Area & Perimeter

SPEC G11 G14 G16 — GEOMETRY & MEASURES

Triangles, quadrilaterals, composite 2D figures and coordinate geometry — the formulae you must memorise and the perpendicular-height trap that catches every year of students.

What You'll Learn

This is a high-priority topic — area and perimeter questions appear on almost every paper, often as part of a longer composite-shape problem.

By the end of this lesson, you will be able to:

Recall and apply the area formulae for rectangles, triangles, parallelograms, trapezia, rhombuses and kites

Identify the perpendicular height of any 2D shape and avoid the slant-height trap

Find the area of composite shapes using the split and subtraction methods

Calculate perimeters by working out missing side lengths first

Find the area of a triangle defined on a coordinate grid using the box-minus-triangles technique

Quote answers in the correct square units (cm², m², mm²)

Why It's High Priority

Area appears every year — sometimes as a quick 2-mark setter, sometimes as a 4–5 mark composite question. The marks lost are nearly always the same ones: forgetting the factor of ½ on a triangle, using a slant length instead of the perpendicular, or omitting the ² on the units. Get those three right and you bank the marks every time.

Core Area Formulae

These four formulae must be memorised — they are not given on the formula sheet.

ShapeArea FormulaWhat to identify

Rectanglelength × widthTwo perpendicular sides

Triangle½ × base × perpendicular heightBase and height at right angles

Parallelogrambase × perpendicular heightBase and the perpendicular distance to the parallel side

Trapezium½(a + b) × hTwo parallel sides a, b and the perpendicular distance h between them

The "Half" Trap

The triangle formula has a factor of ½. The parallelogram formula does not. They look similar — base × height — but the triangle is exactly half of the parallelogram with the same base and height. Forgetting the ½ on a triangle is the single most common error in this topic.

Perpendicular Height — Not Slant

The "height" in every area formula means the perpendicular height — the distance measured at 90° to the base. The slant side of the shape is usually longer, and using it gives the wrong area.

Correct vs Incorrect

Area = base × perpendicular height = 8 × 4 = 32 cm² ✓

Wrong: 8 × 5 = 40 cm² (used the slant — overestimates by 25%)

How to Spot It

If a length is drawn along the side of the parallelogram, it is the slant. If it is drawn as a dashed line at 90° between the two parallel sides — often with a small right-angle square — it is the perpendicular height. Look for the right-angle marker.

01Worked Example — Trapezium

A trapezium has parallel sides of 8 cm and 14 cm. The perpendicular distance between the parallel sides is 6 cm. Find its area. (3 marks)

MMethod. Use the trapezium area formula: Area = ½(a + b) × h, where a and b are the parallel sides and h is the perpendicular distance between them.

CCalculate.

Area = ½ × (8 + 14) × 6

Area = ½ × 22 × 6

Area = 11 × 6 = 66

EEvaluate. Sense-check: a rectangle 14 × 6 = 84 cm². The trapezium is smaller than the rectangle, so 66 cm² is sensible.

Area = 66 cm²

Reverse Problem (cross-reference 2.25)

If the area is given and one length is unknown — e.g. "the trapezium has area 66 cm², parallel sides 8 cm and 14 cm; find h" — substitute into ½(a + b) × h = 66 and solve as an equation: ½ × 22 × h = 66, so 11h = 66, giving h = 6 cm. This combines G16 with the equation-solving from Lesson 2.25.

Rhombus & Kite — Use the Diagonals

Rhombuses and kites have a different formula. Instead of base × height, use the diagonals:

Area = ½ × d₁ × d₂

where d₁ and d₂ are the lengths of the two diagonals.

Why ½ × diagonals works

A rhombus or kite can be split into four right triangles by its two diagonals. The diagonals always cross at 90°, so each triangle has legs of half each diagonal. Adding up the four triangles gives ½ × d₁ × d₂.

02Worked Example — Triangle (slant trap)

The triangle below has a base of 12 cm. The slant side is 13 cm and the perpendicular height is 5 cm. Find the area. (2 marks)

MMethod. Use Area = ½ × base × perpendicular height. Identify which length is at right angles to the base — that is the perpendicular. The 13 cm slant side runs along the edge of the triangle, so it is not the perpendicular height.

CCalculate.

Base = 12, perpendicular height = 5

Area = ½ × 12 × 5 = 6 × 5 = 30

EEvaluate. Common wrong answer: ½ × 12 × 13 = 78. That uses the slant — and the slant is always longer than the perpendicular, so 78 is too big.

Area = 30 cm²

03Worked Example — Kite

A kite has diagonals of length 8 cm and 10 cm. Find its area. (2 marks)

MMethod. Use the kite (and rhombus) formula: Area = ½ × d₁ × d₂, where d₁ and d₂ are the two diagonals.

CCalculate.

Area = ½ × 8 × 10

Area = ½ × 80 = 40

EEvaluate. Square units: cm × cm = cm².

Area = 40 cm²

Composite Shapes — Split Method

A composite shape is built from two or more standard shapes. The split method handles them by cutting the composite into pieces, finding the area of each, and adding the results.

The Split Routine

Draw a line that divides the shape into rectangles, triangles or trapezia

Mark the dimensions of each piece — fill in any missing lengths first

Find the area of each piece using the standard formula

Add the areas together

State the final answer with square units

Choosing the Splits

There is usually more than one way to split a shape — any valid split works, as long as each piece is a shape whose formula you know. Pick the splits that involve the fewest unknown lengths to fill in.

04Worked Example — L-Shape (split)

Find the area of this L-shape. All angles are right angles. (3 marks)

MMethod. Split the L-shape into two rectangles. Rectangle A is the tall left strip (4 cm by 8 cm). Rectangle B is the short bottom-right strip (6 cm by 3 cm). The 3 cm came from 8 − 5 = 3 (the height below the notch).

CCalculate.

Area A = 4 × 8 = 32 cm²

Area B = 6 × 3 = 18 cm²

Total = 32 + 18 = 50 cm²

EEvaluate. Cross-check using the subtraction method: outer rectangle 10 × 8 = 80; notch 6 × 5 = 30; 80 − 30 = 50. ✓

Area = 50 cm²

Composite Shapes — Subtraction Method

When a shape has a piece cut out of it (a hole or a notch), it is usually faster to subtract:

Shaded Area = Outer Shape Area − Cut-Out Area

When to subtract

The shape is described as a "shaded region" with a piece removed

You can fit the whole shape inside a single rectangle (the bounding box)

The cut-out is a single recognisable shape — circle, triangle, rectangle

Both Methods Should Agree

If you have time, run a quick check using the other method. The split-method total and the subtraction-method total must match. If they do not, you have made an arithmetic slip somewhere — find it before moving on.

05Worked Example — Rectangle Minus Triangle

The diagram shows a rectangle 12 cm by 8 cm with a triangular piece cut out of the top edge. The cut-out triangle has a base of 6 cm and a perpendicular height of 4 cm. Find the shaded area. (3 marks)

MMethod. Use the subtraction method. Find the area of the full outer rectangle, then subtract the area of the triangular cut-out.

CCalculate.

Rectangle = 12 × 8 = 96 cm²

Triangle cut-out = ½ × 6 × 4 = 12 cm²

Shaded = 96 − 12 = 84 cm²

EEvaluate. The cut-out is a small fraction of the rectangle (12 ÷ 96 ≈ 12.5%) — so the shaded area being just under 96 is sensible.

Shaded Area = 84 cm²

Perimeter — Find Missing Lengths First

Perimeter is the total distance around the outside of a shape. Just add up every side — but on composite shapes, some side lengths are not labelled. Work them out from the labelled sides before adding.

Filling in Missing Sides

For a shape with all right angles:

The total of all horizontal top sides equals the total of all horizontal bottom sides

The total of all vertical left sides equals the total of all vertical right sides

Use these two rules to fill in any unlabelled side, then sum every side once for the perimeter.

Don't Confuse Area and Perimeter

Area is in square units (cm², m²). Perimeter is in linear units (cm, m). Reading "find the perimeter" but giving a cm² answer is an automatic mark loss.

06Worked Example — L-Shape Perimeter

The same L-shape from Example 4 has an outer width of 10 cm, an outer height of 8 cm, and a notch of 6 cm by 5 cm cut from the top-right corner. Find its perimeter. (3 marks)

MMethod. Find every side length first, then sum them. The 3 cm comes from 8 − 5; the 4 cm comes from 10 − 6. Sum all six sides.

CCalculate.

Going clockwise from the top-left corner:

4 + 5 + 6 + 3 + 10 + 8 = 36

EEvaluate. Check using horizontals and verticals. Horizontal total: 4 + 6 (top notch) + 10 (bottom) = 20, which equals 2 × 10. Vertical total: 5 + 3 (right) + 8 (left) = 16, which equals 2 × 8. ✓ Linear units (cm), not square units.

Perimeter = 36 cm

Area on a Coordinate Grid

If a triangle is given by its three vertex coordinates, you do not need to measure anything. Use the box-minus-triangles technique.

The Routine

Read the three vertex coordinates from the grid

Draw the smallest possible bounding box (rectangle aligned with the axes) that contains the triangle. Its corners are at (x_min, y_min), (x_max, y_min), (x_max, y_max), (x_min, y_max)

Calculate the box area: width × height

Identify the three right triangles sitting between each side of your triangle and the corner of the box

Find the area of each right triangle (½ × horizontal × vertical leg)

Subtract the three right-triangle areas from the box area — the result is the triangle area

Why It Works

The bounding box is split by the triangle into four regions: your target triangle in the middle, plus three right triangles in the corners. Adding all four gives the box. Subtracting the three corner triangles leaves your target.

07Worked Example — Triangle from Coordinates

A triangle has vertices at A(1, 1), B(6, 3) and C(3, 7). Find the area of the triangle. (4 marks)

MMethod. Use the box-minus-triangles technique. The bounding box runs from (1, 1) to (6, 7), giving width 5 and height 6. Three right triangles sit in the corners of the box around the target triangle.

CCalculate.

Box area = 5 × 6 = 30

Right triangle 1 (between A and B, bottom): legs 5 and 2, area = ½ × 5 × 2 = 5

Right triangle 2 (between B and C, top right): legs 3 and 4, area = ½ × 3 × 4 = 6

Right triangle 3 (between C and A, left): legs 2 and 6, area = ½ × 2 × 6 = 6

Triangle ABC = 30 − (5 + 6 + 6) = 30 − 17 = 13

EEvaluate. The target triangle clearly takes up roughly half the bounding box (30 ÷ 2 = 15) — and 13 is close to that, so the answer is sensible.

Area = 13 square units

Examiner Alerts

The same handful of errors lose marks on this topic every series. Memorise the fixes:

Slant Instead of Perpendicular

What students write: Use the slant side of a parallelogram or triangle as the height. What to write instead: Identify the dashed line drawn at 90° between the parallel sides — that, and only that, is h.

Forgetting the Factor of ½

What students write: base × height for a triangle. What to write instead: ½ × base × perpendicular height. The triangle is exactly half the parallelogram with the same base and height.

Missing Square Units

What students write: Area = 66 cm. What to write instead: Area = 66 cm². Area is always in square units. The "²" is the mark for units, and it is one of the easiest marks on the paper to lose.

Mixed Units in the Same Calculation

What students write: Multiply 80 cm by 1.5 m to get an area in "cm·m". What to write instead: Convert everything to a single unit first (e.g. 80 cm × 150 cm = 12 000 cm², or 0.8 m × 1.5 m = 1.2 m²). One mixed-unit slip can invalidate the entire calculation.

Diameter Instead of Radius (lookahead to circles)

Composite questions sometimes mix circle and polygon parts (e.g. a rectangle with a semicircle attached). When the circle bit comes, use the radius in πr² — if you are given the diameter, halve it first. Lesson 4.13 covers this in detail.

Lesson Checkpoint

You should now be able to:

Core Formulae (memorised)

Rectangle: length × width

Triangle: ½ × base × perpendicular height

Parallelogram: base × perpendicular height

Trapezium: ½(a + b) × h

Rhombus / Kite: ½ × d₁ × d₂

Composite Shapes

Split method: divide into rectangles/triangles/trapezia, sum the parts

Subtraction method: outer rectangle − cut-out

Both methods should agree — use the second as a check

Perimeter and Coordinates

Perimeter: sum every side; fill in missing sides using horizontal-equals-horizontal and vertical-equals-vertical

Coordinate triangle area: bounding box minus the three corner right-triangles

Units: area is always cm², m², mm² — perimeter is always cm, m, mm

Up Next

Lesson 4.13 covers Circles & Sectors: circumference, area, arc length and sector area — including how to keep π exact in your final answer.