Every triangle has area equal to half a base multiplied by its corresponding perpendicular height. The triangle’s type determines how you find those measurements; it does not restrict the validity of the basic formula. Any of the three sides can serve as a base.
Match each base with its own height
The height runs from the opposite vertex to the line containing the chosen base and meets it at 90°. In an obtuse triangle, its foot may lie beyond the side itself. Measuring along a sloping edge instead of perpendicular to the base generally overestimates the area.
A right triangle already has perpendicular sides
The two legs meet at a right angle, so they can directly supply base and height. The hypotenuse is opposite that angle; it is not automatically the base for an area calculation. If you choose it as the base, you need its corresponding altitude.
Equilateral and isosceles triangles: find the height first
In an equilateral triangle of side a, the altitude halves the base and creates two right triangles. Applying the Pythagorean theorem gives the height and then the area below.
In an isosceles triangle with equal sides a and base b, the same construction uses half the base but retains a as the hypotenuse. The equal sides must be longer than half the base for a nondegenerate triangle.
Worked example: equal sides of 13 cm and base of 10 cm
The altitude divides the 10 cm base into two 5 cm segments. Subtract the square of 5 from the square of 13, then take the positive square root: the height is 12 cm. Use the full base, not its half, in the area formula.
What if only two sides and their included angle are known?
For sides b and c enclosing angle α, their sine relationship supplies the perpendicular height. Substituting that height gives the following area formula. It works for scalene triangles as well as other triangles, provided α is the angle between the two specified sides.
Use degrees or radians consistently with your calculator’s angle setting. Express the final area in square units. For choosing between different shapes rather than different triangle types, see area formulas for plane shapes.
