` 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 = 1/2*base*height.

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 1/2*b*c*sin A. (See below for a proof of this formula.)

AreaOfTriangleUsingTrig.jpg

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

ThreeTrianglesForTrigWithLetters.jpg

Click here to get a printable version of the picture above


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