Sides 6 and 8 at 90°: find the third side

In a triangle, given two sides of lengths 6 and 8 with an included angle of 90 degrees, find the length of the third side and the area of the triangle.

Step-by-step

1. Given b = 6, c = 8, included angle A = 90°
   Two sides and their included angle are known; this is the most direct application of the Law of Cosines.
2. a² = b² + c² − 2bc·cosA = 6² + 8² − 2 × 6 × 8 × cos90° = 100
   The Law of Cosines relates three sides to one angle, with a minus sign before the cosine of the included angle.
3. a = √100 = 10
   Take the positive square root; side lengths cannot be negative.
4. S = ½·b·c·sinA = ½ × 6 × 8 × sin90° = 24
   For the area from two sides and an angle, the angle must lie between those two sides; you can back-substitute the third side a = 10 to check the magnitude

Why it works

The Law of Cosines generalizes the Pythagorean theorem: when the included angle is a right angle, its cosine is zero and the formula reduces to the Pythagorean theorem. When two sides and their included angle are known, it is more direct than the Law of Sines because it yields the third side in one step. The area uses the formula with two sides and the sine of their included angle; that angle must be between the two given sides.

Common mistakes

• 1) Writing a plus sign instead of a minus sign in the Law of Cosines; it may accidentally work for a right angle but fails completely for acute or obtuse angles.
• 2) Using the sine of the included angle in the area formula but picking the wrong side, using the third side instead.
• 3) Forgetting to take the positive root when evaluating a square root, resulting in a negative side length.

Related

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