Triangle Area Calculator
Calculate a triangle's area using base-height, Heron's formula (three sides), or the two-sides-and-included-angle method. Handy for geometry, surveying, and any problem where you know different combinations of triangle measurements.
About this calculator
A triangle's area can be computed several ways depending on which measurements are known. The base-height method is simplest: A = (base × height) / 2. When all three sides a, b, c are known, Heron's formula applies: first compute the semi-perimeter s = (a + b + c) / 2, then A = √(s(s−a)(s−b)(s−c)). When two sides and the angle between them are known, use the SAS formula: A = (a × b × sin θ) / 2, where θ is the included angle in degrees. All three approaches yield the same result for the same triangle — they simply require different known inputs. Heron's formula is especially useful in surveying where heights are difficult to measure directly.
How to use
Using Heron's formula with sides a = 7, b = 8, c = 9 units: semi-perimeter s = (7 + 8 + 9) / 2 = 12. Then A = √(12 × (12−7) × (12−8) × (12−9)) = √(12 × 5 × 4 × 3) = √720 ≈ 26.83 square units. Alternatively, with base = 7 and height = 7.667: A = (7 × 7.667) / 2 ≈ 26.83 square units — the same answer. Select 'Three Sides', enter 7, 8, and 9, and the calculator confirms the result instantly.
Frequently asked questions
When should you use Heron's formula instead of the base-height formula for triangle area?
Use Heron's formula when you know all three side lengths but do not know the height. This commonly occurs in surveying, navigation, and land measurement where distances between points are easier to measure than perpendicular heights. The base-height formula A = (base × height) / 2 is simpler but requires knowing or calculating the altitude, which isn't always straightforward. Heron's formula is exact and requires only the three side lengths, making it the go-to method for scalene triangles with no obvious altitude.
How do you find the area of a triangle when you know two sides and the included angle?
Use the SAS (side-angle-side) formula: A = (a × b × sin θ) / 2, where a and b are the two known sides and θ is the angle between them. Make sure your calculator or software is set to degrees if θ is given in degrees. For example, with a = 6, b = 10, and θ = 30°: A = (6 × 10 × sin 30°) / 2 = (60 × 0.5) / 2 = 15 square units. This method is powerful in trigonometry and physics problems involving force vectors or angular measurements.
What is the semi-perimeter and why is it used in Heron's formula?
The semi-perimeter s = (a + b + c) / 2 is simply half the total perimeter of the triangle. It was introduced as a notational convenience that makes Heron's formula more compact: A = √(s(s−a)(s−b)(s−c)). Without it, the formula would require writing out (a+b+c)/2 four times, making it unwieldy. The semi-perimeter also appears in other triangle formulas, such as the inradius formula r = A/s. It serves as a normalizing quantity that elegantly captures the 'scale' of the triangle relative to each of its sides.