Real-Life Applications of Surface Area and Volume of Solids

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

1. Explore the volume relationship between cones and cylinders using hands-on discovery and decomposition principles.
2. Calculate cross-sectional areas and internal volumes for real-life structures like buildings, metallic rods, and wooden composite solids.
3. Determine the volume of conical frustum pillars and composite architectural monuments.

Key vocabulary

Decomposition

The process of dividing a complex geometric solid into simpler component solids.

We used decomposition to split the kennel into a cuboid and a triangular prism.

Cross-section

The shape produced by cutting straight through a solid perpendicular to its length.

The building front face forms a composite pentagonal cross-section of uniform length.

Frustum

The remaining base part of a cone or pyramid cut parallel to its base.

The support pillar was constructed in the shape of a truncated conical frustum.

Lesson preview

Slide 1 · Volume Relationships and Solid DecompositionOpening · 12 min

When exploring volumes of real-world structures, hands-on experiments reveal essential geometric relationships. By filling a cone with sand and pouring it into a cylinder of equal base radius and height, we observe that three cone volumes equal one cylinder volume. Furthermore, complex structures like a puppy's kennel are simplified by dividing them along joining planes into elementary solids—such as a cuboid base and a triangular prism roof—and adding their volumes together.

• Hands-on sand transfer experiments show three full cones fill a cylinder of equal radius and height.
• The volume of a right cone is exactly one-third the volume of its corresponding cylinder.
• Real-life composite solids such as a puppy's kennel can be decomposed along their joining planes.
• The total volume of a composite structure is found by summing individual component solid volumes.

Worked example. Step 1: Calculate cuboid volume: $V_{\text{cuboid}}=0.8\times 0.6\times 0.5=0.24{\text{ m}}^3$. Step 2: Calculate triangular prism roof volume: $V_{\text{roof}}=\frac{1}{2}\times \text{base}\times \text{height}\times \text{length}=\frac{1}{2}\times 0.6\times 0.4\times 0.8=0.096{\text{ m}}^3$. Step 3: Sum the volumes: $V_{\text{total}}=0.24+0.096=0.336{\text{ m}}^3$. Answer: $0.336{\text{ m}}^3$.

Check question: A conical container and a cylindrical tin have identical base radii of $6\text{ cm}$ and equal perpendicular heights of $14\text{ cm}$. If the cone is completely filled with liquid and poured into the cylinder, what volume of liquid is in the cylinder? (Use $\pi =\frac{22}{7}$)

1. $528{\text{ cm}}^3$
2. $1584{\text{ cm}}^3$
3. $176{\text{ cm}}^3$
4. $88{\text{ cm}}^3$

Show answer

$528{\text{ cm}}^3$. The volume of liquid poured into the cylinder is equal to the volume of the fully filled cone. The volume of a cone is $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 6^2\times 14=\frac{1}{3}\times \frac{22}{7}\times 36\times 14=528{\text{ cm}}^3$.

Slide 2 · Volumes of Buildings, Rods, and Composite StructuresModel · 14 min

Teach the full lesson on the projector

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

• All 3 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