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# How to Calculate the Height of a Triangle

Updated July 19, 2017

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.

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

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 labelled "c," and the two sides are called "a" and "b."

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

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

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.

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

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

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

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

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

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