Triangle Area Calculator (3 Sides)

When the three sides of a triangle are known, all information needed to calculate its area is present. There is no need to measure angles or determine heights by drawing perpendicular lines. Heron's Formula allows us to directly compute the area from the lengths of the three sides, and this tool performs those calculations for you.

The opposite is also possible: if the area and two arbitrary sides are entered, the tool calculates the missing third side. Once the triangle has been determined unambiguously, it also calculates the perimeter, the three interior angles, the heights to each side, the median axes as well as the inscribed and circumscribed circles.

What is Heron's formula?

This formula was developed in the first century A.D. by Hero of Alexandria and is still used today. The key to this formula is that you must first calculate the semiperimeter, which is half of the perimeter.

Once half the circumference is known, the area can be very easily calculated by multiplying four factors together and then taking the square root.

Consider a well-known triangle with sides of 3-4 and 5 as an example. The semiperimeter is 6, and the subsequent calculations are simple.

This formula can be applied to any valid triangle, as long as the three sides form a closed figure. This includes equilateral, isosceles and scalene triangles alike.

How to use this calculator:

Enter the lengths of sides a, b and c and check the result in the output section. As each field can have a unit specified, you can for example enter one side in centimeters and another in inches. The calculator will first convert all values to the same base unit before performing the calculation.

If you want to calculate the length of a side from back, enter the area and two sides and leave the field for the third side blank. The calculator modifies Heron's formula and returns the length of the missing side. If the three sides do not satisfy the triangle inequality (i.e. the sum of any two sides must be greater than the third), an error message will appear instead of a false result.

Calculating angles from three sides:

The Law of Cosines also allows you to find the angles of a triangle when all three sides are given. For angle A, which is opposite side a, the following formula applies:

By simply switching the letters around, following the same rules, you get B and C. In a 3-4-5 triangle, calculations give approximately 36.87, 53.13, and 90 degrees. This is why it's a right triangle as found in textbooks.

The sum of the three angles is always 180 degrees. The calculator will show each angle and classify the triangle based on that as acute, right or obtuse.

Height, median, two circles.

The height, also called the altitude or perpendicular, is the vertical distance from one side to its opposite vertex. Since area is calculated by (base*height)/2, any height can be derived directly from the area.

A median is a line segment that connects a vertex to the midpoint of the opposite side. By turning on the Altitudes and Medians layers, you can see three altitudes and three medians.

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

Triangle classification

The calculator classifies triangles in two ways simultaneously. In the classification by sides, if all three sides are different then it is a scalene triangle; if two sides are equal, it is an isosceles triangle; and if all three sides are equal, it is an equilateral triangle.

By angle classification 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 more than 90 degrees. Since a triangle can only have one option from each classification, examples include isosceles, acute triangles or right, scalene triangles.

Examples of calculations

The table shows areas for various triangles; results are rounded to two decimal places. The last row shows a triangle with sides of length 9, 6 and 5 inches, which has an area of about 14.14 square inches.

Application examples for the three sides method.

If the length of sides is measured but the angle is unknown, this method can be used. Surveyors walk along irregular boundaries and record the length of each side with a tape measure or GPS surveying device, entering the data into a calculation tool.

Roofers and floor layers also use the same method to determine material requirements for a triangular area. It is usually not practical to use a protractor on a construction site. The same method can be used in building beams and supports as well, since the length of each individual piece will be recorded even if the angles at the joints are not documented.

Tips for accurate results:

Always use the same units when measuring each side if possible. Mixing units is one of the most common causes for incorrect results. If a length seems unusually small or large, you should measure it again since small errors in length can affect area.

Before trusting the result, check the triangle inequality. If the sum of any two sides is less than or equal to the third side, then there is no calculable triangle and the area will be zero.