Area of Basic Shapes

Before memorising any formula, it helps to understand where it comes from. Every area formula connects back to the most basic one: the rectangle.

Rectangle: $A = \ell w$ This is the foundation. A rectangle with length 5 cm and width 3 cm contains exactly 15 unit squares. Area counts those squares.

Triangle: $A = \tfrac{1}{2}bh$ A triangle is exactly half a rectangle with the same base and height. The $\tfrac{1}{2}$ is always there.

Parallelogram: $A = bh$ Slice a triangle off one end and reattach to the other, you get a rectangle with the same base and height.

Trapezium: $A = \tfrac{1}{2}(a + b)h$ The formula averages the two parallel sides ($\tfrac{1}{2}(a + b)$), then multiplies by the height. Think of it as a rectangle with an average width.

Circle: $A = \pi r^2$ The area depends on the square of the radius. Double the radius and the area quadruples, not doubles.

The perpendicular height problem

The single most common error across all of these formulas is using the slant side instead of the perpendicular height. The height $h$ must be measured at 90° to the base.

Every area formula traces back to base × perpendicular height. The height must be perpendicular (90°) to the base, using the slant side gives a larger, wrong answer. Always identify the perpendicular height before substituting.

Pause, copy the perpendicular height rule (height must be 90° to the base) and the origin of each area formula (½bh for triangle, ½(a+b)h for trapezium) into your book.

We just saw that every area formula traces back to base × perpendicular height, and that using the slant side instead of the perpendicular height gives the wrong answer. That raises a question: most real shapes aren't a single triangle or rectangle, how do you find the area of a shape made from several parts? This card answers it → split the composite shape into standard parts, calculate each area separately with the correct formula, then add or subtract.

A composite shape is any shape that is not one of the five standard shapes. The strategy is always the same:

1. Sketch the shape and identify the parts
2. Decide: are you joining (add) or removing (subtract)?
3. Write your plan: e.g. "Area = rectangle + triangle" before touching numbers
4. Calculate each part separately
5. Combine the areas and state the unit

Calculator use for circle problems

Using 3.14 instead of the $\pi$ button introduces rounding error early, and that error compounds through the rest of your calculation.

When the question gives you the diameter, write $r = d \div 2$ as a separate line before substituting into $A = \pi r^2$. Substituting the diameter directly gives an answer four times too large (because $(2r)^2 = 4r^2$).

Composite shapes are solved by adding or subtracting standard shapes. Identify the components, calculate each area separately using the correct formula, then combine. Circles use π, store the full value and round only at the end.

Pause, copy the composite shape strategy (identify parts → calculate each → add or subtract) and the pi rounding rule (store full value, round only the final answer) into your book.