Question

How do you find the area of a triangle given two sides?

Answer 1

Using the Pythagorean Theorem or Special Right Triangles. In this case, it will most likely be Pythag. Theorem.

Explanation:

Let's say you have a triangle,

Both legs are 3.

You would use the equation:
`a^2 + b^2 = c^2`

The hypotenuse is always the sum of the two legs.
Legs = `a,b`
Hypotenuse = `c`

So plug it in:
`3^2 + 3^2 = c^2`

Solve to get your answer (In this case would be `3`).

`9 + 9 = c^2`
`18 = c^2`
`3sqrt(2) = c`

This can also work for finding legs, just make sure to plug in the correct numbers in the correct spots.

Answer 2

You can't; given two sides a`, b` a triangle can have any area from zero to `1/2 ab`, which we get when `a` and `b` are at right angles.

Explanation:

Archimedes' Theorem is a modern form of Heron's Formula. It relates the area of a triangle `mathcal{A}` to the length of its sides `a,b,c:`

`16 mathcal{A} ^2 = 4a^2b^2 - (c^2 - a^2 - b^2)^2`

For a given `a,b` we get a maximum area when the squared term is zero, i.e. when `c^2=a^2+b^2,` i.e. a right triangle.

We can get a degenerate triangle (zero area) when `c= |a \pm b|` as we can verify by plugging into Archimedes. Let's just check the area when `c=a+b`.

`16 mathcal{A}^2 = 4a^2 b^2 - ((a+b)^2-a^2-b^2)^2= 4a^2b^2 - (2ab)^2 = 0 quad sqrt`

A real triangle can't have zero area; it must be positive.