Area of an Oblique Triangle Calculator

An acute triangle is a triangle with no right angles. It can be either isosceles or scalene, but it never has a right angle. The interesting thing about this type of triangle is that the simple formula for finding its area - multiplying the height by the base and dividing by two - usually cannot be used in practice when you have actual measurements to work with.

This tool will calculate the area of any acute triangle where you know either two sides and an included angle, two angles and a side or all three sides. Simply select your method from the drop down list, enter your values and the calculator will do the rest, not only calculating the area but also the perimeter, the three angles, the altitude, as well as the inradius and circumradius of the triangle.

What exactly is an acute triangle?

Triangles can be divided into two categories, depending on the type of angles they have. A right triangle has one angle that is 90 degrees. All other triangles are acute triangles. An acute triangle has three angles that are all less than 90 degrees and fall under the category of acute triangles. An obtuse triangle has one angle that is greater than 90 degrees, which also falls under the category of acute triangles. Only right triangles do not fall into the category of acute triangles. The area of a right triangle can be calculated by taking half of the product of its two legs.

Since an acute triangle has no right angles, two important tools of trigonometry are used to solve it: the law of sines and the law of cosines. The law of sines relates each side to the sine of its opposite angle. The law of cosines relates all three sides to one angle. By combining these two laws, any acute triangle can be solved, meaning that the shape is fully determined by the given conditions.

How to calculate area:

Sides and included angle (SAS)

If two sides and the included angle are known, then the area is equal to half the product of the two sides and the sine of their included angle.

If the angle between two sides with lengths 4 and 5 is 40 degrees, then the area is approximately 6.43. This must be the included angle. If you replace it with another angle, there are multiple possible shapes. So Side-Angle-Side requires that the included angle be explicitly stated.

Two angles and a side (ASA and AAS)

The sum of the three angles is always 180 degrees, so if two angles are known then the remaining angle can be determined. One side establishes the overall scale and with the sine rule the remaining sides and angles can be calculated. If the known side is between the two angles it's called ASA (Angle Side Angle), and if it's on one of the sides it's AAS (Angle Angle Side). In both cases a formula can be derived to directly calculate the area from one side and two angles.

If a side of length 10 lies between two angles of 50 and 60 degrees, then the area is approximately 35.30. This is a typical surveying arrangement where the length of one baseline is measured and from each end the angles to a third point are measured, allowing the area to be calculated without having to enter the terrain.

Three sides (SSS, Heron's formula)

If the lengths of all three sides are known but no angles are known, then Heron's formula can be used to calculate the area by directly feeding the side lengths into the formula. First, half the perimeter s is calculated, which is one-half of the total perimeter. The following formula is used for this:

For a triangle with sides of length 7, 8 and 13 the semiperimeter s is 14 and the area is approximately 24.25. Heron's formula is often used in land surveying, since it does not require any angles to be known and only requires that lengths of the sides be known.

Why the direct use of base and height is avoided:

In a right triangle, the two legs can simply be used as the base and height to calculate the area since they meet at a right angle. For a general triangle, there is no such right angle. The altitude must be constructed by drawing a perpendicular from one vertex to the opposite side, and for an obtuse triangle this perpendicular will fall outside of the side and may even cross over the end of the line segment. Since measuring this altitude in situ can be time consuming, the trigonometric methods described in this section are usually more practical.

Full triangle of input data.

Each of the methods described here will define a triangle completely, so that the calculator returns all remaining elements. The three interior angles can be calculated using the law of cosines. Let A be the angle opposite side a, then:

Any altitude is the value obtained by dividing twice the area by the corresponding side. Any median connects a vertex with the midpoint of the opposite side. Every triangle has two circles: an incircle that lies exactly inside and a circumcircle that passes through all three vertices.

Ambiguity - Why there is no SSA

Another possibility is to consider two sides and a non-included angle, called SSA. This case is intentionally omitted. From two sides and a non-included angle there can sometimes be both an acute triangle and an obtuse triangle, which means the area is not unique. It's a common mistake to feed an SSA calculation tool and rely on a single answer. For SSA you should calculate the missing angle yourself using the law of sines and check whether one or both possible triangles meet the conditions.

Examples of calculations

The table below shows several examples of calculating the area of acute triangles. Each area is calculated using a different method and the results are rounded to two decimal places.

Applications of area calculation for acute triangles.

Since land is rarely a perfect rectangle, surveyors often use such methods. In rough terrain it is often more practical to use an angle-measuring instrument than to drag a tape measure across the ground. By measuring two angles at the ends of a baseline, the area can be calculated using the ASA formula. Navigation and triangulation are based on the same principle: from two directions and one known distance a position is determined.

Construction workers and engineers encounter acute triangles in roof peaks, supports, and any construction where two elements meet at an angle other than 90 degrees. Students learn about these cases in math class. They are usually covered right after the law of sines and cosines, and those tools will be used to check your own calculations.

Tips for accurate results:

Make sure the units of the angles match the values entered. Calculators can switch between degrees and radians but using the wrong unit for sine is the most common cause of incorrect area values. Any length can be entered in any unit used to measure it. The calculator will convert all values into a consistent base unit before calculating, so you can mix centimeters and inches if desired.

For the calculation of a triangle with three sides, this tool first determines if the given lengths can form a triangle or not. The sum of any two sides must be greater than the third side. For the calculation of a triangle with two angles, the sum of the entered angles must be less than 180 degrees, otherwise the third angle is not defined. If a triangle cannot be formed, this calculator will display a warning instead of an incorrect area value.

This tool is for general educational purposes and planning your day to day life. For surveying, construction or other high risk activities these values should be checked against the requirements of the project with tolerances in mind.