How do you find the base of a triangle when you know the area and height?

Use the rearranged area formula: base = (2 x area) / height. For example, if a triangle has an area of 30 square centimetres and a perpendicular height of 6 cm, the base is (2 x 30) / 6 = 10 cm. This works for any triangle, whether it is scalene, isosceles, or equilateral, as long as the height is measured perpendicularly from the base to the opposite vertex.

Can you find the base of a triangle if you only know all three sides?

Yes. If you know all three side lengths (a, b, c) and you want to find the height to one of the sides (so that 'base' is that side), you can use Heron's formula to calculate the area first: s = (a + b + c) / 2, then Area = sqrt(s(s-a)(s-b)(s-c)). Once you have the area, use height = 2 x Area / base. If you want the actual length of a side as the base, the base is simply whichever side you designate as the base. The 'base' of a triangle is any side you choose; height is then the perpendicular distance to it.

How do you find the base of a right triangle?

For a right triangle, the two legs are perpendicular to each other. If you know one leg and the hypotenuse, use the Pythagorean theorem: base = sqrt(hypotenuse squared minus height squared). For example, if the hypotenuse is 13 and one leg (height) is 5, the base is sqrt(169 - 25) = sqrt(144) = 12. If you know both legs, either leg can act as the base.

Base of a Triangle Calculator

This calculator finds the base of a triangle using whichever measurements you have, across three common situations. If you know the area and the perpendicular height, it rearranges the standard area formula, base equals two times area divided by height, to solve for the base directly. If you only know the three side lengths, it applies Heron's formula to work out the area, then converts that into the perpendicular height to whichever side you nominate as the base. If you have a right triangle and know the hypotenuse and one leg, it uses the Pythagorean theorem to find the missing leg, which becomes your base. Choose the matching method from the dropdown, enter your known values, and the tool instantly returns the base length, the area, and the perpendicular height to that base, with full working shown for every step. It also checks your inputs, flagging negative or zero values and warning you if the three sides entered cannot form a valid triangle under the triangle inequality rule. This suits schoolwork, trade measurements such as roofing or fencing, DIY building and landscaping projects, or any time you need to work backwards from area or side lengths to a dimension. The calculator assumes flat, Euclidean geometry, and results are only as accurate as the measurements you enter, so check your figures and units before relying on the answer for anything structural.

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Standard Formula Area formula rearrangement, Heron's formula, and the Pythagorean theorem.

1. Choose Your Method

Known values

Area (square units)

Perpendicular height (same units as base)

Base side (a) the side whose height you want

Side b

Side c

Hypotenuse

Known leg (height/opposite side)

2. Result

Base Length
10
units

Area

square units

Height to Base

units

Please enter valid positive numbers. For three sides, they must satisfy the triangle inequality (the sum of any two sides must exceed the third).

Working:
base = (2 x 30) / 6 = 10.00

How to Find the Base of a Triangle

The "base" of a triangle is any one of its three sides. The height (or altitude) is the perpendicular distance from that base to the opposite vertex. Depending on what information you already know, there are three common methods to calculate the base.

Method 1: From Area and Height

The area of a triangle is half the base times the perpendicular height:

Area = (base x height) / 2

Rearranging to solve for the base:

base = (2 x Area) / height

This is the most direct method. It works for any triangle as long as the height is truly perpendicular to the chosen base (not slanted along another side).

Area	Height	Base
30	6	10.00
50	8	12.50
24	4	12.00
100	5	40.00

Method 2: From Three Sides (Heron's Formula)

If you know all three side lengths (a, b, c) and want to find the perpendicular height to side a, first calculate the area using Heron's formula:

Calculate the semi-perimeter: s = (a + b + c) / 2
Calculate the area: Area = sqrt(s x (s - a) x (s - b) x (s - c))
The perpendicular height to side a is: height = 2 x Area / a

For example, with sides a = 10, b = 8, c = 6: s = 12, Area = sqrt(12 x 2 x 4 x 6) = sqrt(576) = 24, height = 2 x 24 / 10 = 4.80 units.

Method 3: Right Triangle (Pythagorean Theorem)

In a right triangle, the two shorter sides (legs) meet at a right angle. If you know the hypotenuse and one leg, find the other leg (the base) using:

base = sqrt(hypotenuse squared - height squared)

For example, hypotenuse = 13, one leg = 5: base = sqrt(169 - 25) = sqrt(144) = 12.

Worked Example (Default Values)

A triangle has an area of 30 square units and a perpendicular height of 6 units. What is its base?

Apply the rearranged area formula: base = (2 x Area) / height
base = (2 x 30) / 6
base = 60 / 6
base = 10.00 units

Verification: Area = (10 x 6) / 2 = 30 square units. Correct.

Related Calculators

Sources and method: Standard Euclidean geometry. Area formula: Area = (1/2) x base x height. Heron's formula (60 BC). Pythagorean theorem (a^2 + b^2 = c^2). All standard secondary school mathematics.

This calculator assumes Euclidean (flat) geometry. Results are in whatever units you enter for length; area must be in the square of those units. For the three-sides method, the sides must satisfy the triangle inequality.