` Three Ways to Find the Area of a Triangle

Three Ways to Find the Area of a Triangle

We all know the usual formula for the area of a triangle:

(1) area A = 1/2baseheight.

There are two other formulas that are fairly common.
You can find the area of a triangle when you know the lengths of the three sides.
Heron of Alexandria (born about 10 in (possibly) Alexandria, Egypt; died about 75) proved Heron's formula for a triangle with sides a, b, and c:

(2) Area = √(s(s-a)(s-b)(s-c)), where s = semiperimeter = (a + b + c)/2.
(You can find a proof of Heron's formula at http://jwilson.coe.uga.edu/emt725/Heron/Heron.html)

(3) Suppose you know the measure of one angle A in the triangle, and the length of the two sides adjacent to (on either side of) the angle, b and c. Then area of the triangle is 1/2bcsin A. (See below for a proof of this formula.)

Let's try out these three formulas on three congruent triangles and see if we get (approximately) the same area in all three cases.

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Webpage implementation by Elizabeth K. White l last modified : March 24, 2011