How to Calculate the Side Length of an Oblique Triangle

Right triangles, like the one above, are not oblique triangles. All other triangles can be considered oblique.

An oblique triangle is defined as any triangle that does not include a 90-degree angle. This includes acute triangles, where all three angles are less than 90 degrees, as well as obtuse triangles, where one of the angles is greater than 90 degrees. Determining the side length of an oblique triangle without resorting to a ruler appears to be impossible at first glance. It can be done, however, as long as the measurements for two of the angles and one of the sides are known. The angles and sides of a triangle are typically assigned labels for the sake of convenience. The most common labels for angles are A, B and C. The sides are also labeled A, B and C, according to which angle the side is opposite.

Things You'll Need

• Measurements of two angles of the triangle
• Measurement of one side of the triangle
• A scientific calculator

Instructions

1. Add the two known angles together. Subtract this number from 180. The resulting number is the third angle, given in degrees. This is because all triangles have angles adding up to 180 degrees. This holds true no matter the size or type of triangle. For example, if angle A is equal to 20 degrees, and angle B is equal to 100 degrees, then angle C must be equal to 60 degrees.

2. Input one of the known angles into a scientific calculator. Press the "sine" button. This will produce the sine of that particular angle. For example, angle B in the example above is given as 100 degrees. Inputting 100 degrees for the angle gives a rounded figure of 0.98 for the sine. Multiply this sine by the length of side A. For example, using the sine given above and a length of five inches for side A, the figure arrived at is 4.9.

3. Divide this figure by the sine of angle A. The sine of angle A, using the figures from the example above, is 0.34. Dividing 4.9 by 0.34 gives a result of 14.4. This is the length of side B.

4. Repeat the procedure outlined so far to determine the length of side C, substituting angle C for angle B. Angle C, using the examples given above, is equal to 60 degrees. The sine of 60 is 0.87. Multiply this number by the length of side A, which in the example above is five inches. Five multiplied by 0.87 gives a result of 4.35. Divide this by 0.34, the sign of angle A in the example. This gives a result of 12.8. This is the length of side C.

References

• Clark University: Oblique Triangles
• Algebra Lab: Trigonometry: Oblique Triangles - Law of Sines

Resources

• Calculator: Scientific Calculator