Perimeter and area of composite shapes

Bottom span

\(\displaystyle 9\text{ m}\)

Left side

\(\displaystyle 7\text{ m}\)

Top span

\(\displaystyle 4\text{ m}\)

Short right side

\(\displaystyle 3\text{ m}\)

An L-shaped patio is shown. The bottom span is 9 m, the left side is 7 m, the top span is 4 m and the short right side is 3 m. The two inner sides are not labelled. What is the perimeter of the patio?

An 8 cm by 6 cm rectangle has its top-right corner cut out. Change the cut-out width c and height d. Watch the side lengths, the perimeter sum and the area

Compare small and large cut-outs. As the cut-out grew, what happened?

Composite shapes

A composite shape is made from simpler regions joined together or with a piece cut out. Recover any missing outside lengths from matching spans, then trace only the outside for the perimeter and add or subtract regions for the area

1. 1Recover missing lengths Subtract the smaller labelled span from the matching larger span \(\displaystyle 9-4=5\text{ m},\quad 7-3=4\text{ m}\)
2. 2Trace the outside once Add every outside side once and leave out any internal join \(\displaystyle P=9+3+5+4+4+7=32\text{ m}\)
3. 3Add non-overlapping pieces Split an attached shape into standard regions that cover it exactly once \(\displaystyle A=(8\times 9)+(4\times 6)=96\text{ m}^2\)
4. 4Subtract a cut-out Remove the area of a missing region from the enclosing rectangle \(\displaystyle A=12\times 9-4\times 3=96\text{ m}^2\)

Width

\(\displaystyle 12\text{ m}\)

Wall height

\(\displaystyle 5\text{ m}\)

Gable height h

\(\displaystyle 2.5\text{ m}\)

The end wall of a shed is a 12 m wide by 5 m high rectangle with a triangular gable above it. The peak is centred over the midpoint of the rectangle and is 2.5 m above its top. Find the area of the end wall. The dashed vertical line h is the 2.5 m gable height

Find the rectangle and the triangle separately, halving the triangle, then add

Width

\(\displaystyle 12\text{ m}\)

Wall height

\(\displaystyle 5\text{ m}\)

Gable height h

\(\displaystyle 2.5\text{ m}\)

Using the same shed wall, find the perimeter of the end wall. Do not include the join between the rectangle and the gable. The dashed vertical line h is the 2.5 m gable height

Each sloping edge is the longest side of a right triangle with legs 6 m and 2.5 m

Joined curved boundaries

Add the areas only when the component regions do not overlap. For perimeter, each shared edge was counted once by each component, so remove it twice

1. 1A semicircle on a rectangle The shared top edge is the semicircle’s diameter. The outside includes the other three rectangle sides and the semicircular arc
2. 2More than one shared edge A 270° sector based at a rectangle’s top-left corner also meets its left edge. Both common portions are internal
3. 3Remove all shared lengths Place the centre of a 270° sector of radius 5 cm at the top-left corner of a rectangle 5 cm wide and 7 cm high, with one radius along the top edge and the other down the left edge. The two shared lengths are 5 cm each, totalling 10 cm \(\displaystyle \begin{gathered}P=24+(10+7.5\pi)-20\\=14+7.5\pi\text{ cm}\end{gathered}\)
4. 4Keep π until the final step Form the exact expression first. Round only the final requested measure

Rectangle dimensions

\(\displaystyle 6\text{ cm by }4\text{ cm}\)

Sector radius and angle

\(\displaystyle r=6\text{ cm},\quad\theta=270^\circ\)

A rectangle is 6 cm wide and 4 cm high. A 270° sector of radius 6 cm starts along its top edge and extends anticlockwise from the top-left corner. The other radius runs down its left edge. Find the outside perimeter of the combined shape to 2 decimal places

Trace the solid boundary. The second radius shares 4 cm with the rectangle and has 2 cm left outside. Keep π exact, then write exactly 2 decimal places

Outer rectangle

\(\displaystyle 10\text{ cm by }6\text{ cm}\)

Removed corner

\(\displaystyle 3\text{ cm by }2\text{ cm}\)

Which is the first incorrect line when finding the perimeter of a rectangle with its top-right corner removed?