Mensuration | Surface Area & Volume of 3D Solids

Learning Objectives

By the end of this article, you should be able to:

• Recall and apply the formulas for volume of prisms, cylinders, cones, spheres, and pyramids.
• Recall and apply the formulas for total surface area and lateral surface area of these solids.
• Apply the length, area, and volume ratios for similar solids.
• Solve problems involving composite solids (combinations of two or more solids).
• Determine the effect of changing dimensions on volume and surface area.
• Solve DSE-style problems involving 3D mensuration.

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1. Introduction:

3D mensuration appears every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:

• Calculate the volume or surface area of a given solid
• Find an unknown dimension given volume or surface area
• Apply similar solids ratios to find missing measurements
• Solve composite solid problems (e.g., a cylinder with a hemisphere on top)

These are essential skills that also appear in real-world contexts like capacity, packaging, and construction.

2. Volume and Surface Area Formulas

Prisms

• Volume = Area of base × Height
• Total Surface Area = 2 × (Area of base) + (Perimeter of base) × Height (lateral area)

Cylinders

• Volume = \( \pi r^2 h \)
• Lateral Surface Area = \( 2\pi r h \)
• Total Surface Area = \( 2\pi r^2 + 2\pi r h \)

Cones

• Volume = \( \frac{1}{3} \pi r^2 h \)
• Slant height \( l = \sqrt{r^2 + h^2} \)
• Lateral Surface Area = \( \pi r l \)
• Total Surface Area = \( \pi r^2 + \pi r l \)

Spheres

• Volume = \( \frac{4}{3} \pi r^3 \)
• Surface Area = \( 4\pi r^2 \)

Pyramids

• Volume = \( \frac{1}{3} \times \text{Area of base} \times \text{Height} \)
• Surface area = sum of areas of all faces (base + triangular faces)

3. Similar Solids

If two solids are similar, their corresponding lengths are proportional. Let the linear scale factor be \( k \) (ratio of corresponding lengths). Then:

• Area ratio = \( k^2 \)
• Volume ratio = \( k^3 \)

This means if you double the dimensions (scale factor 2), the surface area becomes 4 times larger, and the volume becomes 8 times larger.

Worked Example

Question: Two similar cylinders have heights 5 cm and 15 cm. If the smaller cylinder has volume 200 cm³, find the volume of the larger cylinder.

4. Composite Solids

A composite solid is made up of two or more simple solids. To find the total volume or surface area:

1. Identify the individual solids (e.g., cylinder + hemisphere).
2. Calculate the volume or surface area of each part.
3. For volume: add the volumes.
4. For surface area: add the exposed areas (do not count internal faces where solids meet).

Worked Example

Question: A solid is made of a cylinder of radius 4 cm and height 6 cm, with a hemisphere of the same radius on top. Find the total volume.

5. Worked Examples

Example 1: Volume of a Cone

Question: A cone has radius 6 cm and height 8 cm. Find its volume.

Example 2: Surface Area of a Sphere

Question: Find the surface area of a sphere with radius 7 cm.

Example 3: Similar Solids

Question: Two similar pyramids have volumes 54 cm³ and 432 cm³. Find the ratio of their corresponding heights.

6. DSE-Style Practice Questions

7. Solutions with Explanations

9. Exercise

Click the following link to have
 An Exercise on Mensuration | Surface Area & Volume of 3D Solids

Key Takeaways

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Summary Checklist for Revision

• Volume of prism/cylinder: base area × height
• Volume of cone/pyramid: \( \frac{1}{3} \) base area × height
• Volume of sphere: \( \frac{4}{3}\pi r^3 \)
• Surface area: sum of all faces
• Slant height of cone: \( l = \sqrt{r^2 + h^2} \)
• Similar solids: area ratio = \( k^2 \), volume ratio = \( k^3 \)
• Composite solids: add/subtract parts
• Check units (cm³, m³, etc.)