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Triangle Height Calculator

Range: 0 cm² – 1,000,000,000,000,000,000 cm²

Range: 0 cm – 1,000,000,000 cm

Result

6.0000 cm

Height

A triangle height calculator takes the area of a triangle and the base it was measured against, and returns the perpendicular height: how far it is from that base up to the opposite corner, in centimetres. It is the area formula read backwards. The area page multiplies a base by a height and halves the result; this page doubles the area and divides by the base, which is the same relationship with the unknown moved to the other side. Nothing about the triangle itself is needed — not its angles, not its other two sides, not its perimeter — because the area and the base between them already pin the height down to a single number. That is worth pausing on, since it is the part that surprises people. A triangle with an area of 30 square centimetres and a base of 10 centimetres has a height of 6, and it is the only height it can have, no matter how lopsided the shape is. All the freedom in the triangle is in where the top corner sits along the base, not in how high it is. The one thing to be careful about is the same thing the area page warns about, from the other direction: the base here is the side the height is measured against, and the height has to be perpendicular to it. Enter a sloping side as the base and the answer will be a height that is a little too small, with nothing on the page to suggest anything went wrong. If you are not sure which measurement you have, the area page is the place to settle it first, since it asks for the pair explicitly.

The perpendicular height of a triangle from its area and base

Area (cm²)Base (cm)Height (cm)
30106
2058
7.562.5
1283
844
992
2176
050

Eight triangles and three columns, because the page has one formula and one answer. The first three rows are shared with the area page on purpose, on the same numbers read the other way round: an area of 30 with a base of 10 gives 6, an area of 20 with a base of 5 gives 8, and an area of 7.5 with a base of 6 gives 2.5 — and the area page prints 30, 20 and 7.5 for those same three triangles when you hand it the base and the height. Reading the two tables side by side is the quickest way to see that they are one formula and not two. The fifth row is the isosceles right triangle, where an area of 8 with a base of 4 returns a height of 4: here the base is one leg and the height is the other, which is why the two are equal. That is a different line from the altitude to the hypotenuse, which measures 4 ÷ √2 = 2.8284 and is the one that falls in the middle of the hypotenuse. The sixth row is the self-checking one: an area of 9 with a base of 9 always returns 2, whatever the number, because doubling it and dividing it by itself can only give 2. The seventh has an odd base and still comes back at exactly 6, which is the point of doubling the area before dividing: the division is then the last step, so there is a single rounding and it happens at the exit. The last row is an area of zero, where the triangle has collapsed onto its base and the height is zero — a real answer rather than a missing one. Every value here is recomputed from its two inputs when the page is built, in centimetres, and the four decimals are the same four decimals the results panel uses.

Formula

h = 2 × A ÷ b

Area
How much surface the triangle covers, in square centimetres. This is the number the area page produces, handed back here as a starting point
Base
The side the height is measured against, in centimetres. Any of the three sides will do, as long as the height you want is the one perpendicular to it
Height
What this page works out: the perpendicular distance from the base up to the opposite corner, in centimetres. Not the length of either sloping side
Two
Where the doubling comes from. The area formula halves base times height, so undoing that halving means multiplying the area by two before dividing by the base
2A ÷ b
The whole calculation. Multiplying the area by two first, rather than halving the base first, is what keeps the last digit right when the numbers are not whole
Four decimal places
How wide the reading is written. A height can genuinely be a fraction — an area of 7.5 with a base of 6 gives exactly 2.5, and an area of 1 with a base of 3 gives two thirds
Centimetres
The unit of the answer whatever the dropdown says. This is a length, so unlike the area page the conversion is not squared — a base entered in inches still returns a height in centimetres, but you divide by 2.54 to read it in inches

The obvious use is the one where somebody already knows the area and needs a dimension. Land is the classic case: a plot's area is on the deed or in the survey notes, the base is along a road or a fence line, and the question is how far back the plot runs at its deepest point. That is this calculation exactly, and it is why the formula has been taught alongside the area formula for as long as both have existed. Roofing and carpentry bring a second family of uses, where a triangle's area has already been worked out for pricing and the height is needed as a cutting or setting-out dimension — the rise of a gable from its span, the depth of a tapered panel, the distance from a chord to the far side of an arc chord. Draughting and CAD give a third: a triangle on a drawing whose area is annotated, from which the missing altitude has to be recovered. And there is the school version, which is the one that makes the relationship click. Take any triangle whose area and base you know, work out the height, and then check it against the shape — the number you get is the one the picture shows, which is the whole point of the pairing between this page and the area page. Both directions of the same formula are worth having in front of you at least once, because a formula is much harder to forget once you have run it backwards.

Worked examples

  1. An area of 30 and a base of 10

    1. Double the area: 2 × 30 = 60
    2. Divide by the base: 60 ÷ 10 = 6

    The pair the page loads with, and the row that ties it to the area page: a base of 10 with a height of 6 is exactly what produces an area of 30 over there. Run one page and then the other with the matching numbers and you land back where you started, which is the cheapest possible check that you have understood the formula.

  2. An area of 20 and a base of 5

    1. Double the area: 2 × 20 = 40
    2. Divide by the base: 40 ÷ 5 = 8

    The same triangle the area page uses to make its point about trapezoids: a base of 5 with a height of 8 gives 20, and a trapezoid with a top base of zero, a bottom base of 10 and a height of 4 also gives 20. Nothing here is special — an odd-looking pair of inputs and a whole-number answer.

  3. An area of 7.5 and a base of 6

    1. Double the area: 2 × 7.5 = 15
    2. Divide by the base: 15 ÷ 6 = 2.5

    An area with a half in it, which is what a halved product usually gives, and a base that is even. Doubling first turns the half into a whole number before the division happens, so the answer is exactly 2.5 rather than something that has been rounded to look like it.

  4. An area of 0 and a base of 5

    1. Double the area: 2 × 0 = 0
    2. Divide by the base: 0 ÷ 5 = 0

    An area of zero means the triangle has been flattened onto its own base and has no height left. Zero is a real input rather than an empty box, so this is a real answer — and it is genuinely different from the case where the base is zero, which the page refuses because there is nothing to divide by.

Limitations

This page takes an area and a base and nothing else. It will not work from the three sides of a triangle, which is a different route to the same height and needs the area worked out first. It gives no angles and no side lengths, and the base here is the side the height is measured against rather than any of the other two — feed it a sloping side and the height comes back slightly too small, with nothing on the page to flag it. The base must be greater than zero: an area of zero is accepted and returns a height of zero, but a base of zero has no answer at all rather than an infinite one, and the page says so instead of printing a very large number. The answer is always in centimetres whatever the dropdown is set to, and since a height is a length the conversion is linear — a base entered in inches returns centimetres, and you divide by 2.54 rather than by 6.4516. Four decimal places is a display width rather than a claim about precision, and a height can be a genuine fraction: an area of 1 with a base of 3 gives two thirds, written to four places. Nothing here handles a curved side, a triangle drawn on a sphere, or a shape whose area was measured in a different plane from the base.

Frequently asked questions

Which side is the base here?
The one the height is perpendicular to — and since it is you who is supplying the height's target, it is whichever side you measured the area against. If the area came from a survey or a drawing, the base is the side that drawing measured it from. Getting the pairing wrong is the one way to get a wrong answer out of this page while everything still looks fine.
Why is the area doubled rather than the base halved?
Both give the same answer, and on a computer both are equally exact: doubling a number and halving a number are both exact operations, so either route performs one division and rounds once, at the end. The page doubles because that is the form the formula takes — the area formula halves base times height, and multiplying the area by two is how that halving is undone. It also keeps the division as the last step, so the rounded value is never fed back into anything.
Can the height come out longer than the base?
Yes, and there is nothing wrong with that. A tall narrow triangle has an area that, doubled and divided by a short base, gives a large height. The height and the base are two independent directions and neither constrains the other — only the area ties them together, and the formula is what it is.
What if the base is zero?
The page refuses rather than answering. A base of zero means the division is by zero, and the honest description of that result is that there is no height, not that the height is enormous. Zero area is a different matter and is accepted: a triangle with no area has been flattened onto its base, and its height really is zero.
Where is the three-sides version?
On its own page. Working a triangle's area out from three side lengths uses a different formula and a square root, and once you have the area you can come back here with any of the three sides as the base. The two pages are meant to be used in that order, and the height page does not try to duplicate the area page's job.
Do the units matter?
The dropdown changes what you type in, not what comes out. The reading is always in centimetres. Unlike the area page the conversion is not squared, because a height is a length: divide by 2.54 to read a centimetre answer in inches. Entering a base in inches and reading the answer as inches is the error to watch for.

References

Related calculators

Triangle Area CalculatorHalf a base times the perpendicular height above it.Heron's Formula CalculatorTriangle area from three side lengths, by Heron's formula.Triangle CalculatorThe third side, angles and area from two sides and an angle.Triangle Perimeter CalculatorPerimeter of a triangle from its three sides.