Triangle Area Calculator

What do you know about the triangle?

Choose what you already know about the triangle; the calculator picks the right formula and solves for whatever you leave blank.

Enter the base and the perpendicular height to get the area, or enter the area with one of them to solve for the other.

Base (b)

Height (h)

Area (A)

Please input values into the fields

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If you know the data of a known triangle, you can calculate its area by entering it into this calculator. There are five input possibilities corresponding to different combinations of measurements that may be available: base and height, three sides, two sides and included angle, one side of an equilateral triangle or two equal sides of an isosceles triangle with the base being either of them.

Select the appropriate method, enter data and check area in result section. For methods "three sides", "SAS" (side-angle-side), "equilateral triangle" and "isosceles triangle" also show perimeter, three interior angles, height and median as well as inscribed and circumscribed circle. Also determine if the triangle is scalene, isosceles or equilateral and whether it is acute, right or obtuse.

Five methods for calculating area of a triangle.

Base and height.

This is the method that everyone first learns. If you know the length of one side and the perpendicular height to it then the area is half their product.

A = \tfrac{1}{2} \, b \, h

If the base is 4 and the height is 6, then the area is 12. This method will allow you to find the area of any triangle, but it won't tell you anything about the sides or angles of the triangle. There are many triangles with the same base and height, so this information alone doesn't give you enough information to determine the lengths of the sides or the measures of the angles.

The three sides (Heron's formula)

If the three sides are known but no angles are known, then Heron's formula can be used to directly find the area of the triangle from the length of the sides. First, the semiperimeter is calculated by adding the lengths of the sides and dividing by two. Then the following formula gives the area:

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

For a typical triangle with sides 3-4,5 the semiperimeter is 6 and the result of the area calculation is 6. This method can also be used in reverse. If you enter the area and two sides leaving the third side blank, it will calculate its value.

Two sides and included angle (SAS)

If two sides and the angle between them are known, there is no need to square any numbers. Simply multiply the two sides together, multiply that by the sine of the included angle, and divide by 2.

A = \tfrac{1}{2} \, a \, b \, \sin(C)

If the two sides are 4 and 5, and the included angle is 30 degrees, then the area is 5. The angle must be the one between the two given sides, not some other angle.

equilateral triangle

Since all three sides of an equilateral triangle are the same length, you only need to know the length of one side in order to calculate the area. The formula for calculating the area is as follows:

A = \frac{\sqrt{3}}{4} \, a^2

The value that results from dividing the square root of 3 by 4 is approximately 0.433. To calculate this more quickly, you can multiply the square of the side length by 0.433. If the side length is 6, then the area will be approximately 15.59.

Isosceles triangle

An isosceles triangle has two sides of equal length. If you know the length of the base and the length of the two equal sides, use this formula to calculate its area:

A = \frac{1}{4} \, b \, \sqrt{4a^2 - b^2}

If the base is 6 and the two equal sides are 5, then the area is 12. You can also enter the area and the base to calculate the length of the two equal sides.

This is how this calculation tool works:

Select a method from the top and only the fields required for that will be shown. Enter values and leave the field blank for the value you want to calculate to get the result. Each field has its own units so you can enter one value in centimeters and another in inches for example. The calculator tool will convert all data into the same unit before performing calculations.

Except for the method of base and height, each method also displays the whole triangle in the result area. By opening the levels for the height or medians respectively or the level for the circle you can see further information. A pie chart for the angles or a bar chart for the side lengths help you to recognize the shape at a glance.

Estimate of the entire triangle by two measurements:

A triangle is fully determined by its three sides. For this reason, the "Three Sides" method calculates all other attributes. The three interior angles are found using the law of cosines. Angle A, opposite side a, is calculated as follows:

\cos A = \frac{b^2 + c^2 - a^2}{2bc}

The sum of the three angles in a triangle is always 180 degrees and an evaluating tool uses this information to classify the triangle as acute, right or obtuse. The height of a triangle is the length obtained by dividing twice the area by the length of the side that the height is perpendicular to. A median is a line drawn from a vertex to the midpoint of the opposite side.

Any triangle has two circles associated with it. The incircle is the largest circle that will fit inside the triangle; its radius is equal to the area of the triangle divided by half its perimeter. The circumcircle passes through all three vertices, and its radius is equal to the product of the sides of the triangle divided by four times the area.

r = \frac{A}{s} \qquad R = \frac{a\,b\,c}{4A}

Types of Triangles

Triangles can be classified both by their sides and by their angles. By their sides, if all three sides are the same length then it is an equilateral triangle; if two of the sides are the same length then it is an isosceles triangle; if no sides are the same length then it is a scalene triangle. By their angles, if all three angles are less than 90 degrees then it is an acute triangle; if one angle is exactly 90 degrees then it is a right triangle; if one angle is greater than 90 degrees then it is an obtuse triangle.

Each triangle is designated by each of the two classifications. So a 3-4-5 triangle is both a right and scalene triangle, and a triangle with all three sides equal is both equilateral and acute. When the lengths of the sides are known this calculator will show both designations.

Examples of calculations

The table shows the area of the triangle calculated using various methods. The results are rounded to two decimal places.

Method	You enter	Area
Base and height	base 4 cm, height 6 cm	12.00 sq cm
Three sides	3, 4, 5 cm	6.00 sq cm
Two sides and angle	4 cm, 5 cm, 30 deg	5.00 sq cm
Equilateral	side 6 cm	15.59 sq cm
Isosceles	base 6 cm, equal sides 5 cm	12.00 sq cm

Areas of application for calculating the area of triangles

If it is an irregular non-rectangular space then calculating the area of a triangle may be required. Roofers and floor layers will divide up irregular spaces into triangles to estimate how much material they need. Surveyors will walk along irregular boundaries, note down the lengths of sides and use Heron's formula as using a protractor on site is usually impractical.

Construction workers use the "base and height" method for sloped roofs or pillars. The SAS method is used in fields such as navigation or engineering where two known distances and an angle between them are given. Students learn five different methods to calculate triangle area in math class, and can use these to check their homework.

For accurate results, please follow these tips:

Use the same units for all measurements if possible. Mixing units is one of the most common causes of errors in results. If a conversion is needed, use a calculator to do it. For the "base and height" method, make sure that the height is perpendicular to the base, not just the length of the hypotenuse.

For the three sides method, make sure that the three sides can actually form a triangle. The sum of any two sides must be greater than the third side. Otherwise, the calculator will give you a warning and not an incorrect result.

Frequently asked questions

What is the formula for calculating the area of a triangle?

The most well-known formula for calculating the area is to calculate half of the product of base and height: A = (1/2)*b*h. If the altitude is not given, other formulas can be used. These include Heron's formula which depends on all three sides and the formula (1/2)*a*b*sin(C), which depends on two sides and the included angle. For an equilateral triangle, it is √3 / 4 multiplied by a². For an isosceles triangle, it is one fourth of the product of b and the square root of (4*a2 − b2). This calculator uses all five of these formulas.

How do you find the area of a triangle when given only all three sides?

You use Heron's formula. First you calculate half the perimeter s, which is one-half of the sum of the three sides a, b and c. Then you multiply s with (s - a), (s - b) and (s - c) and take the square root of the result. For a triangle with side lengths 3-4-5, s equals 6, and the area is the product of 6 and the square roots of 3, 2 and 1, which is 6. Select the method for three sides and enter the length of the sides.

How do you find the area of an equilateral triangle?

It is sufficient to give the side length. The area is calculated by dividing the square root of 3 by 4 and multiplying the result by the square of the side length. Since the value of the square root of 3, divided by 4, is approximately 0.433, you can calculate the area more quickly by squaring the side length and multiplying it by 0.433. If the side length is 6, then the area will be approximately 15.59 units squared.

What is the formula for finding the area of a triangle given two sides and the included angle (SAS)?

If two sides and the angle between them are known, then the area is calculated by multiplying one side by another, then multiplying it by the sine of the included angle, and finally dividing the result by 2. The formula therefore reads A = (1/2)*a*b*sin(C). This angle must be the one between the two given sides, not another angle.

Is it possible to calculate the side lengths or height if the area is known?

Yes, it is possible. All methods can be used in reverse. If you enter the area and leave one of the fields blank, the tool will calculate the corresponding value. In the method with three sides, one side can be calculated from the area and two other sides. In the method with base and height, one of them can be calculated from the area and the other side. In the equilateral triangle method, the length of a side can be calculated from the area.

What type of triangle do I have: scalene, isosceles or equilateral? And what type of angles does it have: acute, right or obtuse?

If the data is available for side lengths, then the calculator will classify from both perspectives at once. Looking at sides, a triangle is equilateral if all three sides are equal; isosceles if two sides are equal; and scalene if all three sides are unequal. Looking at angles, a triangle is acute if all angles are less than 90 degrees; right if one angle is exactly 90 degrees; and obtuse if one angle is greater than 90 degrees. For example, a triangle with side lengths of 3-4-5 is a scalene right triangle.

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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: Area of a triangle
The base-times-height, Heron, side-angle-side, and other area formulas, with derivations.
Wikipedia: Heron's formula
The area of a triangle from its three sides, plus the semi-perimeter.
Math is Fun: Area of Triangles
Plain-language walk-through of the base-height and side-angle-side area methods.
Wikipedia: Law of cosines
How the three sides of a triangle determine each interior angle.