Surface area of composite solids | Year 10

01 / LEARN

The idea, step by step

Identify the outside

Surface area measures the outside skin, not the space inside. Sketch each component and mark the regions that touch. A contact face is internal and does not contribute to the final outside area.

Adding complete component areas counts each join twice. Subtract both copies. A labelled face list or net is a useful check, especially when one object sits on another.

A_{o u t s i d e} = A_{1} + A_{2} - 2 A_{c o n t a c t}

Curved and flat surfaces

A cylinder has a curved surface and two circular ends. A hemisphere has one curved surface and one flat circle. At a circular join, the touching circles disappear but the curved surfaces remain.

Use compatible lengths and square units. Retain pi until a decimal answer is requested. Check the result against the appearance of the object rather than counting every component face automatically.

A common trap

Counting the join as part of the exposed skin.

Instead: Remove both copies of a hidden contact region.

02 / WORKED EXAMPLES

See the thinking

Example 1

Joined blocks

A 6 by 4 by 3 cm cuboid joins a 2 by 4 by 3 cm cuboid along their 4 by 3 cm faces. Find the outside area.

Separate areas are 108 and 52 cm².
The contact face has area 12 cm².
Outside area=108+52−2×12=136 cm².

136 cm²

Example 2

Domed cylinder

A cylinder of radius 3 cm and height 10 cm has a matching hemispherical top and an exposed flat base. Find its outside area.

Cylinder curve: $2 π r h = 60 π$ .
Hemisphere: $2 π r^{2} = 18 π$ ; base: $π r^{2} = 9 π$ .
Total: $87 π {cm}^{2}$ .

87 π {cm}^{2}

Check your understanding

Two cubes of edge 2 cm share a square face. Find their outside area.

Show answer

40 cm²

Their areas total 48 cm²; subtract two hidden 4 cm² faces.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Two cubes of edge 3 cm share a whole face. Find the exposed area.
2 marks
Two cubes of edge 4 cm share a whole face. Find the exposed area.
2 marks
Join 10 by 6 by 4 cm and 2 by 6 by 4 cm cuboids along their 6 by 4 faces. Find exposed area.
2 marks
A cube of edge 6 cm has a square-pyramid roof with triangular slant height 5 cm. Find exposed area; the square join is hidden.
3 marks
A radius 2 cm, height 5 cm cylinder has a matching hemisphere on top and an exposed flat bottom. Find outside area.
2 marks
A cube of edge 5 cm supports a square-pyramid roof of slant height 7 cm. Find exposed area.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two cubes of edge 3 cm share a whole face. Find the exposed area.
2 marks
2×54−2×9=90.
90 cm²
Two cubes of edge 4 cm share a whole face. Find the exposed area.
2 marks
The result is an 8 by 4 by 4 cuboid.
2(32+32+16)=160.
160 cm²
Join 10 by 6 by 4 cm and 2 by 6 by 4 cm cuboids along their 6 by 4 faces. Find exposed area.
2 marks
The result is 12 by 6 by 4.
2(72+48+24)=288.
288 cm²
A cube of edge 6 cm has a square-pyramid roof with triangular slant height 5 cm. Find exposed area; the square join is hidden.
3 marks
Five cube faces:180.
Four roof triangles:4×½×6×5=60.
Total=240.
240 cm²
A radius 2 cm, height 5 cm cylinder has a matching hemisphere on top and an exposed flat bottom. Find outside area.
2 marks
Curved cylinder $20 π$ ; hemisphere $8 π$ ; base $4 π$ .
$32 π {cm}^{2}$
A cube of edge 5 cm supports a square-pyramid roof of slant height 7 cm. Find exposed area.
3 marks
Five square faces=125.
Four roof faces=4×½×5×7=70.
Total=195.
195 cm²

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Surface area of composite solids; volume of composite solids; proportion/scaling of objects; related length/area/volume factors; measurement error; the effect of rounding and repeated approximations; practical design models; evaluating assumptions and suitable precision.

Official curriculum source ↗

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