Triangle Height Calculator

Background.

A gable wall 8 metres wide, closed by two roof slopes of 5 metres each, rises 3 metres from the eaves line to the ridge. Nobody had to measure that 3 metres; it follows from the three lengths you can put a tape on. That is the usual position with a triangle's height. The sides are easy to measure and the perpendicular is not, whether the triangle is a gable end, a plot with three surveyed boundaries or one panel of a roof truss.

Every route to the height runs through the area. Euclid showed in Book I, Proposition 41 of the Elements that a triangle is half the parallelogram standing on the same base between the same parallels, which is where area = ½ × base × height comes from. Turned around, height = 2 × area ÷ base, so the real work is getting the area: by Heron's formula from three sides, by ½ab·sin C from two sides and the angle between them, or straight from a plan or title deed if the area is already known.

The point that trips people up is that a triangle has three heights, one for each side you might call the base, and they are usually all different. The gable above is 3 m tall over its 8 m base but 4.8 m tall measured square to either roof slope. The calculator reports the height onto side a, or onto the base you enter, and lists the other two whenever the triangle is fully known.

What is triangle height calculator?

An altitude, or height, of a triangle is the perpendicular segment from a vertex to the straight line through the opposite side, which is that altitude's base. Each of the three sides has its own. Base × height gives the same product, twice the area, whichever side you choose, so the longest side always has the shortest altitude. Extended as lines, the three altitudes meet at one point, the orthocentre. In an obtuse triangle two of them meet their bases outside the figure, on the extension of the side; in a right triangle the two legs are altitudes of each other.

How to use this calculator.

1. 01Pick the mode that matches what you measured: all three sides, two sides with the angle between them, or the area with one base.
2. 02With three sides, put the side you want the height onto in side a. Use one unit throughout. Each side must be shorter than the other two added together, so 13, 14 and 27 is refused: laid end to end, 13 and 14 only just reach 27 and enclose no area.
3. 03With two sides and an angle, the angle must be the one at the corner where sides a and b meet. If the angle you know faces one of those sides instead, that is the side-side-angle case; use the SSA triangle calculator, which deals with the chance of two different triangles.
4. 04With area and base, enter the area in the square of the base's unit, for example m² with a base in metres.
5. 05Read the height and the base it belongs to, then the other two altitudes. In area-and-base mode they are reported as not determined.

The formula.

The height onto side a is h = 2Δ ÷ a, where Δ is the area, and the heights onto b and c are 2Δ ÷ b and 2Δ ÷ c. With three sides, the area comes from Heron's formula: s = (a + b + c) ÷ 2 and Δ = √(s(s − a)(s − b)(s − c)). With two sides and their included angle C, Δ = ½ab·sin C, and the third side follows from the law of cosines, c² = a² + b² − 2ab·cos C, so all three altitudes are available. That route also shows why h onto a is simply b·sin C: the height depends only on the side leaning away from the base and how steeply it leans. In area-and-base mode the division is the whole calculation, and nothing more can be recovered, because sliding the apex along a line parallel to the base keeps both the area and that height fixed while the other two sides change. A built-in check holds for every triangle: 1/h_a + 1/h_b + 1/h_c = 1/r, where r = Δ ÷ s is the radius of the inscribed circle.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Altitude’ (live revision retrieved 2026-07-29). Defines the altitudes as the cevians perpendicular to the opposite sides, states that the three altitudes are concurrent at the orthocenter, gives h_a = bc/(2R) = c·sin B = b·sin C, and records formula (5), 1/h₁ + 1/h₂ + 1/h₃ = 1/r, which this page uses as an internal consistency check. Open access.
2. [2]Wolfram Research, MathWorld — ‘Heron's Formula’ (live revision retrieved 2026-07-29): Δ = √(s(s−a)(s−b)(s−c)) with the semiperimeter s = ½(a+b+c). The entry gives the derivation and alternative forms but carries no worked numeric example, so a published example was taken from a separate source for the cross-check. Open access.
3. [3]Wolfram Research, MathWorld — ‘Triangle’ (live revision retrieved 2026-07-29): Equation (18) Δ = ½ah, the base-and-height area form this page inverts, and Equation (19) Δ = ½ac·sin B, the two-sides-and-included-angle form used in the second mode. Open access.
4. [4]Euclid, Elements, Book I, Proposition 41 — ‘If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.’ This is the classical origin of area = ½ × base × height. David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29.