Triangle Area Calculator (base and height)

The base-times-height formula works for every triangle, not just right triangles.

A = ½ × b × h

Where b is any side (the base) and h is the perpendicular distance from that side to the opposite vertex. The trick is the word “perpendicular.” If you measure h along the slanted side instead of straight down to the base, you get the wrong number.

Two common ways to mess this up:

1. Using a side length instead of the perpendicular height. Take an isosceles triangle with two 5 cm sides and a 6 cm base. The height is not 5, it is exactly 4, because the apex sits above the midpoint of the base and 3-4-5 closes the right triangle. Use 5 and the area comes out 15 instead of 12, a quarter too high.

2. Picking a height that falls outside the triangle. On an obtuse triangle the perpendicular from the apex can land beyond the end of the base. That is still the right height. Extend the base line and measure to where the perpendicular meets it.

Where this comes up in real work:

• Roofing: the triangular gable end of a house. A 30 ft wide gable with a 12 ft peak height has area 0.5 × 30 × 12 = 180 sq ft. That’s the siding or paint area you need for one gable.
• Sail-making: triangular sails are sized this way. A 25 ft luff (the vertical leading edge) and a 12 ft foot give roughly 150 sq ft, though a real sail is cut with curved edges rather than straight ones.
• Garden beds: a corner garden bed that’s a right triangle 6 ft along the fence and 4 ft out has 12 sq ft of planting area.

If you don’t know the perpendicular height:

Use Heron’s formula instead, which works from the three sides alone. Or use the SAS formula: A = ½ × a × b × sin(C), where C is the angle between sides a and b.

Worked example: banner cut

You are cutting a triangular party banner. Base 36 in, height 24 in. Area = 0.5 × 36 × 24 = 432 sq in = 3 sq ft. Add 2 inches of seam allowance on each edge and the fabric you actually cut is larger.

Why the ½

A triangle is exactly half the rectangle drawn around it, using the base as the width and the perpendicular height as the height. For a right triangle you can see it straight away: cut the rectangle along its diagonal and the two halves are congruent.

For any other triangle the picture takes one more step. Drop the perpendicular from the apex and it splits the triangle into two right triangles, each of which is half of its own sub-rectangle. Add them and you still have half the whole. That works for an obtuse triangle too, where one of the two sub-rectangles is subtracted rather than added.

The consequence is worth carrying: any two triangles with the same base and the same height have the same area, no matter how far the apex slides sideways. Shear a triangle and nothing changes. It looks wrong and it is the basis of half of classical geometry.