Scalene Triangle Area Calculator
Calculate the area of a scalene triangle using whichever measurements you already have. Choose from four methods: base and height, all three side lengths (Heron's formula), two sides and the included angle, or two angles and the side between them. The perimeter, all three altitudes, and the inradius update alongside the area, and the Show Your Work panel walks through every step with your actual numbers.
What is a scalene triangle?
A scalene triangle has three sides of different lengths and three angles of different measures. Unlike equilateral triangles (all sides equal) or isosceles triangles (two sides equal), no two parts of a scalene triangle are the same. This makes it the general case: the formulas that work for a scalene triangle work for any triangle, so understanding how to calculate its area covers every scenario you will encounter in geometry, trigonometry, land surveying, or engineering.
How to calculate the area of a scalene triangle
There are four standard methods depending on which measurements you have. If you know the base and perpendicular height, multiply them together and divide by two: Area = 0.5 x base x height. If you have all three side lengths, Heron's formula works: compute the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s(s - a)(s - b)(s - c)). If you know two sides and the angle squeezed between them, use the SAS formula: Area = 0.5 x a x b x sin(C). Finally, if you know two angles and the side between them, the ASA formula gives: Area = c^2 x sin(A) x sin(B) / (2 x sin(A + B)). This calculator supports all four methods and shows the full working for each.
Perimeter, altitude, and inradius
When you use the SSS, SAS, or ASA method, the calculator derives all three side lengths and also computes the perimeter (a + b + c), the altitude to side a (the perpendicular height from vertex A to side a, equal to 2 x Area / a), and the inradius (the radius of the largest circle that fits inside the triangle, equal to Area / s where s is the semi-perimeter). The base-height method gives only the area and that one altitude because the other side lengths are not determined without more information.
Practical applications
Scalene triangle area calculations appear in land surveying when plot boundaries form irregular triangles, in carpentry and roofing when cuts are not square, in navigation when computing the area covered by triangulated routes, and in physics when resolving vector cross-products into magnitudes. The SAS formula in particular underpins the cross-product area formula used throughout engineering. The ASA method is used in triangulation surveys where two observation angles and the baseline between the observers are known.
Scalene triangle formulas by known values
| Known values | Formula | Method name |
|---|---|---|
| Base b, Height h | Area = 0.5 x b x h | Base-Height |
| Three sides a, b, c | Area = sqrt(s(s-a)(s-b)(s-c)), s = (a+b+c)/2 | Heron's formula |
| Two sides a, b; included angle C | Area = 0.5 x a x b x sin(C) | SAS |
| Two angles A, B; included side c | Area = c^2 x sin(A) x sin(B) / (2 sin(A+B)) | ASA |
Choose the row that matches the measurements you have. All formulas give the same area for a valid triangle.
Frequently asked questions
What makes a triangle scalene?
A scalene triangle has three sides all of different lengths, which means its three angles are also all different. The name comes from the Greek word for "unequal." Any triangle that is not equilateral (all sides equal) or isosceles (exactly two sides equal) is scalene.
Which formula should I use if I only have the three side lengths?
Use Heron's formula. First calculate the semi-perimeter s = (a + b + c) / 2, then Area = sqrt(s(s - a)(s - b)(s - c)). This formula works for any triangle and requires no angle measurements. Select the "Three sides" method in the calculator above and it will work through every step for you.
Can I calculate the area if I know two sides and an angle?
Yes, as long as the angle is the one between those two sides (the included angle). The SAS formula is Area = 0.5 x a x b x sin(C), where C is the angle between sides a and b. Select the SAS method, enter the two sides and the included angle in degrees, and the calculator returns the area plus the perimeter and inradius.
What is the inradius and how is it calculated?
The inradius is the radius of the largest circle that fits entirely inside the triangle, touching all three sides. It equals the area divided by the semi-perimeter: r = Area / s. For a triangle with area 20 m^2 and semi-perimeter 12 m, the inradius is 20 / 12 = 1.667 m. The calculator shows this value whenever all three sides are known.
How does the altitude differ from the height I enter in the base-height method?
In the base-height method, the height you enter is exactly the perpendicular distance from the chosen base to the opposite vertex. That IS the altitude for that base. A triangle has three altitudes, one for each side acting as the base. When all three sides are known, the calculator shows the altitude to side a: h_a = 2 x Area / a. The other two altitudes can be found similarly using h_b = 2 x Area / b and h_c = 2 x Area / c.
Does it matter which side I call side a, b, or c?
In the SSS method, labeling the sides a, b, and c is arbitrary because Heron's formula treats them symmetrically and gives the same area regardless of the order you enter them. In the SAS method, sides a and b must be the two sides that border the angle C you enter. In the ASA method, side c must be the side that lies between angles A and B.
What units does the calculator use?
The calculator is unit-agnostic for the math: if you enter sides in metres, the area comes out in square metres. If you enter sides in feet, the area is in square feet. Select metric or imperial at the top to keep the unit labels correct in the results and the worked steps. The angle inputs are always in degrees.