The area of a triangle formula can be derived using a combination of geometry and the area of a rectangle formula. The three cases of the derivation are shown below corresponding to the three types of triangles: right triangle, acute triangle and obtuse triangle.
The first case of a triangle is the right triangle case. Here we can observe that the rectangular region defined by the base and height can be divided into two congruent right-triangles.
Solving for the area of the right triangle we are left with the area of triangle formula verifying the first case.
The second case is an acute triangle where all three angles of the triangle are acute. In this case, we can divide the acute triangle into two right triangles as shown in the illustrations.
We can represent the area of the acute triangle as the sum of the two right triangles, shown below.
Then we can simplify factoring out the common factor of leaving us the expression below.
Then since we can substitute the variable back into the equation leaving us with the formula. This verifies the second case.
The final case corresponds to the obtuse triangle shown in the illustration above. Here we can represent the area of the triangle as the area of the rectangle defined by the base and a height of minus the area of the two right triangles colored green. This is shown in the expression below.
Expanding the expression gives us the following expression.
Combining like expressions
Finally, we are left with the formula for area of the triangle.
This completes the derivation of the formula for the three types of triangles: right, acute and obtuse.