# Volume and surface area of 3D solids

> IB Mathematics: Applications and Interpretation SL · IB AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-volume-and-surface-area-of/

This module covers formulas for and calculation of volume and total surface area of common 3D solids including prisms, pyramids, cones, spheres and composite solids, core skills for both Paper 1 and Paper 2 of IB AI SL.

**Prerequisites:** [Area of 2D shapes](https://www.owlsprep.com/study/ib-math-ai-sl-u2-area-of-2d-shapes/); [Unit conversion for length and area](https://www.owlsprep.com/study/ib-math-ai-sl-u1-unit-conversion/)

## Learning objectives

- Recall formulas for volume and surface area of common 3D solids
- Calculate unknown dimensions given volume or surface area
- Solve volume and surface area problems for composite solids
- Apply correct unit conversion for area and volume measurements

## Formulas for Common 3D Solids

IB AI SL requires you to use formulas for volume and surface area of six common 3D solids. Some formulas are provided in the formula booklet, but memorizing common ones saves critical time in exams.

**Volume** — The total amount of space occupied by a 3D solid, measured in cubic units ($\text{cm}^3$, $\text{m}^3$ etc.)

*Notation:* V

*Example:* A 1 cm cube has a volume of 1 $\text{cm}^3$

**Total Surface Area** — The total area of all outer faces of a 3D solid, measured in square units

*Notation:* SA

*Example:* A 1 cm cube has a total surface area of 6 $\text{cm}^2$

| Solid | Volume Formula | Total Surface Area Formula |
| --- | --- | --- |
| Cube (side $a$) | $a^3$ | $6a^2$ |
| Cuboid ($l \times w \times h$) | $lwh$ | $2(lw + lh + wh)$ |
| Prism ($A$ = cross-section area, $l$ = length) | $Al$ | $2A + Pl$ ($P$ = cross-section perimeter) |
| Pyramid ($A$ = base area, $h$ = height) | $\frac{1}{3}Ah$ | Base area + sum of lateral faces |
| Cone ($r$ = radius, $s$ = slant height) | $\frac{1}{3}\pi r^2 h$ | $\pi r^2 + \pi r s$ |
| Sphere ($r$ = radius) | $\frac{4}{3}\pi r^3$ | $4\pi r^2$ |

> **info**
>
> Cone and sphere formulas are provided in the IB formula booklet, but prism and pyramid formulas are not, so memorize these.

**Worked example:** Calculate the volume and total surface area of a right circular cone with radius 3 cm and perpendicular height 4 cm. Give your answer to 3 significant figures.

1. First calculate the slant height $s$ of the cone using Pythagoras' theorem:

   $$s = \sqrt{r^2 + h^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5 \text{ cm}$$
2. Calculate volume using the cone volume formula:

   $$V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (3)^2 (4) = 12\pi \approx 37.7 \text{ cm}^3$$
3. Calculate total surface area using the cone SA formula:

   $$SA = \pi r^2 + \pi r s = \pi (3)^2 + \pi (3)(5) = 24\pi \approx 75.4 \text{ cm}^2$$
4. Final answer (3 s.f.): Volume = 37.7 cm³, Total Surface Area = 75.4 cm²

## Finding Unknown Dimensions

Common exam questions give you the volume or surface area of a solid and ask you to find an unknown dimension (radius, height, side length etc.). This requires rearranging the formula to isolate the unknown variable.

**Worked example:** The total volume of a sphere is 100 cm³. Calculate the radius of the sphere, correct to 2 decimal places.

1. Start with the standard sphere volume formula:

   $$V = \frac{4}{3}\pi r^3$$
2. Substitute $V = 100$ and rearrange to isolate $r^3$:

   $$100 = \frac{4}{3}\pi r^3 \\ r^3 = \frac{100 \times 3}{4\pi} = \frac{75}{\pi} \approx 23.873$$
3. Take the cube root of both sides to solve for $r$:

   $$r = \sqrt[3]{23.873} \approx 2.88 \text{ cm}$$

**Check your understanding**

Check your understanding:

1. A cube has a total surface area of 150 cm². What is its side length?

   - 5 cm
   - 12.5 cm
   - 25 cm
   - 150/6 = 25 cm

   *Why:* Total SA of a cube = $6a^2$, so $6a^2 = 150 \to a^2 = 25 \to a = 5$ cm

## Volume and Surface Area of Composite Solids

Most extended response exam questions for this topic involve composite solids, which are made by joining two or more simpler solids. There is a key difference between calculating total volume and total surface area for these shapes.

> **tip**
>
> For volume: simply add the volumes of the individual solids, no adjustment needed. For surface area: subtract twice the area of any overlapping face (once from each solid), because overlapping faces are internal and not part of the outer surface.

**Worked example:** A solid sculpture is made by attaching a hemisphere (half a sphere) of radius 10 cm to the top of a solid cylinder of radius 10 cm and height 20 cm. Calculate the total volume of the sculpture, to 3 significant figures.

1. Add the volume of the cylinder and the volume of the hemisphere:

   $$V_{\text{cylinder}} = \pi r^2 h = \pi (10)^2 (20) = 2000\pi \\ V_{\text{hemisphere}} = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2000}{3}\pi$$
2. Calculate total volume:

   $$V_{\text{total}} = 2000\pi + \frac{2000}{3}\pi = \frac{8000}{3}\pi \approx 8380 \text{ cm}^3$$

## Unit Conversion for Volume and Area

Unit conversion is a frequent exam test point for this topic, because it is an area of common mistake. Remember: area is a squared unit, volume is a cubed unit, so conversion factors must match.

> **warning**
>
> If 1 m = 100 cm, then 1 m² = 100² = 10 000 cm², and 1 m³ = 100³ = 1 000 000 cm³. Never use the length conversion factor directly for area or volume.

**Worked example:** Convert a volume of 2.5 m³ to cm³.

1. Start with the length conversion, then cube it for volume:

   $$1 \text{ m} = 100 \text{ cm} \\ 1 \text{ m}^3 = (100 \text{ cm})^3 = 1 000 000 \text{ cm}^3$$
2. Multiply by 2.5 to get the final converted volume:

   $$2.5 \times 1 000 000 = 2 500 000 \text{ cm}^3$$

## Common pitfalls

- **Wrong:** Forgetting to subtract overlapping area when calculating composite solid surface area
  - Why it fails: Overlapping faces between joined solids are internal and not part of the outer surface
  - Correct: Subtract twice the area of any overlapping face from the total sum of individual surface areas
- **Wrong:** Using the same conversion factor for area/volume as for length
  - Why it fails: Area is squared length and volume is cubed length, so conversion factors must also be squared/cubed
  - Correct: Square the length conversion factor for area, cube it for volume
- **Wrong:** Forgetting the $\frac{1}{3}$ factor in pyramid and cone volume formulas
  - Why it fails: Prism and pyramid formulas are similar, but pyramids have a 1/3 factor that is often omitted
  - Correct: Memorize that all point-topped solids (pyramids, cones) have a 1/3 volume factor
- **Wrong:** Using perpendicular height instead of slant height for cone surface area
  - Why it fails: The lateral surface of a cone is calculated along the slanted edge, not the perpendicular height
  - Correct: Calculate slant height with Pythagoras' theorem before substituting into the SA formula
- **Wrong:** Calculating total surface area when the question asks for lateral surface area
  - Why it fails: Students often rush and miss the keyword 'lateral' in the question
  - Correct: Always read the question carefully to confirm if total or lateral surface area is required

## Cheatsheet

| Solid | Volume | Total Surface Area |
| --- | --- | --- |
| Cube (side $a$) | $a^3$ | $6a^2$ |
| Cuboid ($l,w,h$) | $lwh$ | $2(lw+lh+wh)$ |
| Prism ($A,l$) | $Al$ | $2A + Pl$ |
| Pyramid ($A,h$) | $\frac{1}{3}Ah$ | Base + lateral |
| Cone ($r,s$) | $\frac{1}{3}\pi r^2 h$ | $\pi r^2 + \pi r s$ |
| Sphere ($r$) | $\frac{4}{3}\pi r^3$ | $4\pi r^2$ |
| Unit conversion | Cube length factor | Square length factor |

## What's next

Mastering volume and surface area of 3D solids is a foundational skill for all geometry topics in IB AI SL. You will use these core skills when solving problems involving density, similar 3D solids, and 3D coordinate geometry, all of which appear regularly in both Paper 1 and Paper 2. These skills are also often combined with other topics in extended, context-based questions on Paper 2, so building accuracy here will earn you easy marks in the exam.

- [Right-angled triangle trigonometry](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-angled-triangle-trigonometry/)
- [Sine rule, cosine rule for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sine-rule-cosine-rule-for/)
- [Bearings and trigonometric applications](https://www.owlsprep.com/study/ib-math-ai-sl-u3-bearings-and-trigonometric-applications/)

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