A triangular corner lot is listed at 3,000 square feet with 100 feet of street frontage — how deep is it? Rearranging the area formula gives the answer: height = 2 × area ÷ base. Double the area (6,000), divide by the base (100), and the lot runs 60 feet deep. The same rearrangement sizes flagpoles for triangular banners, sets the altitude of roof trusses, and checks whether a triangular garden bed is deep enough for the planned rows. Height here always means the altitude: the perpendicular distance from the chosen base to the opposite vertex, measured at a right angle, never along a slanted side.

The formula, and why the doubling comes first

Every triangle's area is half of base times height, so the height is hidden inside that half. Undoing the half is the first move: double the area, then divide by the base. In symbols, with area A and base b, height h = 2A ÷ b. The doubling is not a fussy detail; skipping it is the single most common height error, and it always gives exactly half the true height. Think of it as rebuilding the rectangle the triangle was cut from: a triangle is half a parallelogram with the same base and height, so doubling the area reconstructs that parallelogram, and dividing by the base recovers its height. If you remember only one sentence about triangle height, make it this one: double first, divide second.

Worked example: a triangular garden bed

A gardener's triangular bed covers 432 square feet along a 36-foot fence line that serves as the base. Double the area to 864, divide by 36, and the bed is 24 feet deep from fence to far corner. That depth decides the layout: rows running parallel to the fence can be spaced 3 feet apart, giving 8 rows, and a 24-foot-deep bed with 3-foot paths between every second row still leaves room for a wheelbarrow turnaround at the point. The check runs backward just as cleanly: half of 36 times 24 is 18 times 24, which is 432, the listed area. Whenever a height answer looks suspicious, run this backward check; a mismatch means the base and the area came from different triangles, or the doubling was skipped.

Worked example: a banner on a pole

A print shop needs a triangular banner with 24 square feet of fabric on a 6-foot pole. Double 24 to 48, divide by 6, and the banner hangs 8 feet tall. The shop cares because the pole, the grommets, and the wind load all scale with that 8-foot drop: a banner twice as tall catches roughly twice the wind, so the pole rating matters as much as the fabric. The same arithmetic sizes sail battens, triangular trade-show flags, and pennant strings, where the pole side is the base and the drop is the height. Note the unit discipline: square feet of area divided by feet of base leaves feet of height, and mixing square meters with feet would silently corrupt the answer.

A height ladder to memorize

A few area-and-base pairs are worth knowing by heart, because they recur in homework and on job sites. An area of 50 with a base of 10 gives a height of 10. An area of 96 with a base of 16 gives 12. An area of 210 with a base of 28 gives 15. An area of 360 with a base of 45 gives 16. An area of 1,000 with a base of 80 gives 25. Sanity-check any new answer against this ladder: a height near these anchors for similar inputs is plausible, while a height off by a factor of two almost always means the doubling step was skipped or applied twice. The ladder also trains estimation: a 500-square-foot triangle on a 50-foot base must be about 20 feet deep, so an answer of 10 or 40 should trigger a recheck before it reaches a materials order.

Three sides and no area? Heron first

Often you know all three sides and no area at all, and the height still falls out in two steps. Take a triangle with sides 13, 14, and 15 feet. First find the semi-perimeter: half of 13 plus 14 plus 15 is 21. Heron's formula gives the area as the square root of 21 times 8 times 7 times 6, the square root of 7,056, which is exactly 84 square feet. Now the height to each side follows from the same doubling rule. To the 13-foot side: 168 divided by 13 is about 12.92 feet. To the 14-foot side: 168 divided by 14 is exactly 12 feet. To the 15-foot side: 168 divided by 15 is exactly 11.2 feet. One triangle, three heights, and the longest side gets the shortest height, which is always true: height and base are inversely related for a fixed area.

The isosceles shortcut

An isosceles triangle hides its height inside a right triangle, and that gives a shortcut that never needs the area first. Take equal sides of 17 feet and a base of 16 feet. The altitude to the base splits the base into two 8-foot halves and splits the triangle into two right triangles with hypotenuse 17 and one leg 8. The height is the square root of 289 minus 64, the square root of 225, which is exactly 15 feet. The area check confirms it: half of 16 times 15 is 120 square feet, and doubling 120 and dividing by 16 returns 15. Roof trusses use this constantly: a symmetric gable with 17-foot rafters spanning 16 feet rises 15 feet, and the truss builder who knows the shortcut never waits for an area figure. The same right-triangle split works for any isosceles triangle, and it is also why the altitude to the base of an isosceles triangle is simultaneously a median and an angle bisector.

The equilateral shortcut

Equilateral triangles are even tidier, because all three heights are identical. For a side of 12 inches, the height is 12 times the square root of 3 divided by 2, which is 12 times 0.8660, about 10.39 inches. The area follows as half of 12 times 10.39, about 62.35 square inches, and the doubling rule returns the same height: 2 times 62.35 divided by 12 is about 10.39. The 0.866 factor is worth memorizing: every equilateral height is about 87 percent of its side, so a 20-foot equilateral sign panel rises about 17.32 feet. Quilters cutting equilateral triangles, engineers sizing triangular gusset plates, and teachers drawing 30-60-90 triangles all lean on this one factor, and it is the fastest height estimate in the whole triangle family.

Right triangles: altitude to the hypotenuse

A right triangle has a special altitude: the one dropped from the right-angle vertex to the hypotenuse. For legs of 9 and 12 feet, the hypotenuse is 15 feet by the Pythagorean theorem, and the area is half of 9 times 12, which is 54 square feet. The altitude to the hypotenuse is 2 times 54 divided by 15, which is 108 divided by 15, exactly 7.2 feet. There is a shortcut here too: the altitude equals the product of the legs divided by the hypotenuse, 9 times 12 divided by 15, which is also 7.2. This altitude matters in construction because it is the shortest distance from the right-angle corner to the long side, the measurement a framer needs when bracing a corner or cutting a diagonal support. Note that the two legs are themselves altitudes to each other: the altitude to the 9-foot leg is 12 feet, and to the 12-foot leg is 9 feet, which is why right triangles always carry two obvious heights and one hidden one.

Two sides and the angle between them

When you know two sides and the angle sandwiched between them, the area comes from the sine formula and the height follows. Take sides of 20 and 26 feet with a 40-degree angle between them. The area is half of 20 times 26 times the sine of 40 degrees: 260 times 0.6428 is about 167.12 square feet. The height to the 26-foot side is then 2 times 167.12 divided by 26, about 12.86 feet. Surveyors meet this case constantly: two measured sides from a total station plus one angle, and the height to either side is one sine away. The trap is the calculator mode: the sine of 40 degrees is 0.6428, but the sine of 40 radians is 0.7451, and a calculator left in radian mode silently inflates every area and height that follows. Check the mode before trusting any sine-based height.

When the altitude falls outside the triangle

In an obtuse triangle, the altitude to one of the short sides lands outside the triangle, on the extension of that side, and students often distrust the answer because of it. Trust it anyway: the formula does not care where the foot of the altitude lands, only that the distance is perpendicular to the base line. A triangle with a 120-degree angle at one corner has two altitudes that fall outside and one that falls inside, and all three are still real lengths that still satisfy the doubling rule. The three altitudes meet at a single point called the orthocenter, which sits inside acute triangles, on the right-angle vertex of right triangles, and outside obtuse triangles. If a problem asks for "the height" without naming a base, it usually means the altitude to the longest side or to the base drawn horizontally in the diagram, so name the base before you compute.

The three-heights check

Every triangle carries a built-in error check: each side times its own altitude equals twice the area, so all three products must agree. For the 13-14-15 triangle above, 13 times 12.92 is about 168, 14 times 12 is exactly 168, and 15 times 11.2 is exactly 168. If your three products disagree, one height is wrong, and the usual suspect is a mismatched pairing: the altitude must be paired with the side it is perpendicular to, not with the side that happens to be drawn at the bottom of the page. Rotating the triangle does not change any height; it only changes which side you call the base. This check is the fastest way to catch a Heron arithmetic slip before it propagates into a cut list or a land record.

How to use this calculator, honestly

This calculator performs the doubling step at the heart of the height formula. Type the triangle's area in the Variable A box and 2 in the Variable B box, and the Result box shows twice the area. For the corner lot, entering 3000 and 2 gives 6,000; divide that by the 100-foot frontage on paper and the 60-foot depth is confirmed. For the garden bed, 432 and 2 give 864, and 864 divided by 36 is 24 feet. For the banner, 24 and 2 give 48, and 48 divided by 6 is 8 feet. The tool does not ask for the base, so the division is a one-step finish you do by hand, and keeping that division visible is what stops the classic half-height mistake. Keep area and base in matching units: square feet with feet gives feet of height, square meters with meters gives meters, and square feet with a base in inches gives nonsense until one of them is converted.

Five mistakes that cost real answers

The first mistake is forgetting to double, which halves every height. The second is measuring along a slanted side instead of perpendicular to the base; the slant is always longer than the altitude, so the answer inflates. The third is pairing a height with the wrong side, which breaks the three-heights check. The fourth is mixing units, square feet of area with a base in inches, which corrupts the answer by a factor of 12 before any division happens. The fifth is using the sine of an angle in the wrong calculator mode, degrees versus radians, which quietly poisons every SAS height. Run the backward check (half of base times height must equal the area) after every calculation, and most of these mistakes announce themselves immediately.

Related calculators on this site

If the height is one step in a bigger job, three companion tools carry the next step. The triangle area calculator works the forward direction, base times height divided by 2, whenever the height is known and the area is the goal. The right triangle calculator handles the Pythagorean routes, legs and hypotenuse, that feed the isosceles and right-triangle shortcuts above. The triangle angles calculator recovers the missing angles when only sides are known, and the slope calculator converts a height-over-run pair into a grade or an angle for ramps and roofs. Each one uses the same discipline: name the base before you enter anything, keep units consistent, and check the answer against a known anchor.

A classroom note on altitude versus median

Students often confuse the altitude with the median and the angle bisector, because all three lines run from a vertex toward the opposite side. They coincide only in special triangles: in an isosceles triangle they coincide on the symmetry axis, and in an equilateral triangle they coincide everywhere. In a general scalene triangle they are three different lines with three different lengths, and only the altitude is perpendicular to the base. Drawing all three on one scalene triangle, say the 13-14-15 triangle, makes the distinction visible in a way no definition can: the median lands at the midpoint, the bisector splits the angle, and the altitude alone meets the base at a right angle. That right angle is the whole job of a height, and it is why "how tall" always means "how far, perpendicularly."

With the pre-filled defaults — Variable A 10, Variable B 5 — the Triangle Height Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.