A triangle’s height (altitude) is the perpendicular distance from a vertex to the opposite side. Every triangle has three, one for each side. Heights are needed for areas, for roof pitches and for the strength of triangular frames.
How to use the Triangle Height Calculator
- Triangle type
- Side a
- Side b
- Side c
Results update as you type, and everything is calculated in your browser.
How it works
Pick the type of triangle so only the necessary sides are asked for. The area is computed with Heron’s formula and each height is twice the area divided by its base.
Tips
- The angles of a triangle always add up to 180°.
- Any side must be shorter than the sum of the other two.
- Label sides opposite their angles: side a is opposite angle α.
- For right triangles, the hypotenuse is always the longest side.
Common mistakes to avoid
- Using a non-perpendicular side as the height.
- Mixing degrees and radians.
- Applying the Pythagorean theorem to a triangle without a right angle.
Inputs explained
| Field | What to enter |
|---|---|
| Triangle type | Choose one: General (three sides), Equilateral, Isosceles (legs a, base b), Right (legs a and b) |
| Side a | Units: cm, m, mm, in, ft, yd … |
| Side b | Units: cm, m, mm, in, ft, yd … |
| Side c | Units: cm, m, mm, in, ft, yd … |
What each result means
- Area = sqrt(([a] + [b] + [c])×(-[a] + [b] + [c])×([a] – [b] + [c])×([a] + [b] – [c]))÷4
- Height to side a = 2×[Area]÷[a]
- Height to the second side = 2×[Area]÷[b]
- Height to the third side = 2×[Area]÷[c]
- Perimeter = [a] + [b] + [c]
- Angle α (opposite a) = deg(acos(([b]^2 + [c]^2 – [a]^2)÷(2×[b]×[c])))
- Angle β (opposite b) = deg(acos(([a]^2 + [c]^2 – [b]^2)÷(2×[a]×[c])))
- Angle γ (opposite c) = 180 – [Angle α (opposite a)] – [Angle β (opposite b)]
Example values
How the main result changes when Side a changes and everything else stays at the example values above:
| Side a (cm) | Height to side a (cm) |
|---|---|
| 2.5 | 5.8462 |
| 3.75 | 5.9983 |
| 5 | 5.8788 |
| 6.25 | 5.6261 |
| 7.5 | 5.2618 |
| 10 | 4.1325 |
Formula
hₐ = 2A/a, h_b = 2A/b, h_c = 2A/c, A from Heron's formulaWorked example
Inputs — Triangle type: General (three sides); Side a: 5 cm; Side b: 6 cm; Side c: 7 cm.
Results — Area = 14.696938 cm²; Height to side a = 5.878775 cm; Height to the second side = 4.898979 cm; Height to the third side = 4.199125 cm; Perimeter = 18 cm; Angle α (opposite a) = 44.415309 °; Angle β (opposite b) = 57.12165 °; Angle γ (opposite c) = 78.463041 °.
Key terms
- Altitude
- A perpendicular segment from a vertex to the line of the opposite side.
Frequently asked questions
How many heights does a triangle have?
Three, one to each side.
Can a height be outside the triangle?
Yes – in an obtuse triangle two heights fall outside.
What is the height of an equilateral triangle?
a√3/2.
Where do the three heights meet?
At the orthocenter.
What formula does the Triangle Height Calculator use?
hₐ = 2A/a, h_b = 2A/b, h_c = 2A/c, A from Heron's formula.