Geometry
Area of a Triangle Calculator
Find a triangle's area from whatever you know — a base and height, three sides, two sides and an angle, or the corner coordinates. Pick a method, type your numbers, and the answer and full working appear instantly.
Enter what you know
A base and the perpendicular height to it
Area
30
Base
10 cm
Height
6 cm
Recomputes as you type
No sign-up · Free
The basics
What the area of a triangle means
A triangle's area is the amount of flat space it covers. Every method below returns the same number for the same triangle — they just start from different known quantities. The foundation is one idea: a triangle is exactly half of a rectangle sharing its base and height.
A = ½ · b · h
Half a rectangle
Copy any triangle, flip it, and the two pieces tile a rectangle of the same base and height. The rectangle covers b × h, so one triangle covers half of it.
½ · 10 · 6 = 30
height ⟂ base
Height is perpendicular
The height is the straight-line distance from the base to the opposite corner — measured at a right angle to the base, not along a sloping edge.
the straight drop, not a slanted side
Pick your method
Four ways, one area
Which formula you reach for depends on what you can measure. Here is the whole map at a glance — each row is a tab in the calculator above.
A = ½ · b · h
Base & height
A base and the perpendicular height to it.
A = √(s(s − a)(s − b)(s − c))
Three sides
All three side lengths (Heron's formula).
A = ½ · a · b · sin C
Two sides + angle
Two sides and the angle between them.
A = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Vertex coordinates
The (x, y) coordinates of all three corners.
Method
Find the area by hand
The base-and-height route is three quick steps. The other methods follow the same shape: write the formula, substitute, evaluate.
- 1
Identify the base and its height
Choose any side as the base, then find the perpendicular height dropped to it from the opposite corner.
b = 10, h = 6 - 2
Write the formula and substitute
Put the two numbers into the half-base-times-height formula.
A = ½ · 10 · 6 - 3
Multiply through
Half of base times height is the area, in square units of your measurement.
A = 30 square units
Worked examples
See every method worked through
The same triangle problem, solved four different ways depending on what you start with. Each tab computes the result live from the formula.
Given base b = 10, height h = 6
A = ½ · b · h
A = ½ · 10 · 6
A = 30
A = 30
Quick chart
“Area with base X and height Y”
The fastest lookup for the base-and-height case. Read the area straight off the grid for common base and height pairs — every cell is ½ × base × height.
| base ＼ height | h = 3 | h = 4 | h = 5 | h = 6 | h = 8 | h = 10 |
|---|---|---|---|---|---|---|
| b = 4 | 6 | 8 | 10 | 12 | 16 | 20 |
| b = 5 | 7.5 | 10 | 12.5 | 15 | 20 | 25 |
| b = 6 | 9 | 12 | 15 | 18 | 24 | 30 |
| b = 8 | 12 | 16 | 20 | 24 | 32 | 40 |
| b = 10 | 15 | 20 | 25 | 30 | 40 | 50 |
| b = 12 | 18 | 24 | 30 | 36 | 48 | 60 |
| b = 15 | 22.5 | 30 | 37.5 | 45 | 60 | 75 |
| b = 20 | 30 | 40 | 50 | 60 | 80 | 100 |
Each cell = ½ · base · height
Where it shows up
Why triangle area matters
The half-base-height rule quietly underpins construction, land surveying, and computer graphics.
roof & gable
Building & trades
Pitched roofs, gable ends, and triangular trusses are sized by their area to estimate materials and load.
shoelace on a plot
Land surveying
Surveyors split an oddly shaped plot into triangles and total their areas from corner coordinates — exactly the shoelace method.
meshes of triangles
3D graphics
Every rendered surface is a mesh of triangles; their areas drive lighting, texturing, and physics in games and CAD.
Reference
Area by triangle type
The general formulas always work, but some triangles have a tidy shortcut worth remembering.
Right triangleA = ½ · leg₁ · leg₂the two legs are the base and height
Equilateral (side s)A = (√3 ÷ 4) · s²all three sides equal
Isosceles (base b, equal side s)A = (b ÷ 4) · √(4s² − b²)two sides equal
Any triangle (base, height)A = ½ · b · hthe universal formula
Any triangle (three sides)A = √(s(s−a)(s−b)(s−c))Heron's formula
Worth knowing
Things that trip people up
height ≠ a slanted side
Use the perpendicular height
The most common mistake is plugging in the length of a sloping edge instead of the perpendicular height. Only the straight drop to the base counts.
a 5-slope with a 4 vertical drop → use 4
a + b > c
Three sides must close up
For Heron's formula the lengths have to satisfy the triangle inequality — each side shorter than the other two combined — or no triangle exists.
2, 3, 9 → no triangle
sin C, not the side
The angle goes between the sides
In the two-sides method C must be the angle enclosed by the two sides you used — not some other angle of the triangle.
½ · 8 · 6 · sin 40° ≈ 15.43
units² always
Area is in square units
Whatever length unit you measure in, the area comes out squared. Mixing units (a base in metres, a height in centimetres) is the silent source of wrong answers.
cm → cm², ft → ft²
FAQ
Common questions
The most common way is half the base times the perpendicular height: A = ½ × b × h. For a base of 10 and a height of 6, A = ½ × 10 × 6 = 30. If you don't have the height, use the three sides (Heron's formula), two sides and the angle between them, or the corner coordinates — this calculator does all four.
The base formula is A = ½ × base × height. From three sides use Heron's formula A = √(s(s − a)(s − b)(s − c)) where s is the semi-perimeter. From two sides and the included angle use A = ½ × a × b × sin C. From vertex coordinates use the shoelace formula.
Heron's formula finds a triangle's area from its three side lengths a, b, and c. First compute the semi-perimeter s = (a + b + c) ÷ 2, then A = √(s(s − a)(s − b)(s − c)). For sides 3, 4, 5 the semi-perimeter is 6, so A = √(6 × 3 × 2 × 1) = √36 = 6.
You have three options. If you know all three sides, use Heron's formula. If you know two sides and the angle between them, use A = ½ × a × b × sin C. If you know the coordinates of the corners, use the shoelace formula. Pick the matching tab above and the calculator handles it.
Use Heron's formula. Add the three sides and halve the total to get the semi-perimeter s, then take A = √(s(s − a)(s − b)(s − c)). It works for any triangle as long as the three lengths can actually form one — each side must be shorter than the sum of the other two.
A right triangle's two legs are perpendicular, so they are the base and height. The area is just half their product: A = ½ × leg₁ × leg₂. For legs of 3 and 4, A = ½ × 3 × 4 = 6 — no separate height measurement needed.
Any triangle is exactly half of a parallelogram (or rectangle) with the same base and height — copy the triangle, rotate it, and the two halves tile the full rectangle. The rectangle's area is base × height, so one triangle is half of that.
Area is always in square units of whatever length unit you used. Measure the sides in centimetres and the area is in cm²; use feet and it is ft². Set the unit in the calculator and every result and step is labelled accordingly.
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Author
Ugo CharlesBuilt and fact-checked against our editorial standards · Last updated June 2026
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