The area of a triangle is ½ × base × height, where the height is the perpendicular distance from the base to the opposite corner. A triangle with a 10 cm base and a 6 cm height has an area of ½ × 10 × 6 = 30 cm². When you do not know the height, other formulas work from three sides (Heron’s formula), from two sides and the angle between them, or from the corners’ coordinates.
Choose the right formula
| What you know | Formula | Example result |
|---|---|---|
| Base and perpendicular height | A = ½bh | b = 10, h = 6 → 30 |
| All three sides | Heron’s formula | 7, 8, 9 → 26.83 |
| Two sides and the included angle | A = ½ab sin C | 9, 12, 40° → 34.71 |
| Coordinates of the corners | Shoelace formula | (1, 2), (7, 3), (4, 9) → 19.5 |
| Equilateral, side a | A = (√3 ÷ 4)a² | a = 6 → 15.59 |
1. Base and height
Any side can be the base, as long as the height is measured perpendicular to that side, from the opposite vertex. The slanted side is not the height unless the triangle has a right angle there.
Right triangles are the easiest case: the two legs are a base and its height. Legs of 5 and 12 give an area of ½ × 5 × 12 = 30.
Obtuse triangles have heights that fall outside the triangle for two of their sides. Extend the base line and measure from the far corner straight down to it. The formula is unchanged.
Why it works: two identical copies of any triangle fit together into a parallelogram with area b × h, so one triangle is exactly half.
A practical example: a gable wall
The triangular end of a gable roof is 24 feet wide at the base and rises 6 feet to the peak. Its area is ½ × 24 × 6 = 72 ft², which you would add to the rectangular wall area when estimating siding or paint. See how much paint do I need for the full wall calculation.
2. Three sides: Heron’s formula
When you can measure the sides but not the height, as with a triangular garden bed or a plot of land, use Heron’s formula:
Sides 7, 8 and 9: s = (7 + 8 + 9) ÷ 2 = 12
A = √[12 × 5 × 4 × 3] = √720 ≈ 26.83
A nicely whole-number case: sides 13, 14 and 15 give s = 21 and A = √(21 × 8 × 7 × 6) = √7,056 = 84.
If s − a, s − b or s − c comes out zero or negative, the three lengths cannot form a triangle. Each side must be shorter than the other two combined.
3. Two sides and the included angle
The angle must be the one between the two known sides.
a = 9, b = 12, C = 40°
A = ½ × 9 × 12 × sin 40° = 54 × 0.6428 ≈ 34.71
This formula is really ½bh in disguise: b × sin C is the height measured from side a. Set your calculator to degree mode before using it; the trig functions calculator shows sine values, and SOHCAHTOA explained covers why b sin C is the height.
If you know two angles and one side instead, find the missing sides with the law of sines first. The triangle calculator solves every SSS, SAS, ASA, AAS and SSA case and reports the area.
4. Coordinates: the shoelace formula
With corners at (x₁, y₁), (x₂, y₂) and (x₃, y₃):
Corners (1, 2), (7, 3), (4, 9)
½ |1(3 − 9) + 7(9 − 2) + 4(2 − 3)| = ½ |−6 + 49 − 4| = ½ × 39 = 19.5
The absolute value bars keep the answer positive regardless of the order you list the points. If the result is zero, the three points lie on a straight line. The same method extends to polygons with any number of corners; the polygon area calculator handles those.
5. Equilateral triangles
When all three sides equal a:
A side of 6 gives (√3 ÷ 4) × 36 ≈ 15.59. The height of an equilateral triangle is (√3 ÷ 2) × a, so this is just ½bh with that height substituted. The equilateral triangle calculator gives every measurement from one side.
Irregular shapes: split them into triangles
Any polygon can be divided into triangles by drawing diagonals from one corner, which is how surveyors have long measured irregular lots. Measure each triangle’s three sides with a tape, apply Heron’s formula to each, and add the results. A four-sided garden bed, for example, becomes two triangles sharing one diagonal; measure the four edges and that diagonal, and you have everything you need. For room floors with straight walls, splitting into rectangles is usually faster, as shown in how to calculate square footage.
Finding a height from the area
Rearranging ½bh lets you find a height you cannot measure directly:
For the 7-8-9 triangle, the height to the side of length 8 is 2 × 26.83 ÷ 8 ≈ 6.71.
Common mistakes
- Using a slanted side as the height. The height must meet the base at 90°.
- Forgetting the ½. Base × height is the parallelogram, twice the triangle.
- Using a non-included angle in ½ab sin C.
- Calculator in radian mode when the angle is in degrees.
- Mixed units. Convert every length to the same unit before multiplying; the area comes out in square units.
For right triangles specifically, the right triangle calculator gives area, perimeter and angles together, and how to use the Pythagorean theorem finds a missing side first. The general-purpose triangle area calculator accepts any of the inputs above.
Frequently asked questions
What is the formula for the area of a triangle?
Area = ½ × base × height, where the height is measured perpendicular to the base. A triangle with a 10-inch base and a 6-inch height has an area of ½ × 10 × 6 = 30 square inches.
How do you find the area of a triangle with only three sides?
Use Heron's formula. Compute the semi-perimeter s = (a + b + c) ÷ 2, then area = √[s(s − a)(s − b)(s − c)]. For sides 7, 8 and 9, s = 12 and the area is √720 ≈ 26.83.
How do you find the area with two sides and the angle between them?
Area = ½ × a × b × sin C, where C is the angle between sides a and b. For sides 9 and 12 with a 40° angle between them, the area is ½ × 9 × 12 × sin 40° ≈ 34.71.
Why is the triangle area formula half of base times height?
Any triangle is exactly half of a parallelogram with the same base and height. Two copies of the triangle, one rotated, fit together into that parallelogram, whose area is base × height.
How do I find the height of an obtuse triangle?
For an obtuse triangle, the height to one of the shorter sides falls outside the triangle. Extend that side with a dashed line and measure perpendicular from the opposite corner to the extended line. The formula ½ × base × height still works.