2.4 Perimeter, Area & Compound Shapes
Key Takeaways
- Perimeter is a length measured in linear units and area is a surface measured in square units, so an answer given in the wrong unit is wrong even when the number is right.
- The four area formulas that cover almost every item are rectangle (length x width), triangle (half base x height), parallelogram (base x height) and trapezium (half the sum of the parallel sides x height).
- Triangle and parallelogram areas use the perpendicular height, never the slanted side, which is the single most common error in this topic.
- Compound shapes are solved by finding the missing edges from opposite parallel sides first, then splitting the figure into non-overlapping rectangles or subtracting an unshaded region.
- Shapes with the same perimeter can have very different areas, so questions asking for the largest area from a fixed length of fencing are testing that relationship directly.
Perimeter, Area & Compound Shapes
Perimeter and area appear on every 11+ maths paper, and the questions are almost always diagram-led: a shape with some edges labelled and one measurement missing. The arithmetic is easy. The marks are lost on units, on perpendicular height, and on missing edges that had to be worked out first.
Unit conversion is covered in section 2.7, and angles, coordinates and volume in section 2.8. This section is about the flat-shape measurements themselves.
Perimeter and Area Are Different Kinds of Measurement
- Perimeter ($P$) is the total distance around the outside boundary of a 2D shape. It is a length, so it is measured in linear units: $\text{mm}, \text{cm}, \text{m}, \text{km}$.
- Area ($A$) is the amount of surface enclosed inside the boundary. It is measured in square units: $\text{cm}^2, \text{m}^2$.
Writing $\text{cm}$ where $\text{cm}^2$ belongs is treated as an error, and in a multiple-choice paper the wrong unit is usually offered as a distractor beside the right number. Say the unit out loud as you choose the option.
The Standard Formulas
| Shape | Area formula | Perimeter |
|---|---|---|
| Square | $A = s \times s$ | $P = 4s$ |
| Rectangle | $A = l \times w$ | $P = 2(l + w)$ |
| Triangle | $A = \frac{1}{2} b \times h$ | Add the three sides |
| Parallelogram | $A = b \times h$ | $2(\text{base} + \text{slant side})$ |
| Trapezium | $A = \frac{a + b}{2} \times h$ | Add all four sides |
Perpendicular Height Is Not the Slanted Side
In the triangle and parallelogram formulas, $h$ is the perpendicular height — the straight-up distance from the base to the opposite vertex or side, measured at a right angle. Diagrams deliberately label the slanted side as well, because that number is the distractor.
A triangle has a base of $10\text{ cm}$, a perpendicular height of $6\text{ cm}$, and a slanted side of $8\text{ cm}$. $A = \frac{1}{2} \times 10 \times 6 = \mathbf{30\text{ cm}^2}$. Using the $8$ gives $40$, which will be one of the options.
The same trap works in reverse for perimeter: perimeter needs the actual sides, including the slant, and never the internal perpendicular height.
Compound and L-Shaped Figures
Compound shapes come with two or three edges deliberately unlabelled. Find them first.
Rule for missing edges: on a rectilinear shape, the two horizontal runs on one side add up to the single horizontal run on the other, and the same is true vertically.
Worked example. An L-shaped floor has a bottom edge of $10\text{ m}$, a total left height of $8\text{ m}$, a top edge of $4\text{ m}$, and a right vertical edge of $3\text{ m}$.
- Missing inner vertical edge: $8 - 3 = 5\text{ m}$.
- Missing inner horizontal edge: $10 - 4 = 6\text{ m}$.
- Perimeter: $10 + 8 + 4 + 5 + 6 + 3 = \mathbf{36\text{ m}}$.
- Area by splitting: left rectangle $4 \times 8 = 32\text{ m}^2$; right rectangle $6 \times 5 = 30\text{ m}^2$; total $\mathbf{62\text{ m}^2}$.
- Area by subtracting (the check): whole rectangle $10 \times 8 = 80\text{ m}^2$, minus the missing corner $6 \times 3 = 18\text{ m}^2$, gives $\mathbf{62\text{ m}^2}$. ✓
Doing it both ways takes a few extra seconds and catches almost every arithmetic slip. Choose splitting when the shape divides into obvious rectangles, and subtracting when a rectangular bite has been taken out of a larger rectangle.
Shaded Regions
A shaded-region question gives an outer shape with an inner shape removed. The method is always the same: outer area minus inner area.
A $12\text{ cm}$ by $9\text{ cm}$ rectangle has a $4\text{ cm}$ square cut from it. $108 - 16 = \mathbf{92\text{ cm}^2}$.
A rectangular lawn $14\text{ m}$ by $10\text{ m}$ has a path $1\text{ m}$ wide running all the way around the inside edge. What area is grass? The grass is a rectangle inset by $1\text{ m}$ on every side, so its dimensions are $12\text{ m}$ by $8\text{ m}$, giving $\mathbf{96\text{ m}^2}$. Subtracting only $1\text{ m}$ once from each dimension is the trap.
Working Backwards From a Known Area
The harder items give the area and ask for a length. Rearrange the formula rather than guessing.
A rectangle has an area of $84\text{ cm}^2$ and a width of $7\text{ cm}$. Find its perimeter. Length $= 84 \div 7 = 12\text{ cm}$. Perimeter $= 2(12 + 7) = \mathbf{38\text{ cm}}$.
A triangle has an area of $54\text{ cm}^2$ and a base of $12\text{ cm}$. Find its perpendicular height. $54 = \frac{1}{2} \times 12 \times h$, so $54 = 6h$ and $h = \mathbf{9\text{ cm}}$.
A square has an area of $169\text{ cm}^2$. Find its perimeter. Side $= \sqrt{169} = 13\text{ cm}$, so $P = \mathbf{52\text{ cm}}$. Square numbers to $144$ and the value $169$ are worth knowing on sight.
Perimeter and Area Do Not Move Together
This relationship is examined directly and surprises most children the first time.
| Rectangle | Perimeter | Area |
|---|---|---|
| $1 \times 11$ | $24\text{ cm}$ | $11\text{ cm}^2$ |
| $3 \times 9$ | $24\text{ cm}$ | $27\text{ cm}^2$ |
| $5 \times 7$ | $24\text{ cm}$ | $35\text{ cm}^2$ |
| $6 \times 6$ | $24\text{ cm}$ | $36\text{ cm}^2$ |
All four have the same perimeter. The area more than triples. The general result is that, for a fixed perimeter, the area is largest when the shape is closest to a square — which is exactly what "a farmer has 24 m of fencing and wants the largest possible rectangular pen" is asking.
The reverse also holds: shapes with equal areas can have very different perimeters, so a question comparing two figures must be read carefully to see which measure it wants.
The Routine
- Label every edge on the diagram, including the ones you work out. Rough work in the test booklet is allowed and is not marked.
- Decide whether the question wants a length or a surface, and fix the unit before calculating.
- Check that the height you are using is perpendicular.
- For compound shapes, find missing edges before choosing a method, then split or subtract.
- Sanity-check the size: an area answer smaller than one of the shape's sides, or a perimeter larger than the area of a big shape, usually means a formula was misapplied.
An L-shaped room has a long bottom edge of 10 m, a total left height of 8 m, a top edge of 4 m, and a right vertical edge of 3 m. What is the total area of the room?
A triangle has a base of 10 cm, a perpendicular height of 6 cm and a slanted side of 8 cm. What is its area?
A rectangular lawn 14 m by 10 m has a path 1 m wide running all the way around the inside edge. What area is left as grass?
A farmer has 24 m of fencing for a rectangular pen with whole-metre sides. Which dimensions enclose the largest area?