Triangle Height Calculator

Background.

The Quanta triangle height calculator finds the altitude of a triangle — the perpendicular distance from a vertex down to the line containing the opposite side — from any of three starting points: all three side lengths, two sides with the angle between them, or the area together with the base you want the height measured against. It returns that height, the area behind it, the formula path it took, and the other two altitudes when they can be worked out. With the default 13-14-15 triangle it returns an area of 84 and a height of 12.9230769231 onto the side of length 13, with the remaining altitudes coming out at exactly 12 and 11.2.

Every mode is the same identity read backwards. Euclid's Elements, Book I, Proposition 41 established that a parallelogram sharing a base with a triangle and lying between the same parallels is double the triangle, which is where area = ½ × base × height comes from. Rearranged, height = 2 × area ÷ base. So the calculator's real job is finding the area, and each mode does that differently: Heron's formula from three sides, half the product of two sides times the sine of the angle between them, or simply accepting the area you already have.

The first thing to be clear about is that a triangle has three heights, not one. Each side can serve as the base, and each gives a different altitude — in the 13-14-15 triangle they are 12.9230769231, 12 and 11.2. That is why this page names the base beside the answer instead of leaving you to guess, and why it prints the other two underneath. If you were told "the height of the triangle" without a base attached, the question was incomplete, and the usual intended meaning is the altitude onto the longest side or onto whichever side is drawn horizontally.

The second thing is a limitation worth stating before you use the area-and-base mode. Knowing an area and one base is enough to fix that one height, but it is not enough to fix the triangle. Infinitely many triangles share an area of 84 and a base of 13 — slide the apex along a line parallel to the base and neither the base nor the area changes, while the other two sides and the other two altitudes change continuously. So in that mode the calculator returns the height you asked for and says outright that the other two are not determined, rather than manufacturing numbers for them.

The third thing matters for obtuse triangles. When one angle exceeds 90°, two of the three altitudes fall outside the triangle: the perpendicular from a vertex meets the extension of the opposite side rather than the side itself. The length is still correct and still satisfies area = ½ × base × height; it is only the picture that changes. The 2-3-4 triangle in the test suite is exactly this case, and its altitudes come back as 2.9047375097, 1.9364916731 and 1.4523687548.

Heights are what turn a shape into a quantity in practice. A gable end needs its rise to work out sheet material; a triangular garden bed needs its perpendicular depth to plan rows; a truss needs the height of each web triangle to size the members; a plot of land with three measured boundary lines needs Heron's formula and then an altitude to be divided sensibly. In all of those the sides are the thing you can measure with a tape and the height is the thing you cannot, which is precisely the gap this page fills.

What is triangle height calculator?

The height, or altitude, of a triangle is the perpendicular segment from one vertex to the line containing the opposite side. That opposite side is called the base for that altitude, and since any of the three sides can be chosen as the base, every triangle has three altitudes. MathWorld's Altitude entry defines them as the cevians perpendicular to the opposite sides and notes that all three are concurrent at a single point, the orthocenter; it also records the relation 1/h₁ + 1/h₂ + 1/h₃ = 1/r, where r is the inradius, which this calculator uses as an internal consistency check. The altitude is tied to the area by the oldest identity in plane geometry: Euclid's Elements Book I, Proposition 41 shows that a parallelogram on the same base and between the same parallels as a triangle has double its area, so the triangle's area is ½ × base × height. MathWorld states the same as Equation (18) of its Triangle entry, Δ = ½ah. Reversing it gives h = 2Δ ÷ a, which is the calculator's engine. Getting the area depends on what you know: with three sides it is Heron's formula, Δ = √(s(s − a)(s − b)(s − c)) with s the semiperimeter ½(a + b + c); with two sides and the angle they enclose it is Δ = ½ab sin C, MathWorld's Equation (19); and if the area is already known no derivation is needed at all. In an obtuse triangle two of the three altitudes land outside the figure, meeting the extended base — the length is unaffected, only the drawing changes. In a right triangle the two legs are themselves altitudes, which is why a 3-4-5 triangle has altitudes of 4, 3 and 2.4.

How to use this calculator.

1. 01Choose what you know. "All three sides" is the most common — it is the case where you can measure with a tape but cannot measure the height directly.
2. 02In three-sides mode, enter the sides in any consistent unit. Side a is the base the main result is measured against, so put the side you care about there. Each side must be shorter than the sum of the other two, or the lengths cannot form a triangle at all.
3. 03In two-sides-and-angle mode, the angle must be the one enclosed by the two sides you entered — the angle at the vertex where they meet — not one of the other two. Enter it in degrees.
4. 04In area-and-base mode, enter the area and the side you want the height measured perpendicular to. This is the direct rearrangement of area = ½ × base × height.
5. 05Read the height at the top and check the base beside it. Because a triangle has three heights, the answer only means something once you know which side it belongs to.
6. 06Read the other two altitudes underneath. In area-and-base mode they are not determined and the output says so — that is a real limitation, not a missing feature.
7. 07For an obtuse triangle, expect two of the three altitudes to fall outside the shape. Measure to the extension of the base, not to the base itself.

The formula.

Everything on this page rests on area = ½ × base × height, so the calculator's task in each mode is to establish the area and then divide it out. Rearranged, the altitude onto side a is h = 2Δ ÷ a, and by the same argument the altitudes onto b and c are 2Δ ÷ b and 2Δ ÷ c.

In three-sides mode the area comes from Heron's formula: with s = (a + b + c) ÷ 2, the area is √(s(s − a)(s − b)(s − c)). For the 13-14-15 default, s = (13 + 14 + 15) ÷ 2 = 21, and the area is √(21 × 8 × 7 × 6) = √7056 = 84 exactly. The altitude onto side a is 2 × 84 ÷ 13 = 168 ÷ 13 = 12.9230769231. Onto side b it is 2 × 84 ÷ 14 = 12 exactly, and onto side c it is 2 × 84 ÷ 15 = 11.2 exactly.

In two-sides-and-angle mode the area is ½ab sin C, MathWorld's Equation (19), and the third side comes from the law of cosines, c² = a² + b² − 2ab cos C, so that all three altitudes are still available. With a = 13, b = 14 and C = 37°, the area is ½ × 13 × 14 × sin 37° = 54.7651671068, the third side is 8.6195518644, and the altitude onto side a is 2 × 54.7651671068 ÷ 13 = 8.4254103241.

In area-and-base mode the division is all there is: 2 × 84 ÷ 13 = 12.9230769231 again, which is a useful check that the two routes agree. What that mode cannot do is recover the rest of the triangle. Slide the apex of a triangle along any line parallel to the base and the base and the area both stay fixed while the other two sides change, so an area and a base describe an infinite family of triangles with different remaining altitudes. The calculator says so in the output rather than returning a number it cannot justify.

An independent structural check runs on every result: MathWorld's Altitude entry gives 1/h₁ + 1/h₂ + 1/h₃ = 1/r, where r is the inradius. For the 13-14-15 triangle the inradius is 84 ÷ 21 = 4, and 13/168 + 1/12 + 1/11.2 = 0.0773809524 + 0.0833333333 + 0.0892857143 = 0.25, which is exactly 1 ÷ 4. The test suite asserts that identity on every sample triangle.

Rounding stage: all arithmetic runs in Decimal at 40 significant digits and is rounded once, at the return boundary, to 10 decimal places. The area is never rounded before the altitudes are divided out of it — doing that would move the ninth decimal place of the height. The sine and cosine in two-sides-and-angle mode are the only double-precision step, evaluated once each and immediately carried back into Decimal, so the error floor is near 1e-16. Nothing on this page is sorted into bands or grades, so no threshold depends on display rounding; the two text outputs name the formula path and report the other altitudes, and neither changes because a value crossed a boundary.

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.