Lesson 79: Area of a Triangle

New Concept

The area of a triangle can be determined by finding half of the product of its base and height.

Why it matters

Mastering the area of a triangle is a key step toward deconstructing any complex shape, from rocket fins to architectural designs. This simple formula introduces the powerful idea that geometric properties can be described and manipulated with symbolic language.

What’s next

Next, you'll apply this formula to calculate the area of different types of triangles, including right triangles and those within parallelograms.

The area of a triangle is half the area of a parallelogram with the same base and height. We calculate this with the formula $A = \frac{1}{2}bh$, where 'b' is the base and 'h' is the perpendicular height.

A triangle has a base of 10 cm and a perpendicular height of 6 cm. Its area is $A = \frac{1}{2}(10 \text{ cm})(6 \text{ cm}) = 30 \text{ cm}^2$.
For a triangle with a base of 5 inches and a height of 8 inches, the area is $A = \frac{1}{2}(5 \text{ in})(8 \text{ in}) = 20 \text{ in}^2$.

Think of a parallelogram—it's made of two identical triangles! To find a triangle's area, you find the area of its 'big brother' parallelogram (base times height) and slice it in half. It’s that simple and a neat geometric trick.

When calculating the area of any triangle, the base and height must be perpendicular measurements. This means the lines representing the base and height must intersect to form a right angle ($90^\circ$), like the corner of a square.

• In a right triangle, the two sides forming the right angle are the base and height. For legs of 3 m and 4 m, the area is $A = \frac{1}{2}(3 \text{ m})(4 \text{ m}) = 6 \text{ m}^2$.
• For a triangle with a slanted side of 12 cm but a perpendicular height of 9 cm from its 10 cm base, use the correct height: $A = \frac{1}{2}(10 \text{ cm})(9 \text{ cm}) = 45 \text{ cm}^2$.

You can't measure a triangle's height along a slanted side! The height must be the straight-up distance from the base to the opposite vertex. Think of it as measuring a person's height—they have to stand up straight, not lean over.

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!