Question

How do you find the height of a scalene triangle when the base and the two side lengths are different?

The measurements are base - 12 cm
one side - 8 cm
the other side - 7 cm

I need the height to find out the area

Answer 1

Although the easy formula for the area is
A = (1/2)b h
that does not help much if you do not know the altitude h

Use Heron's formula
http://www.mathopenref.com/heronsformula.html

Answer 2

if you want the height, drop the altitude to the base. It divides the base into two parts, x and 12-x.
Now, using the Pythagorean Theorem, you can find the height h using

x^2 + h^2 = 8^2
(12-x)^2 + h^2 = 7^2

subtract and you get rid of h, leaving

x^2 - (12-x)^2 = 64-49
24x - 144 = 15
24x = 159
x = 53/8

now use that to find h, and you get

3/8 √143

You can check that against Heron's formula (26.9061).

(1/2)(3/8 √143)*12 = 26.9061