How to Find the Height of a Triangle When You Know the Area and Base

The area of a triangle is half its base times its height. If you already know the area and you know the base, you can run that formula backwards and recover the height. It is a two step rearrangement and there is really only one place people trip, which is the half. If you are not sure the area formula is the one you need, start with area of a triangle, which covers all four formulas and when each applies.

Where the formula comes from

Start with the area of a triangle, which is A = ½bh. You want h on its own. Multiply both sides by 2 and the half disappears, leaving 2A = bh. Now divide both sides by b, and you have h = 2A / b. That is the whole derivation.

The half is worth understanding rather than memorising. Draw a rectangle around any triangle so that they share the same base and the same height. The triangle always fills exactly half of it, no matter how slanted the triangle is, because the two leftover corner pieces fold in to fill the other half precisely. So the half is not a correction factor someone invented. It is the fixed relationship between a triangle and the rectangle it sits inside, and that is why undoing it means multiplying by two.

A worked example

A triangular flower bed covers 24 square meters. The side along the fence, which you are treating as the base, measures 8 meters. How tall is the bed measured square to that fence?

Double the area first: 2 times 24 is 48. Divide by the base: 48 divided by 8 is 6. The height is 6 meters. Check it by going forwards, ½ times 8 times 6 is 24, which is the area you were given, so the answer holds.

Why the factor of two catches people out

Height means square to the base, not along a side

In this formula, height means the perpendicular distance from the base to the opposite corner. It is the shortest way from that corner down to the line the base sits on, and it meets the base at a right angle. On most triangles it is not the length of any side, which is where the confusion starts.

There is a second consequence that surprises people. Every triangle has three sides, so it has three possible bases, and each one comes with its own height. All three pairs give the same area, which means the height you calculate depends entirely on which side you called the base. Pick a longer base and you get a shorter height. Neither answer is wrong. They are answers to different questions.

When the height falls outside the triangle

On an obtuse triangle, the perpendicular from the far corner can miss the base completely and land on an extension of it, out past the end. That looks wrong the first time you see it, and students often assume they have drawn something incorrectly. They have not.

The formula does not change and needs no special case. Height is measured to the line that the base lies on, not to the segment you drew, so h = 2A / b still returns the correct number. If you are working the other way and measuring a height off a sketch, that is the moment to be careful, because the perpendicular you need may sit entirely outside the shape.

Watch the units before you divide

The height comes out in whatever length unit your base is in, provided the area is in the matching square unit. Square meters with meters gives meters. Square feet with feet gives feet. The trouble starts when the two do not match, and mixed units produce an answer that is wrong by a factor of a hundred or more while still looking like a plausible number.

Say the area is 2.4 square meters and the base was written down as 150 centimeters. Convert the base to 1.5 meters before you do anything else. Then h = 2 times 2.4 divided by 1.5, which is 3.2 meters. Divide by 150 without converting and you get 0.032, a number that is not the height in any unit at all.

Mistakes that catch people out

• Forgetting to double the area. Dividing area straight by base returns half the height. Check by feeding your answer back into ½bh.
• Using a slanted side as the height. The height is perpendicular to the base. A slanted side is always longer, so the area you calculate from it comes out too big.
• Pairing the height with the wrong base. Each base has its own height. Mixing a height from one side with the length of another gives a meaningless result.
• Mixing units. Convert the base and the area into the same family first, meters with square meters, feet with square feet.
• Assuming the height must fit inside the triangle. On an obtuse triangle it lands outside, on the extension of the base, and the formula still works.

Practice questions

Common questions

What is the formula for the height of a triangle from the area and base?

The formula is h = 2A / b, where A is the area and b is the base. Multiply the area by two, then divide by the base length.

Why do you multiply the area by two?

Because the area formula already contains a half. A triangle covers exactly half the rectangle that shares its base and height, so recovering the height means undoing that half by doubling the area first.

Does it matter which side I use as the base?

It changes the answer but not its correctness. Every triangle has three bases and each one has its own matching height. A longer base gives a shorter height, and all three pairs produce the same area.

Can the height be longer than the base?

Yes. A tall narrow triangle has a height much greater than its base. There is no rule that one must be larger than the other.

What if the triangle is obtuse and the height falls outside it?

Nothing changes. Height is measured to the line the base lies on, not to the drawn segment, so on an obtuse triangle the perpendicular lands on an extension of the base and h = 2A / b still gives the right answer.

What units does the height come out in?

The same length unit as the base, as long as the area uses the matching square unit. Square meters with meters gives meters. Convert both into one system before dividing, or the answer will be out by a large factor.

Related lessons and calculators

• Area of a Triangle for the four formulas this rearrangement comes from, and how to choose between them.
• Triangle Solver for when you know sides or angles instead of the area, and want everything else at once.
• Area of 2D Shapes Calculator to go the other way and find the area first, including from three sides with Heron’s formula.
• Distance and Pythagoras Calculator for working out a perpendicular height on a right triangle from the sides you can measure.
• Area Converter if your area arrived in acres or hectares and the base is in meters or feet.
• Volume of 3D Shapes Calculator for the same rearranging idea applied to prisms, cones and pyramids.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.
• Height of a trapezoid, the same rearrangement on a shape with two parallel sides instead of one.