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

Geometry Lessons

The short answer

b = 2A / h

Double the area, then divide by the height. The base you get belongs to the particular height you divided by, and no other.

This is the mirror image of finding a height. The area formula has two unknowns you might be missing, and whichever one it is, the rearrangement works the same way: undo the half by doubling, then divide by whatever you were given.

Where the formula comes from

Start from the area of a triangle, A = ½bh, and this time solve for b instead. Multiply both sides by 2 and the half disappears, leaving 2A = bh. Divide both sides by h and you have b = 2A / h.

It is worth noticing that h and b are interchangeable in the original formula. Neither one is special. Whichever of the two you know, doubling the area and dividing by it returns the other, which is why the same two step move solves both problems.

1

Double the area

Multiply A by 2. This cancels the half built into the area formula.

2

Divide by the height

Use the perpendicular height you were actually given, measured square to the base you want.

3

Check it backwards

Put your b and h into half base times height. You should land back on the area you started with.

A worked example

A triangular sail has an area of 36 square meters, and the mast gives you a perpendicular height of 9 meters. Double the area first: 2 times 36 is 72. Divide by the height: 72 divided by 9 is 8. The base measures 8 meters.

Check by going forwards. Half of 8 times 9 is 36, which is the area you were given, so the answer holds.

A triangle of area 36 with a height of 9 marked by a dashed perpendicular and an unknown base, beside the three steps rearranging the area formula to give a base of 8.
The working on the right is the whole method. Doubling first is what turns an equation containing a half into one you can divide straight through.

Dividing area by height is not the base

This is the same trap that catches people finding a height, and it is quiet for the same reason. Area divided by height returns exactly half the base every time, and half of a plausible number is still a plausible number.

A / hgives36 / 94

the half in the area formula was never undone

2A / hgives2 x 36 / 98

doubling first cancels the half, so this is the real base

Each height comes with its own base

A triangle has three sides, so it has three base and height pairs, and every pair multiplies out to the same area. That means the base you calculate depends entirely on which height you were handed. Divide by a larger height and you get a shorter base. Neither answer is more correct than the other, they are answers about different sides.

Where this actually bites is when a drawing gives you several dimensions and you pair the wrong two. The height must be the perpendicular dropped onto the side you are trying to measure. If it lands on a different side, the result means nothing.

Watch the units before you divide

The base comes out in whatever unit the height is in, as long as the area is in the matching square unit. Square meters with meters gives meters. If the area is in square meters and the height was written in centimeters, convert first. An area of 2.4 square meters with a height of 60 centimeters means dividing by 0.6, giving a base of 8 meters, not by 60.

Mistakes that catch people out

  • Forgetting to double the area. Dividing straight through returns half the base. Check by feeding your answer back into half base times height.
  • Using a slanted side as the height. The height is perpendicular. A slanted side is longer, so the base you get comes out too short.
  • Pairing the height with the wrong side. Each height belongs to one base. Mixing them gives a meaningless number.
  • Mixing units. Convert the area and the height into the same family before dividing.
  • Assuming the base must be the bottom. Any of the three sides can be the base. It is a choice, not a property of the shape.

Practice questions

  1. An area of 30 square centimeters with a height of 6 centimeters. Find the base.
  2. An area of 45 square meters with a height of 10 meters. Find the base.
  3. An area of 7.5 square feet with a height of 3 feet. Find the base.
  4. An area of 84 square centimeters with a height of 7 centimeters. Find the base.
  5. An area of 2.4 square meters with a height of 60 centimeters. Find the base in meters.
Show the answers
  1. b = 2 x 30 / 6 = 10 cm
  2. b = 2 x 45 / 10 = 9 m
  3. b = 2 x 7.5 / 3 = 5 ft
  4. b = 2 x 84 / 7 = 24 cm
  5. convert 60 cm to 0.6 m, then b = 2 x 2.4 / 0.6 = 8 m

Common questions

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

The formula is b = 2A / h. Multiply the area by two, then divide by the perpendicular height.

Why do you multiply the area by two?

Because the area formula already contains a half. Doubling the area cancels it, leaving the base times the height, which you can then divide through.

Does the height have to be perpendicular?

Yes. It must be measured square to the base you are trying to find. A slanted side is longer, and using it returns a base that is too short.

Can I use any height in the triangle?

You can, but the base you get will be the one that pairs with that particular height. Each of the three heights returns its own matching base.

What units does the base come out in?

The same length unit as the height, provided the area uses the matching square unit. Convert everything into one system before dividing.

Is this the same as finding the height?

It is the same rearrangement. The base and the height sit in identical positions in the area formula, so doubling and dividing works either way round.

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