Area, perimeter & composite shapes

What you will learn

break a composite shape into rectangles and triangles (decompose),
build a complex outline by starting from a bigger rectangle and subtracting (compose),
approximate the area of an irregular shape using a grid,
handle trapeziums and L-shapes confidently.

1. Core area formulas (recap)

Standard area formulas

Rectangle

A = L \times W.

Triangle

A = \tfrac{1}{2} \times b \times h,

where $b$ is the base and $h$ the perpendicular height.

Parallelogram

A = b \times h.

Trapezium

A = \tfrac{1}{2}(a + b) \times h,

where $a, b$ are the parallel sides and $h$ the distance between them.

2. Composite shapes: decompose and add

Worked example 1 L-shape area

An L-shape is made by joining a $6$ m by $4$ m rectangle and a $3$ m by $2$ m rectangle.

L-shape: two rectangles joined. Area = sum of the two rectangles.

Area $= 6 \times 4 + 3 \times 2 = 24 + 6 = 30$ m $^2$ .

3. Composite shapes: start big and subtract

Worked example 2 Cut corners

A rectangular piece of cardboard $40$ cm by $30$ cm has a $5$ cm square cut from each of its four corners. Find the remaining area.

Start with a big rectangle; subtract the four removed squares.

Original rectangle area: $40 \times 30 = 1200$ cm $^2$ .
Four squares removed: $4 \times (5 \times 5) = 100$ cm $^2$ .
Remaining area: $1200 - 100 = 1100$ cm $^2$ .

4. Approximating an irregular shape on a grid

Worked example 3 Grid estimate

An irregular lake is drawn on a $1$ km grid. Count the fully-covered squares (say $28$ ) and the partial squares (say $14$ , counted as half).

Area estimate:

28 + \tfrac{1}{2} \times 14 = 35 \text{ km}^2.

Finer grids (say $100$ m instead of $1$ km) give a more accurate estimate.

5. Perimeter of composite shapes

Trace the outline. Do not miss any hidden sides when two rectangles meet.

Worked example 4 Staircase perimeter

A shape is formed by attaching a $3$ m by $2$ m rectangle to the corner of a $6$ m by $4$ m rectangle, forming a staircase. The visible outline has sides $6, 4, 3, 2, 3, 2$ (walking around the L).

Perimeter $= 6 + 4 + 3 + 2 + 3 + 2 = 20$ m.

Practice: Year 8 core

Fluency

Simple composite areas

A rectangle $8$ m by $5$ m has a $2$ m by $3$ m rectangle cut from one corner. Find the remaining area.
Two rectangles, $4 \times 2$ and $3 \times 5$ , are joined to form a single L. Find the total area.
A trapezium has parallel sides $8$ cm and $14$ cm and height $5$ cm. Find the area.
A floor plan is a $10$ m by $6$ m rectangle with a triangular bay window (base $6$ m, height $2$ m) added to one long side. Find the total floor area.
A shape is made of a rectangle $12 \times 5$ with a semicircle of diameter $5$ on one short side (use $\pi \approx 3.14$ ). Find the area.

Fluency

Perimeter

A rectangle $12$ by $7$ has a $3$ by $2$ notch cut into one long side (flush with the top-right). Find the new perimeter.
An L-shape has outer dimensions $10$ m by $8$ m, with a $4$ m by $3$ m notch removed from the top-right corner. Find the perimeter.
A rectangle $20$ by $15$ has a square of side $5$ cut from each corner. Find the perimeter of the remaining cross shape.

Fluency

Approximate by grid

On a $1$ cm grid, a shape covers $18$ full squares and $10$ partial squares. Estimate its area (count partial squares as half).
Using a finer $0.5$ cm grid, a shape covers $70$ full small squares and $20$ partial ones. Estimate its area.
Explain why approximating with a finer grid gives a better area estimate.

Reasoning

Explain and spot the mistake

Sam says the area of an L-shape is always the sum of the two rectangles that make it. Is that right? Give an example where it fails.
A student computes the perimeter of a shape by adding the two outer dimensions and multiplying by $2$ , ignoring the notch. Explain why this can be wrong.
Without calculating, decide which has the larger area: a $10 \times 10$ square with a $3 \times 3$ square cut from one corner, or an $11 \times 9$ rectangle. Justify.
A trapezium has parallel sides $3$ m and $7$ m and height $4$ m. Another has parallel sides $5$ m and $5$ m (so a rectangle) and height $4$ m. Which has the larger area? What does this tell you about the average of the parallel sides?

Problem solving

Real contexts

A backyard is L-shaped: a $15$ m by $12$ m rectangle with a $6$ m by $4$ m rectangle cut out of one corner (a shed). Grass seed covers $1$ m $^2$ per gram. How much seed is needed (in kg)?
A concrete slab is a rectangle $8$ m by $6$ m with a semicircular pool alcove (radius $1.5$ m) cut out of one long side. Find the slab area ( $\pi \approx 3.14$ ).
A picture frame has outer $25$ cm by $20$ cm and an inner window $19$ cm by $14$ cm. What is the area of the frame material?
A paddock on a map has shape of a trapezium (parallel sides $400$ m and $600$ m, height $250$ m). Find its area in hectares ( $1$ ha $= 10\,000$ m $^2$ ).

Challenge

Reasoning

Harder composites

A shape consists of a semicircle ( $r = 4$ cm) sitting atop a rectangle $8$ cm by $5$ cm (semicircle on an $8$ cm side). Find the area and the perimeter.
A square of side $20$ cm has a circle inscribed in it (touching all four sides). Find the area between the circle and the square (use $\pi \approx 3.14$ ).
Find the area of a hexagonal stop-sign-like shape with side $6$ cm, treated as a rectangle $12 \times 6\sqrt{3}/2$ with two triangles bolted to the short sides. (Extension: this requires $\sqrt{3}$ ; accept $\sqrt{3} \approx 1.73$ .)