How to Calculate Acute Angles

A right triangle has one 90-degree angle and two acute angles.
••• Wooden Triangle image by Towards Ithaca from

A right triangle is any triangle with a right, or 90-degree, angle. Because the angles in a triangle must total 180 degrees, the remaining two angles are acute, which means they are less than 90 degrees. Trigonometry primarily concerns itself with measurements and ratios of this special type of triangle. Sine, cosine and tangent are ratios that center on the acute angles of a right triangle. You can use these ratios to calculate the angles.

    Orient the triangle so that one leg of the 90-degree angle is vertical. Label this leg "a." The other leg of the 90-degree angle will be horizontal. Label that leg "b." Label the third side, the hypotenuse, "c."

    Measure the length of the three sides. In some applications, you will only be able to measure sides "a" and "b." If this is the case, use the Pythagorean Theorem to calculate side "c."

    Example: a = 3; b = 4 a^2 + b^2 = c^2 (Pythagorean Theorem) 9 + 16 = 25 c = 5

    Divide the length of side "a" by the length of the hypotenuse, side "c." This is the sine of the acute angle that shares the horizontal leg with the right angle. Enter this ratio into your scientific calculator and use the inverse sine function to determine the angle.

    Example: a = 3; c = 5 sine = 3/5 angle 1 = 36.87 degrees

    Add 90 degrees to this angle and subtract the result from 180. This will be the value for the second acute angle in the right triangle.

    Example: 90 + 36.87 = 126.87 180 - 126.87 = 53.13 angle 2 = 53.13 degrees

    Things You'll Need

    • Ruler
    • Scientific calculator


    • Scientific calculators can calculate angles in different units, such as degrees and radians. Your calculator's units must be set to "degrees" for this operation.

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