A triangle covers exactly half of the parallelogram that two copies of it would form, which is why every area formula contains a ½ somewhere. What changes is the starting information. This calculator offers four methods (base and height, three sides, two sides with the angle between them, and corner coordinates) so you can work from what you actually measured instead of hunting for a height you don’t have.
How to use the triangle area calculator
- In What do you know?, choose Base and height, Three sides (Heron’s formula), Two sides and the angle between them (SAS) or Coordinates of the three corners.
- Enter the values shown. For coordinates, type x₁, y₁, x₂, y₂, x₃ and y₃ for the three corners in any order.
- Pick the Units, plus the Angle unit for SAS. Coordinate answers come out in square units of whatever grid you used.
- Read the area at the top. The side-based methods also list the angles, perimeter, semiperimeter and height to side c (SAS adds the third side); the coordinate method adds side lengths, angles and whether the corners run clockwise or counterclockwise.
Triangle area formulas
Base and height:
Three sides (Heron’s formula), with s as the semiperimeter:
Two sides and the included angle:
Corner coordinates (shoelace formula):
Which method fits your measurements
| Method | Best when | Watch out for |
|---|---|---|
| Base and height | You can measure straight across from a side to the opposite corner | The height must be at a right angle to the base |
| Heron’s formula | You can tape all three sides, as with a garden bed or a lot | The two shorter sides must add up to more than the longest |
| SAS | You know two sides and the angle where they meet | The angle must be the one between those two sides |
| Coordinates | Corners come from a site plan, a survey or a graph | Points on one line give zero area |
Worked example: mulch for a triangular bed
A corner flower bed has sides of 8 ft, 11 ft and 13 ft. Choose Three sides and enter them.
- Semiperimeter: s = (8 + 11 + 13) ÷ 2 = 16 ft
- Area = √(16 × 8 × 5 × 3) = √1,920 ≈ 43.82 ft²
- Cross-check: the calculator gives a height of 6.741 ft to the 13 ft side, and ½ × 13 × 6.741 ≈ 43.82 ft².
- Mulch 3 in (0.25 ft) deep: 43.82 × 0.25 ≈ 10.95 ft³, or about 0.41 yd³.
Coordinates work the same way. Corners plotted on a site plan at (2, 1), (9, 3) and (5, 8), in meters, give ½ × |2(3 − 8) + 9(8 − 1) + 5(1 − 3)| = ½ × |−10 + 63 − 10| = 21.5 m². Use the cubic feet to cubic yards converter when ordering bulk material, and the area converter to switch between ft², m² and acres.
Heron’s formula and needle-thin triangles
Heron of Alexandria gave the three-side formula, with a proof, in his Metrica in the first century CE. It is elegant, but it has a numerical weak spot. When one side is tiny compared with the other two, a long, needle-thin sliver, one of the factors s − a, s − b or s − c is the difference of two nearly equal numbers, and floating-point rounding erases most of its significant digits. Any calculator that works in standard double precision, this one included, starts losing trailing digits once the longest side is roughly a million times the shortest, and the error grows quickly beyond that. Numerical analyst William Kahan published a sorted rearrangement of Heron’s formula that avoids most of the loss. For a truly thin triangle, the practical fix is to measure the short base and the perpendicular height and use ½bh.
Common mistakes
- Measuring the height along a slanted side. The height is perpendicular to the base; a sloped edge is always longer and overstates the area.
- Using the wrong angle in SAS. The angle must sit between the two sides you entered. If it does not, the triangle calculator can solve the triangle first.
- Forgetting square units. Area is in square units: 43.82 ft² is not 43.82 ft, and 1 ft² equals 144 in², not 12.
Frequently asked questions
What is the formula for the area of a triangle?
The basic formula is half the base times the perpendicular height, A = ½bh. A triangle with a 10 cm base and a 6 cm height covers 30 cm². Heron's formula, ½ab sin C and the shoelace formula reach the same area from other measurements.
How do I find the area of a triangle from its three sides?
Use Heron's formula. Work out the semiperimeter s = (a + b + c) ÷ 2, then take √(s(s − a)(s − b)(s − c)). Sides of 8, 11 and 13 give s = 16 and an area of √1,920 ≈ 43.82.
How do I find the area of a triangle from coordinates?
Use the shoelace formula: half the absolute value of x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂). Corners at (0, 0), (4, 0) and (1, 3) give ½ × |12| = 6 square units. A result of 0 means the three points lie on one straight line.
Does the order of the corners matter in the shoelace formula?
Not for the area, because the formula takes an absolute value. The sign of the raw sum shows the direction instead: positive means the corners are listed counterclockwise and negative means clockwise. The calculator reports which order you entered.
Why can't I use a slanted side as the height?
The height must be the perpendicular distance from the base to the opposite corner. Only in a right triangle does a side, one of the legs, double as the height. In an obtuse triangle the height can even fall outside the shape, measured to the extended base line.