Surface Area of Composite Solids

By the end of this lesson, learners should be able to

1. Identify external and internal faces of a composite solid.
2. Visualise and sketch unfolded nets of composite solids.
3. Calculate the total surface area of composite solids including combined cuboids, pyramids, cylinders, cones, and semi-cylinders.

Key vocabulary

Composite solid

A three-dimensional shape constructed by joining two or more basic geometric solids together.

The grain store model is a composite solid made of a cuboid and a pyramid.

Internal face

A boundary surface where two component solids join, hidden from the solid's exterior.

The top square face of the cuboid is an internal face and is omitted.

Semi-cylinder

A three-dimensional solid formed by cutting a cylinder in half along its longitudinal axis.

The curved roof of the grain barn is shaped like a semi-cylinder.

Lesson preview

Slide 1 · Visualising and Calculating Area of Composite SolidsModel · 18 min

A composite solid is formed by combining two or more simple 3D shapes, such as a cuboid base topped by a pyramid roof or a cylinder joined to a cone. When determining the total surface area of a composite solid, we only sum the areas of all visible external faces. Any face where two component solids join is internal and hidden, so its area must be excluded from the total sum. Use Figure 1 to see how the hidden internal face is excluded from the unfolded net.

• A composite solid is created by joining two or more standard geometric 3D shapes together.
• Visualising the unfolded net helps identify every individual exposed outer face of the composite solid.
• Internal contact faces where the component solids connect must be excluded from the total surface area.
• Calculate the area of each outer face separately before summing them to find the total area.

Worked example. 1. Identify visible outer faces: 1 square floor ($6\text{ m}\times 6\text{ m}$), 4 rectangular side walls ($6\text{ m}\times 4\text{ m}$), and 4 triangular roof faces. Exclude the internal $6\text{ m}\times 6\text{ m}$ joint face. 2. Calculate the floor area: $A_{\text{floor}}=6\text{ m}\times 6\text{ m}=36{\text{ m}}^2$. 3. Calculate total wall area: $A_{\text{walls}}=4\times (6\text{ m}\times 4\text{ m})=96{\text{ m}}^2$. 4. Calculate pyramid slant height: $s=\sqrt{3^2+4^2}=5\text{ m}$. 5. Calculate total roof area: $A_{\text{roof}}=4\times (\frac{1}{2}\times 6\text{ m}\times 5\text{ m})=60{\text{ m}}^2$. 6. Sum outer face areas: $A_{\text{total}}=36+96+60=192{\text{ m}}^2$. Answer: $192{\text{ m}}^2$.

Check question: A closed-bottom cylindrical tank ($r=7\text{ cm}$, height $h=10\text{ cm}$) is topped by a cone ($r=7\text{ cm}$, slant height $s=25\text{ cm}$). What is the total outer surface area of the composite solid? Take $\pi =\frac{22}{7}$.

1. $924{\text{ cm}}^2$
2. $990{\text{ cm}}^2$
3. $1144{\text{ cm}}^2$
4. $1298{\text{ cm}}^2$

Show answer

$1144{\text{ cm}}^2$. The outer surface comprises the circular base ($\pi r^2=\frac{22}{7}\times 7^2=154{\text{ cm}}^2$), the curved cylinder wall ($2\pi rh=2\times \frac{22}{7}\times 7\times 10=440{\text{ cm}}^2$), and the curved cone face ($\pi rs=\frac{22}{7}\times 7\times 25=550{\text{ cm}}^2$). Summing gives $154+440+550=1144{\text{ cm}}^2$.

Slide 2 · Composite Solids with Curved SurfacesCheck · 22 min

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This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• Both slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 8 lessons in this sub-strand

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All 9 lessons in Surface Area and Volume of Solids

1. 1Surface Area of Prisms and PyramidsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
2. 2Surface Area of Pyramids, Cones, and FrustumsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
3. 3Surface Area of Frustums, Spheres, and HemispheresDetermine the surface area of prisms, pyramids, cones, frustums and spheres
4. 4Surface Area of Composite SolidsCalculate the surface area of composite solids
5. 5Volume of Prisms, Pyramids, and ConesCalculate the volume of prisms, pyramids, cones, frustums and spheres
6. 6Volume of Frustums and SpheresCalculate the volume of prisms, pyramids, cones, frustums and spheres
7. 7Volume of Composite SolidsDetermine the volume of composite solids
8. 8Surface Area and Volume of Solids: Real-Life Applications (Part 1)Explore the use of the surface area and volume of solids in real-life situations
9. 9Real-Life Applications of Surface Area and Volume of SolidsExplore the use of the surface area and volume of solids in real-life situations