Does the area formula work for every type of triangle?

Yes. ½ × base × height works for right-angled, acute (sharp) and obtuse (wide) triangles, as long as you use the perpendicular height.

What if the height is outside the triangle?

For an obtuse triangle the perpendicular height can fall outside, measured to an extended base line. The formula still works with that perpendicular distance.

Area of a Triangle | Math

Why a triangle is half a rectangle

Take any rectangle and draw a line from one corner to the opposite corner. You have cut it into two identical triangles. Each triangle is therefore exactly half of the rectangle.

Because the area of a rectangle is base × height, the area of a triangle must be:

Area of a triangle = ½ × base × height

This single formula works for every triangle, no matter its shape.

What counts as the base and the height

The base is any one side you choose.
The height (also called the perpendicular height or altitude) is the straight distance from that base to the opposite corner, measured at a right angle to the base.

The height is not the slanted side. This is the most common mistake. Imagine a flagpole standing straight up from the ground: the ground is the base and the pole's length is the height.

Described diagram: picture a triangle sitting on a flat base 8 cm wide. From the top vertex, a dashed line drops straight down to the base, meeting it at a perfect square corner. That dashed line is the height. If it measures 5 cm, the slanted sides are longer, but you only use the 5 cm dashed line.

Worked example 1: a simple triangle

A triangle has base 8 cm and perpendicular height 5 cm.

Area = ½ × base × height
Area = ½ × 8 × 5
Area = ½ × 40 = 20 cm²

Worked example 2: a right-angled triangle

In a right-angled triangle, the two sides that meet at the right angle are already perpendicular, so one is the base and the other is the height.

A right-angled triangle has short sides 6 cm and 9 cm.

Area = ½ × 6 × 9 = ½ × 54 = 27 cm²

No extra work is needed — the right angle gives you the height for free.

Worked example 3: working backwards

Sometimes you know the area and must find a missing length.

A triangle has area 30 cm² and base 12 cm. Find the height.

Area = ½ × base × height
30 = ½ × 12 × height
30 = 6 × height
height = 30 ÷ 6 = 5 cm

Choosing the easiest base

Any side can be the base, as long as you pair it with the height measured to that side. Choose the base whose height you actually know. If a question gives you a base of 10 cm and a perpendicular height of 4 cm, use those two numbers together — never mix a base from one pair with a height from another.

Activity: count the squares

Draw a triangle on squared (grid) paper with a base of 6 squares and a height of 4 squares. Now:

Use the formula: ½ × 6 × 4 = 12 square units.
Count the full squares inside, then combine the half-squares along the slanted edges.

You should reach 12 both ways. Seeing the count match the formula proves why the ½ is there.

Where this is useful

Builders use triangle areas for roofs and gable ends; designers use them for flags and warning signs. This builds directly on area and perimeter, and it helps to know your types of triangles so you can spot the right base and height every time.

Area of a Triangle

Key takeaways