve the perimeter to get s, then A = √(s(s − a)(s − b)(s − c)). Sides of 7, 8 and 9 give s = 12 and A = √(12 × 5 × 4 × 3) = √720 = 26.8328.
How do I find the area of a triangle with two sides and an angle?
Multiply the two sides, multiply by the sine of the angle between them, and halve it: A = ½ × a × b × sin(C). Sides of 5 and 7 with a 60° angle give 15.1554. The angle must sit between those two sides.
What is the area of a right triangle?
Half the product of the two legs, since one leg is the base and the other is the height. Legs of 3 and 4 give ½ × 3 × 4 = 6. The hypotenuse is not used.
Does the height have to be at a right angle to the base?
Yes. The height is the perpendicular distance from the base to the opposite corner, not the length of a slanted side. In an obtuse triangle it falls outside the triangle, but A = ½ × b × h still works.
Why does the calculator say my sides cannot make a triangle?
The longest side must be shorter than the other two added together. Sides of 3, 4 and 10 fail because 3 + 4 = 7 is less than 10, and sides of 2, 3 and 5 only lie flat because 2 + 3 = 5.
Sources: Wolfram MathWorld, Triangle Area; Wolfram MathWorld, Heron's Formula; W. Kahan, Miscalculating Area and Angles of a Needle-like Triangle.