How to Figure the Height of a Triangle

All triangles' areas are proportional to their heights.

In some triangles, you can easily calculate the height from the shape's other dimensions. In right triangles, the height is the square root of the difference between the base and hypotenuse's squared lengths. When you know the angles in the triangle, you can calculate the height using trigonometry. But if you know the triangle's area, you can calculate the height from that measurement and the shape's base. Heron's formula lets you find this area without previously knowing the triangle's height.

Instructions

1. Add together the three sides' lengths and divide the sum by 2. If, for instance, the base is 6 inches and the other lengths are 7 and 11 inches, the equation would be (6 + 7 + 11) ÷ 2 = 12.

2. Subtract each length from this answer: 12 - 6 = 6, 12 - 7 = 5 and 12 - 11 = 1.

3. Multiply together these three values: 6 --- 5 --- 1 = 30.

4. Multiply this answer by the answer from Step 1: 30 --- 12 = 360.

5. Find the square root of this answer: √360 = 18.97 square inches. This is the triangle's area.

6. Divide the area by the length of the triangle's base: 18.97 ÷ 6 = 3.16.

7. Multiply this answer by 2: 3.16 --- 2 = 6.32 inches. This measurement is the triangle's height.

References

• University of Georgia; Problem Solving With Heron's Formula; James W. Wilson; 1986
• University of Kentucky; Heron's Formula for Triangular Area; Christy Williams, et al.

Resources

• University of Illinois: Triangle Area Calculator
• University of Virginia: Geometrical Proof of Heron's Formula