Perimeter 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?

Predict: a small rectangle is removed from the top-right corner of a larger rectangle. Does the outside perimeter become smaller?

Test the prediction by changing the width c and height d of a corner cut out of an 8 cm by 6 cm rectangle. Trace the replacement boundary sides

Why does the perimeter stay at 28 cm?

Trace the exposed boundary

A composite boundary can contain straight edges and arcs. Internal joins are not exposed

1. 1Walk the entire boundary Count each exposed edge exactly once. Find missing horizontal and vertical lengths by subtraction
2. 2Exclude internal joins For two joined pieces with perimeters P₁ and P₂, subtract twice their total shared length L. Each piece counted it once \(\displaystyle P=P_1+P_2-2L\)
3. 3Use the required fraction of a circle For a central angle θ measured in degrees, the arc is θ/360 of the full circumference. Add any exposed straight edges \(\displaystyle \ell=\frac{\theta}{360}\,2\pi r\)
4. 4Count shared portions, not whole sides When two sides overlap only partly, subtract only the actual shared length twice. Trace the outline to check
5. 5Keep units and precision consistent Convert dimensions before adding; keep π exact until the final rounding step. Perimeter uses length units

A semicircle is joined to the full top edge of an 8 m by 5 m rectangle. Which expression gives the outside perimeter?

A semicircle is joined to the full top edge of a rectangle 10 m wide and 6 m high. Find the outside perimeter to 2 decimal places

Count the bottom, two vertical sides and one semicircular arc. Exclude the dashed join

Joined curved boundaries

When adding the perimeters of joined pieces, 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 For a rectangle 5 cm wide and 7 cm high with a 270° sector of radius 5 cm, the shared lengths total 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?