How to Calculate the Height of a Triangle

Equilateral Triangle

Triangles are everywhere and learning how to measure them is helpful in the everyday world. Use the Pythagorean theorem to measure the height of right triangles. If you have a triangle with all sides equal (equilateral triangle), or an isosceles triangle (two sides equal), then divide it into half vertically to get two right triangles. Use the method suited to the information given.

Things You'll Need

• Calculator
• Protractor
• Ruler
• Pencil

Instructions

Equilateral and Isosceles Triangles Using Pythagorean Theorem

1. Draw a straight line from the highest point of the triangle to the base.

2. Apply the Pythagorean theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides. As an equation it is written as: c^2=a^2+b^2. Per convention, the hypotenuse is labeled "c," and the two sides are called "a" and "b."

3. Solve for "a": a^2=c^2-b^2, or "b": b^2=c^2-a^2.

4. Input the information you have. For example, let "b" = the line drawn from the top to the base, let "c" = 10 cm and "a" = 6 cm. Then:
   b^2=10^2-6^2
   b^2=100-36
   b^2=64

5. Find the square root of "b": √b=8. We need to find the square root because we are looking for the length of one side.

Triangle Using Heron's Formula

1. Write down Heron's formula: a=√p(p-a)(p-b)(p-c) where "p" is the half perimeter: a+b+c/2.

2. Find the value of "p." For example, let us take the sides to measure 4 cm, 5 cm and 6 cm, with the base being 4 cm. Then:
   p=4+5+6/2
   p=15/2
   p=7.5

3. Plug in the value of "p" in the formula.
   a=√7.5(7.5-4)(7.5-5)(7.5-6)
   a=√7.5(3.5)(2.5)(1.5)
   a=√98.4375
   a=9.9216

4. Rewrite the area formula to solve for height: a=1/2b*h. So h=2*a/b.

5. Plug in the value of area in the formula.
   h=2*9.9216/4
   h=19.8432/4
   h=4.9068

References

• Clark University: Right Triangles
• Math Open Reference: Heron's Formula