Area of a Triangle
The area of a triangle equals one-half the product of its base and perpendicular height: A = ½bh. In Grade 6 Saxon Math Course 1 (Chapter 8: Advanced Topics in Geometry and Number Operations), students see that every triangle is exactly half of a parallelogram with the same base and height. For a triangle with base 14 cm and height 9 cm: A = ½ × 14 × 9 = 63 cm². The height must be perpendicular to the base (forming a 90° angle), not the length of a slanted side. Students apply the formula in both straightforward calculations and reverse problems where area and one dimension are given.
Key Concepts
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.
Common Questions
What is the formula for the area of a triangle?
A = ½ × base × height (A = ½bh). The height must be the perpendicular distance from the base to the opposite vertex.
Why is the area of a triangle half that of a rectangle or parallelogram?
Any triangle is exactly half of a parallelogram with the same base and height. Two congruent triangles can always be arranged to form a parallelogram.
Find the area of a triangle with base 10 m and height 6 m.
A = ½ × 10 × 6 = 30 m².
A triangle has area 45 cm² and base 9 cm. What is its height?
45 = ½ × 9 × h → 45 = 4.5h → h = 10 cm.
Can you use a slanted side as the height in the triangle area formula?
No. The height must be perpendicular to the base. Using the slant side gives an incorrect (larger) area.