Area of a Triangle

The space inside a triangle equals half of its base multiplied by its height.

A = 12 b h

The height h is measured straight up from the base b, meeting it at a right angle.

What each symbol means

A: the area (space inside the triangle)
b: the base (any one side you choose to sit it on)
h: the height (the straight-up distance from that base to the opposite corner)

A triangle takes up exactly half the space of a rectangle that shares its base and height. That is the whole idea behind the formula, and once you see it, the "half" stops looking mysterious.

The formula is A = 12 b h. Pick a side to be the base. Measure the height straight up from that base to the far corner. Multiply them, then take half.

Height is not a slanted side: The height must meet the base at a right angle (straight up). It is not the length of a leaning side. For a tall thin triangle, the height can even fall outside the triangle, but it is still measured straight up from the base line.

A worked example

A triangle has base 8 cm and height 5 cm

Write the formula: A = 12 b h.
Put in the numbers: A = 12 × 8 × 5.
Multiply the base and height: 8 × 5 = 40.
Take half: 12 × 40 = 20.
The area is 20 square cm.

Order does not matter: You can multiply in any order. Half of 8 is 4, times 5 is 20. Or 8 times 5 is 40, halved is 20. Same answer. Pick the order that keeps the numbers easy.

The area of a triangle is A = 12 b h.
The height is measured straight up from the base, at a right angle.
A triangle is half of the matching rectangle, which is where the 12 comes from.
Area is in square units.

Why it is half the base times height

Draw any triangle sitting on its base. Now draw the smallest rectangle that fits around it, with the same base and the same height.
The rectangle has area base times height: b × h.
The triangle cuts that rectangle so that the parts left over exactly match the parts of the triangle. Every triangle fills precisely half of its surrounding rectangle.
Half of b × h is 12 b h, and that is the area of the triangle.

Practice questions

Question 1

Find the area of a triangle with base 10 cm and height 6 cm.

Write the formula: A = 12 b h.
Put in the base and height: A = 12 × 10 × 6.
Multiply the base and height first: 10 × 6 = 60.
Take half of that: 12 × 60 = 30.
The area is 30 square cm.

Answer: 30 square cm

Question 2

Find the area of a triangle with base 7 m and height 4 m.

Write the formula: A = 12 b h.
Put in the base and height: A = 12 × 7 × 4.
Multiply the base and height first: 7 × 4 = 28.
Take half of that: 12 × 28 = 14.
The area is 14 square m.

Answer: 14 square m

Question 3

A triangle has base 12 cm and height 9 cm. Find its area.

Write the formula: A = 12 b h.
Put in the base and height: A = 12 × 12 × 9.
Multiply the base and height first: 12 × 9 = 108.
Take half of that: 12 × 108 = 54.
The area is 54 square cm.

Answer: 54 square cm

Question 4

A triangle has area 20 square cm and base 8 cm. What is its height?

Start from the formula and fill in what we know: 20 = 12 × 8 × h.
Work out the number part first: 12 × 8 = 4, so 20 = 4h.
Undo the multiplication by dividing both sides by 4: h = 20 ÷ 4.
That gives h = 5, so the height is 5 cm.

Answer: 5 cm

Question 5

A triangular sail has base 3 m and height 5 m. How much fabric does it need?

The fabric needed is the area of the triangle, so use A = 12 b h.
Put in the base and height: A = 12 × 3 × 5.
Multiply the base and height first: 3 × 5 = 15.
Take half of that: 12 × 15 = 7.5.
The sail needs 7.5 square m of fabric.

Answer: 7.5 square m