Perpendicular Base and Height
When calculating the area of a triangle, the base and height must be perpendicular—they must form a 90° angle with each other. In Grade 6 Saxon Math Course 1 (Chapter 8: Advanced Topics in Geometry and Number Operations), students learn that the height is the straight-line distance from the base to the opposite vertex measured at a right angle, not the length of a slanted side. A triangle with base 12 cm and perpendicular height 8 cm has area = ½ × 12 × 8 = 48 cm². Students are cautioned not to use the slant side as the height, which is a common error.
Key Concepts
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.
Common Questions
What does perpendicular mean for base and height in a triangle?
Perpendicular means the base and height meet at a 90° angle. The height is always measured straight up from the base, not along a slanted side.
How do you identify the correct height in a triangle?
The height is the straight-line distance from the base to the opposite vertex that forms a right angle with the base. It is often shown with a small square symbol at the corner.
A triangle has base 10 m and perpendicular height 6 m. What is its area?
A = ½ × 10 × 6 = 30 m².
Why can you not use a slanted side as the height?
The area formula A = ½bh requires the height to be perpendicular to the base. A slanted side is longer than the true perpendicular height and gives an incorrect area.
What happens to the perpendicular height of an obtuse triangle?
For an obtuse triangle, the perpendicular height may fall outside the triangle. An altitude line extends from the vertex straight down to a point on an extension of the base.