Area, volume and surface area
The area of a triangle
Draw a rectangle round any triangle and the triangle fills exactly half of it. That is the whole reason the formula has a half in it.
Half of the surrounding rectangle
Draw the smallest rectangle that fits around a triangle. The triangle takes up exactly HALF of it — every time. The rectangle's area is base × height. So: Area of a triangle = ½ × base × height Or in algebra: A = ½bh The half is not a rule to memorise — it is what the picture shows.
The height must be perpendicular
The height in the formula is the PERPENDICULAR HEIGHT: measured at right angles (90°) to the base, from the base to the opposite vertex. On a slanted triangle, the sloping side is NOT the height. It is longer than the perpendicular height, so using it gives an answer that is too big. Look for the little right-angle square in the diagram — that marks the height.
Using the formula
A triangle has a base of 8 cm and a height of 6 cm. A = ½ × b × h A = ½ × 8 × 6 A = 24 cm² Write the formula, substitute, then work it out. And do not forget the units — area is always in SQUARE units.
Compound shapes
A COMPOUND SHAPE is made of simpler shapes joined together. Split it, do each piece, then add. A shape splits into a 5 by 3 rectangle and a triangle with base 5 and height 4: Rectangle: 5 × 3 = 15 cm² Triangle: ½ × 5 × 4 = 10 cm² Total: 15 + 10 = 25 cm²
Worked examples
A triangle with base 8 cm and height 6 cm
- A = ½ × b × h.
- A = ½ × 8 × 6.
- 8 × 6 = 48, and half of 48 is 24.
Answer: 24 cm²
A compound shape: 5 by 3 rectangle plus a triangle base 5, height 4
- Rectangle: 5 × 3 = 15 cm².
- Triangle: ½ × 5 × 4 = 10 cm².
- 15 + 10 = 25.
Answer: 25 cm²
A triangle with base 10 cm and height 7 cm
- ½ × 10 × 7.
- 10 × 7 = 70, and half of 70 is 35.
Answer: 35 cm²
Why is there a half in the formula?
- Draw the rectangle around the triangle.
- The triangle fills exactly half of it.
- The rectangle is base × height.
Answer: Because a triangle is half its surrounding rectangle
A triangle of area 30 cm² with base 12 cm — find the height
- 30 = ½ × 12 × h, so 30 = 6h.
- 30 ÷ 6 = 5.
Answer: 5 cm
Which length is the height?
- The height must be at right angles to the base.
- The sloping side is longer than that.
Answer: The perpendicular height, not the sloping side
Where you see this in real life
- A gable end of a roof is a triangle, and its area decides how much cladding to order.
- Sailmakers cost a sail from its area, which is why the same formula turns up on every sail plan.
Key words
- Base
- the side of a triangle you measure the height from
- Perpendicular height
- the height measured at right angles to the base
- Formula
- a rule written with letters, such as A = ½bh
- Compound shape
- a shape made from simpler shapes joined together
- Vertex
- a corner of a shape