How to Find Degrees in a Triangle
A triangle is composed of three sides that intersect to create three angles. The total number of degrees inside of a triangle is 180 degrees, of which each of the triangle's angles is a part. When all that you know about a triangle is its side measurements, you can use the cosine rule and the sine rule, which are both based on trigonometric properties, to calculate its angles.
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Instructions
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First Angle
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1
Select two sides of the triangle, multiply their lengths together and then double that product. For this example, two of the sides measure 8 and 10. Multiplying 8 and 10 together results in 80, and 80 doubled is 160.
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2
Square the same two sides, and then add the squares together. For this example, 8 squared is 64 and 10 squared is 100. Adding 100 and 64 together results in 164.
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3
Square the length of the third side, and then subtract that square from the sum of the other two squares calculated in Step 2. For this example, the third side is 9. The square of 9 is 81, and 81 subtracted from 164 equals 83.
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4
Divide the difference of squares calculated in Step 3 by the two sides doubled in Step 1. For this example, 83 divided by 160 equals 0.51875.
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5
Find the arccosine, which is the inverse of the cosine function, of the amount calculated in Step 4 to calculate the degrees of the first angle. For this example, the arccosine of 0.51875 is 58.75156 degrees. This is the measurement of one angle of the triangle.
Second and Third Angles
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6
Calculate the sine of the known angle. For this example, the sine of 58.75156 degrees is 0.85493.
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7
Multiply the sine calculated in Step 1 by the measurement of one of the other two sides. For this example, the sine of 0.85493 multiplied by 10 results in 8.5493.
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8
Divide the amount from Step 2 by the length of the third side. For this example, 8.5493 divided by 9 results in 0.94992.
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9
Calculate the arcsine, which is the inverse of the sine function, of the amount calculated in Step 3. For this example, the arcsine of 0.94992 is 71.19004 degrees. This is the measurement of another angle.
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10
Add the two angles together, and then subtract that sum from 180 to calculate the last angle. For this example, 71.19004 added to 58.75156 is 130.5416, and that sum subtracted from 180 results in 49.4584. The last angle is 49.4584 degrees.
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Tips & Warnings
If one angle is already known, then skip to Section 2. If two angles are known, then skip to the last step.
