How do you find the base of a triangle?

If the area and height are known, then the base can be calculated as twice the area divided by the height, or b = 2A/h. If the area is not known, other methods may be used such as using a side or an angle. For a right triangle or an isosceles triangle, the Pythagorean theorem can be applied. If two sides are known, then the third side can be calculated by subtracting from the perimeter. If one side and two angles are known, then the sine rule can be used.

What is the length of the base of a triangle with an area of 10 square centimeters?

The length of the base is determined by the height. Using the formula b = 2A / h, if the area is 10 square centimeters and the height is 4 centimeters, then the length of the base would be 5 centimeters. Changing the height will change the length of the base. The area is only determined by the product of the base and the height, so neither the base nor the height can be determined independently.

Can any side of a triangle be used as its base?

Yes. The altitude can be measured as the vertical distance from one side to its opposite vertex, and any of the three sides can be used as a base. For each chosen base there is an associated altitude, and regardless of which combination is used, the area will always come out the same.

How to calculate base of isosceles triangle?

In an isosceles triangle the height divides the base into two halves and forms two identical right triangles. Each of these triangles has one of the equal sides as its hypotenuse and half of the base as one of its legs. By applying Pythagoras' theorem to one of these triangles, you can calculate half of the base and then double it. If you enter the same side and height in the corresponding mode, the calculator will perform the calculations and provide the area, perimeter, and vertex angle.

What is the difference between base and height?

The base is one side of the triangle while the height is the vertical distance from that side to the opposite vertex. The two are always perpendicular to each other. The hypotenuse of a triangle is not a height, and it is a common mistake to use the hypotenuse when calculating the base.

Base of a Triangle Calculator

Base of a triangle calculator

What do you know about the triangle?

Pick whichever facts you already have about the triangle. The classic method needs only the area and the height; the other four find the base from side lengths or angles instead.

Area

Height

Base

Enter any two of area, height and base and leave the third blank; the calculator fills it in for you.

Please input values into the fields

The base of your triangle is shown above, along with the area and the other measurements the chosen method provides.

Any of the three sides can serve as the base, as long as the height is measured straight out to the opposite corner. Base and height always pair with the same perpendicular direction.

The base of a triangle is the side chosen as the reference for calculating area. The size of the whole triangle depends on this side combined with the height which is perpendicular to that side from its opposite vertex.

Regardless of what information is given, this calculator can find the base. The simplest case requires only area and height. When two equal sides, hypotenuse of a right triangle, perimeter or one side along with two angles are known, the calculator switches to an appropriate method and calculates the base from there.

What is a base of a triangle?

A triangle has three sides, each of which can be considered the base. The base is the side from which the height is measured. If you rotate the triangle so that a different side touches the ground, then that side becomes the new base and there will be a corresponding height.

The only unchanging rule is that the base and height must be perpendicular to each other. The height is the straight-line distance from the base to the opposite vertex, forming a right angle with the base. If they are combined correctly, all of the following formulas can be used without any problems:

Basic formula for calculating base area and height.

The area of a triangle is half of the product of its base and height.

A = \frac{b \times h}{2}

By slightly reformatting this equation three questions can be answered at the same time. In the first mode if any one value is left blank then the calculator will determine that value.

b = \frac{2A}{h} \qquad h = \frac{2A}{b} \qquad A = \frac{b \times h}{2}

In this formula b is the base, h is the height and A is the area. Because the area is determined by multiplying the base by the height, these two values are inversely proportional to each other. A tall, narrow triangle can have the same area as a short, wide triangle.

A full example:

Say you're making a triangular shelf and need to cover an area of 60 square centimeters with a height of 15 centimeters. Double the area to get 120, then divide by the height:

b = \frac{2 \times 60}{15} = \frac{120}{15} = 8 \text{ cm}

Therefore, this shelf plate needs a base of 8 centimeters. By changing the order of numbers, with the same tool it is also possible to calculate the height given the base or the surface area given the base and the height.

Determine the base side without using the area.

The area is not the only clue. Depending on what type of triangle you have, there are four ways to use length or angle alone to find the base.

Isosceles triangle: two sides of equal length and the height.

An isosceles triangle has two sides of equal length that meet at the vertex point and a base that connects the other ends of those sides. The altitude is drawn perpendicular from the vertex to the midpoint of the base, bisecting the triangle into two identical right triangles. Since each half has one leg as the altitude and the other leg being half the length of the base, by Pythagorean theorem we can derive the following formula:

b = 2\sqrt{\,L^{2} - h^{2}\,}

In this case, L is the length of each equal side and h is the height. For the same triangle, the calculator also provides area, perimeter and vertex angles.

Right triangle: hypotenuse and one leg.

In a right triangle the two legs meet at the right angle and the hypotenuse is the longest side opposite the right angle. If the hypotenuse c and one of the legs a are known, then the other leg can be used as the base to calculate it directly using the Pythagorean theorem:

b = \sqrt{\,c^{2} - a^{2}\,}

Since the two legs were originally perpendicular to each other, one of them is the height of the other. Therefore, the area is half their product.

Three sides: Calculation from the circumference

The perimeter is the sum of all three sides. If the perimeter P and two of the sides are known, then the third side, which is the base, is what remains:

b = P - (a + c)

If all three sides are known, it is not necessary to calculate the altitude separately in order to find the area of the triangle - Heron's formula can be used instead. A calculator will show both the base and the area together.

A = \sqrt{s\,(s-a)(s-b)(s-c)}, \qquad s = \frac{P}{2}

To form a true triangle each side must be shorter than the sum of the other two sides. If the numbers do not meet this rule then the calculation tool will indicate this rather than outputting a nonsensical base.

Sine rule: one side and two angles

If one side and two angles are known, trigonometry comes into play. The law of sines states that the ratio of any side to the sine of its opposite angle is equal. So if a side and both corresponding angles are known, then the base can be calculated:

\frac{b}{\sin B} = \frac{a}{\sin A} \;\Longrightarrow\; b = a\,\frac{\sin B}{\sin A}

Here a is the known side length, A is the angle opposite it, and B is the angle opposite the base b. The third angle is what's left after subtracting the two given angles from 180 degrees, because of the rule that the sum of all interior angles in any triangle must be 180 degrees.

What method should I use?

What you know	How the base is found	Example
Area and height	Twice the area, divided by the height	60 cm sq, 15 cm -> 8 cm
Two equal sides and height	Pythagoras on each half, then doubled	sides 13, height 12 -> 10
Hypotenuse and a leg	Pythagoras on the hypotenuse and known leg	hyp 5, leg 3 -> 4
Perimeter and two sides	Perimeter minus the two known sides	P 12, sides 3 and 4 -> 5
A side and two angles	Law of sines from the side and its angles	side 7, angles 40 and 60 deg -> 9.4

Units and hints

Use the same units for both base and height, and use the corresponding area unit for your answer. Even if a calculator will do conversions for you, it is best to get a reasonable answer by using square inches for the area and centimeters for the height. Make sure that the height is perpendicular to the base. The hypotenuse of a triangle is longer than the height, so using the wrong side as the height will give an overestimate for the base.

This tool is for learning purposes and to make rough estimates. When used in engineering, construction or surveying fields the results of calculations should be checked against tolerances required by the project.

Frequently asked questions

How do you find the base of a triangle?: If the area and height are known, then the base can be calculated as twice the area divided by the height, or b = 2A/h. If the area is not known, other methods may be used such as using a side or an angle. For a right triangle or an isosceles triangle, the Pythagorean theorem can be applied. If two sides are known, then the third side can be calculated by subtracting from the perimeter. If one side and two angles are known, then the sine rule can be used.
What is the length of the base of a triangle with an area of 10 square centimeters?: The length of the base is determined by the height. Using the formula b = 2A / h, if the area is 10 square centimeters and the height is 4 centimeters, then the length of the base would be 5 centimeters. Changing the height will change the length of the base. The area is only determined by the product of the base and the height, so neither the base nor the height can be determined independently.
Can any side of a triangle be used as its base?: Yes. The altitude can be measured as the vertical distance from one side to its opposite vertex, and any of the three sides can be used as a base. For each chosen base there is an associated altitude, and regardless of which combination is used, the area will always come out the same.
How to calculate base of isosceles triangle?: In an isosceles triangle the height divides the base into two halves and forms two identical right triangles. Each of these triangles has one of the equal sides as its hypotenuse and half of the base as one of its legs. By applying Pythagoras' theorem to one of these triangles, you can calculate half of the base and then double it. If you enter the same side and height in the corresponding mode, the calculator will perform the calculations and provide the area, perimeter, and vertex angle.
What is the difference between base and height?: The base is one side of the triangle while the height is the vertical distance from that side to the opposite vertex. The two are always perpendicular to each other. The hypotenuse of a triangle is not a height, and it is a common mistake to use the hypotenuse when calculating the base.

Related calculators

Triangle Area CalculatorFind the area of a triangle five ways: from the base and height, from three sides with Heron's formula, from two sides and the angle between them, from one side of an equilateral triangle, or from the base and equal sides of an isosceles one. Also gives the perimeter, angles, heights, medians, and circles.

Triangle Perimeter CalculatorFind the perimeter of a triangle three ways: from one side of an equilateral triangle, from the equal sides and base of an isosceles triangle, or from the three sides of a scalene triangle. Also gives the area, semi-perimeter, angles, heights, medians, and circles, and estimates a fencing cost.

Angle of a Right Triangle CalculatorFind the two acute angles of a right triangle from any two parts, or solve the whole triangle, with sides, area, perimeter, altitude, and special-triangle recognition.

Triangle Area Calculator (3 Sides)Find the area of a triangle from its three side lengths with Heron's formula, or enter the area with two sides to solve for the third. Also gives the angles, heights, medians, and inscribed and circumscribed circles.

Area of an Obtuse Triangle CalculatorFind the area of an obtuse triangle four ways - from the base and height, three sides, two sides and the angle between them, or two angles and the side between them - and check that the triangle really is obtuse.

ABC Triangle CalculatorRight-triangle solver: enter any two measurements - two sides, a side and an acute angle, or the area with one side - and it finds the missing sides, both acute angles, the area, perimeter, height to the hypotenuse, circumradius, and inradius, with a built-in Pythagorean-triple check.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Triangle
Definitions of base and height, the area formula and the main triangle types.
Wolfram MathWorld: Triangle
Formal treatment of triangle area, side and angle relations.
Wikipedia: Heron's formula
Area of a triangle from its three side lengths, used by the perimeter method.
Wikipedia: Law of sines
The side-to-sine ratio used to find a side from a known side and two angles.