19.6 Composite shapes and solids — Learn

Prerequisites: use area, circle and solid formulae; split a figure into known shapes.

Objectives: find areas, perimeters, volumes and exposed surface areas of composite figures.

Split, calculate, combine

A composite shape joins familiar shapes. Draw or imagine the joins, then add areas or volumes. For an opening or cut-out, subtract its area. For surface area, count exposed faces only.

Example 1. A $12$ cm by $8$ cm rectangle has a $4$ cm by $3$ cm corner removed.

Find the large area: $12\times8=96\text{ cm}^2$.

Find and subtract the cut-out: $4\times3=12$, then $96-12=84$.

The remaining area is $84\text{ cm}^2$.

Example 2. A semicircle with radius 4 cm is joined to a rectangle 8 cm by 5 cm along an 8 cm side. Work out the outside perimeter.

The outside includes the 8 cm side, two 5 cm sides and the semicircle arc.

The arc is half a circumference: $\frac12\times2\pi\times4=4\pi$.

So the perimeter is $8+5+5+4\pi=18+4\pi\text{ cm}$.

Example 3. Join cuboids $4\times3\times2$ cm and $5\times3\times2$ cm along a $3\times2$ cm face.

Add their volumes: $4\times3\times2+5\times3\times2=54\text{ cm}^3$.

Find each cuboid's surface area: $2(4\times3+4\times2+3\times2)=52$ and $2(5\times3+5\times2+3\times2)=62$.

Subtract the two hidden join faces: $52+62-2(3\times2)=102$.

So the joined solid has volume $54\text{ cm}^3$ and outside surface area $102\text{ cm}^2$.

Example 4. A cylinder with radius 3 cm and height 4 cm has a hemisphere (half a sphere) of radius 3 cm on top. Work out the combined volume.

Find the cylinder volume: $\pi\times3^2\times4=36\pi$.

Find the hemisphere volume: $\frac12\times\frac43\pi\times3^3=18\pi$.

Add the volumes: $36\pi+18\pi=54\pi\text{ cm}^3$.

Watch out. Adding the surface areas counts the joined faces twice. Remove both copies of every internal face.

Progress check

The pond cover is a cylinder of radius $4$ m and height $1$ m with a hemisphere of radius $4$ m on top. Work out the total volume in terms of $\pi$. Which measurements did you need?

You should now be able to

• split a composite area into familiar shapes or subtract a cut-out;
• add volumes of joined solids;
• remove internal faces when finding outside surface area;
• explain which dimensions and formulae you chose.