An Equivalent Area Formula
An equivalent area formula for triangles shows that A = ½ × base × height is the same as A = (base × height) ÷ 2 in Grade 6 math (Saxon Math, Course 1). Multiplying by ½ and dividing by 2 are equivalent operations, so both forms give the correct answer. The form A = bh/2 is often easier to calculate, especially when base × height gives an even number. For example, a triangle with base 8 and height 5: A = (8 × 5) / 2 = 40 / 2 = 20 square units. This formula also shows that a triangle is exactly half the area of a rectangle with the same base and height — a key geometric insight for visual proofs.
Key Concepts
Because multiplying by $\frac{1}{2}$ and dividing by 2 are equivalent operations, the area formula for a triangle can also be written as $A = \frac{bh}{2}$. This form can be easier to calculate with, especially with even numbers.
A triangle has a base of 8 meters and a height of 5 meters. Its area is $A = \frac{(8 \text{ m})(5 \text{ m})}{2} = \frac{40 \text{ m}^2}{2} = 20 \text{ m}^2$. Using a right triangle with a base of 6 inches and a height of 8 inches, the area is $A = \frac{(6 \text{ in})(8 \text{ in})}{2} = \frac{48 \text{ in}^2}{2} = 24 \text{ in}^2$.
If fractions make your head spin, just use this formula instead. Multiply the base and height together first to get one solid number, then simply divide that result by two to get the final area. Same destination, different road!
Common Questions
What are the two equivalent forms of the triangle area formula?
A = ½ × b × h and A = (b × h) ÷ 2. Both give the same result because multiplying by ½ equals dividing by 2.
Why is A = bh/2 sometimes easier to use than A = ½bh?
When b × h is an even number, dividing by 2 is simple. For example, 8 × 5 = 40; 40 ÷ 2 = 20. Working with halves of odd products requires fractions.
What is the area of a triangle with base 12 cm and height 7 cm?
A = (12 × 7) / 2 = 84 / 2 = 42 cm².
Why is a triangle's area half of a rectangle's area?
Any triangle can be completed into a rectangle (or parallelogram) with the same base and height. The triangle is exactly half that shape.
Does the height in the triangle formula have to be perpendicular to the base?
Yes. The height must be the perpendicular distance from the base to the opposite vertex, not the slant height of a side.