How to Find the Leg of an Isosceles Triangle When the Altitude Isn't Known

The hypotenuse is opposite the right angle of a right triangle.

An isosceles triangle has two equal sides called legs. The third side is the base, and the perpendicular line that connects the base to the point at which the two legs meet is the altitude. The altitude cuts the base and the isosceles triangle in half to form two right triangles within the isosceles triangle -- of which each leg is a hypotenuse. If you don't know the altitude of an isosceles triangle, you can use trigonometry to find the length of a leg if you know the base and one of the angles between the base and leg.

Instructions

1. Determine the base of an isosceles triangle and one of the angles between the leg and base. For example, assume the base of an isosceles triangle is 49 inches and the angle between the base and leg is 30 degrees.

2. Divide the length of the base by 2. This will represent one of the sides of one of the right triangles in the isosceles triangle. In the example, divide 49 by 2, which equals 24.5 inches.

3. Substitute your result for "adjacent" and substitute the angle into the cosine equation, which is: cos(angle) = adjacent/hypotenuse. In the equation, "cos" represents the trigonometric function cosine; "angle" represents the angle of a right triangle; "adjacent" represents the side that is adjacent to the angle; "hypotenuse" represents the side of the right triangle opposite the right angle. In the example, substitute your result and the angle, which results in cos(30) = 24.5/hypotenuse.

4. Calculate the cosine of the angle on a scientific calculator. In the example, the cosine of 30 degrees is 0.87. This leaves 0.87 = 24.5/hypotenuse.

5. Divide the number on the right side of the equation by the number on the left side of the equation to solve for the hypotenuse. In the example, 24.5 divided by 0.87 equals 28.2. This is the length of the hypotenuse, which is also the length of the leg of the isosceles triangle.

References

• "Elementary Geometry for College Students"; Daniel C. Alexander and Geralyn M. Koeberlein; 2010
• Math Open Reference; Isosceles Triangle; John Page; 2009
• MathsIsFun.com; Sine, Cosine and Tangent

Resources

• Math.com; Scientific Calculator