Surface area and volume

1. Overview

Surface area and volume are about measuring 3D shapes. Volume tells you how much space a 3D object takes up (its capacity), while surface area is the total area of all the faces on the outside. You'll need to calculate these for cuboids, prisms, cylinders, spheres, pyramids, and cones. Knowing these calculations is essential for many real-world problems.

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Key Definitions

• Volume: The amount of 3D space an object occupies, measured in cubic units (e.g., $cm^3, m^3$).
• Surface Area: The total area of the outer surfaces of a 3D object, measured in square units (e.g., $cm^2, m^2$).
• Cross-section: The shape exposed by making a straight cut through a 3D object.
• Prism: A 3D shape with a constant cross-section throughout its length.
• Net: A 2D pattern that can be folded to make a 3D shape.
• Capacity: The volume of fluid a container can hold (often measured in Litres or ml).

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Core Content

Cuboids

A cuboid has 6 rectangular faces. Opposite faces are identical.

Volume of a Cuboid: $\qquad \bf{V = l \times w \times h}$

Surface Area of a Cuboid: $\qquad \bf{SA = 2(lw + lh + wh)}$

Worked example 1 — Surface area of a cuboid

Question: Find the surface area of a cuboid with length 7 cm, width 5 cm, and height 4 cm.

1. Identify dimensions: $l = 7\text{ cm}, w = 5\text{ cm}, h = 4\text{ cm}$
2. Write down the formula: $SA = 2(lw + lh + wh)$
3. Substitute the values: $SA = 2(7 \times 5 + 7 \times 4 + 5 \times 4)$
4. Calculate inside the brackets: $SA = 2(35 + 28 + 20)$
5. Sum the values inside the brackets: $SA = 2(83)$
6. Multiply by 2: $SA = 166\text{ cm}^2$
7. Final answer: $\bf{SA = 166\text{ cm}^2}$

Worked example 2 — Volume of a cuboid

Question: A rectangular tank has a length of 1.5 m, a width of 80 cm, and a height of 50 cm. Calculate the volume of the tank in $m^3$.

1. Identify dimensions: $l = 1.5\text{ m}, w = 80\text{ cm}, h = 50\text{ cm}$
2. Convert width and height to meters: $w = 0.8\text{ m}, h = 0.5\text{ m}$
3. Write down the formula: $V = l \times w \times h$
4. Substitute the values: $V = 1.5 \times 0.8 \times 0.5$
5. Calculate the volume: $V = 0.6\text{ m}^3$
6. Final answer: $\bf{V = 0.6\text{ m}^3}$

Prisms

The volume of any prism is the area of its cross-section multiplied by its length.

Volume of a Prism: $\qquad \bf{V = \text{Area of cross-section} \times \text{length}}$

Worked example 1 — Volume of a triangular prism

Question: A prism has a triangular cross-section with base 8 cm and height 5 cm. The prism is 12 cm long. Find the volume.

1. Find Area of Triangle: $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 5 = 20\text{ cm}^2$
2. Write down the formula: $V = \text{Area of cross-section} \times \text{length}$
3. Substitute the values: $V = 20 \times 12$
4. Calculate the volume: $V = 240\text{ cm}^3$
5. Final answer: $\bf{V = 240\text{ cm}^3}$

Worked example 2 — Volume of a complex prism

Question: A prism has a cross-section in the shape of a trapezium. The parallel sides of the trapezium are 4 cm and 6 cm, and the distance between them is 3 cm. The length of the prism is 9 cm. Calculate the volume of the prism.

1. Find the area of the trapezium: $A = \frac{1}{2}(a+b)h = \frac{1}{2}(4+6) \times 3 = \frac{1}{2} \times 10 \times 3 = 15\text{ cm}^2$
2. Write down the formula: $V = \text{Area of cross-section} \times \text{length}$
3. Substitute the values: $V = 15 \times 9$
4. Calculate the volume: $V = 135\text{ cm}^3$
5. Final answer: $\bf{V = 135\text{ cm}^3}$

Cylinders

A cylinder is a special type of prism with a circular cross-section.

Volume of a Cylinder: $\qquad \bf{V = \pi r^2 h}$

Curved Surface Area of a Cylinder: $\qquad \bf{CSA = 2\pi rh}$

Total Surface Area of a Cylinder: $\qquad \bf{TSA = 2\pi r^2 + 2\pi rh}$

Note: The curved surface area is the area of the side of the cylinder, as if you unrolled it. The total surface area includes the top and bottom circular faces.

Worked example 1 — Total surface area of a cylinder

Question: Find the total surface area of a cylinder with radius 4 cm and height 10 cm. Leave your answer in terms of $\pi$.

1. Area of two circular ends: $2 \times \pi \times r^2 = 2 \times \pi \times 4^2 = 2 \times \pi \times 16 = 32\pi$
2. Curved surface area: $2 \times \pi \times r \times h = 2 \times \pi \times 4 \times 10 = 80\pi$
3. Add them together: $32\pi + 80\pi = 112\pi\text{ cm}^2$
4. Final answer: $\bf{112\pi\text{ cm}^2}$

Worked example 2 — Volume of a cylinder

Question: A cylindrical water tank has a radius of 0.7 m and a height of 1.2 m. Calculate the volume of water the tank can hold, giving your answer to 3 significant figures.

1. Write down the formula: $V = \pi r^2 h$
2. Substitute the values: $V = \pi \times (0.7)^2 \times 1.2$
3. Calculate the volume: $V = \pi \times 0.49 \times 1.2 = 1.847256... \text{ m}^3$
4. Round to 3 significant figures: $V = 1.85\text{ m}^3$
5. Final answer: $\bf{V = 1.85\text{ m}^3}$

Spheres, Pyramids, and Cones

The formulas for these are often provided on the exam formula sheet, but you must know how to apply them.

Volume of a Sphere: $\qquad \bf{V = \frac{4}{3}\pi r^3}$

Surface Area of a Sphere: $\qquad \bf{SA = 4\pi r^2}$

Volume of a Cone: $\qquad \bf{V = \frac{1}{3}\pi r^2 h}$

Volume of a Pyramid: $\qquad \bf{V = \frac{1}{3} \times \text{base area} \times \text{vertical height}}$

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Extended Content (Extended Only)

While all the shapes are covered in the core syllabus, Extended students will face more complex problems, including:

• Compound Shapes: Problems involving combinations of two or more of the basic shapes (e.g., a cone on top of a cylinder, a hemisphere attached to a cuboid). The key here is to carefully identify which surfaces are on the exterior of the combined shape and should be included in the total surface area. "Hidden" surfaces where the shapes join are not included.

• Algebraic Manipulation: Extended questions may require you to rearrange the volume or surface area formulas to find an unknown dimension (e.g., finding the radius of a sphere given its volume). This involves using your algebra skills to isolate the variable you're trying to find.

Worked example — Hemisphere on a cylinder

Question: A solid is made up of a hemisphere of radius 6 cm on top of a cylinder of radius 6 cm and height 10 cm. Calculate the total surface area of the solid.

1. Surface area of hemisphere: Half the surface area of a sphere, so $\frac{1}{2} \times 4\pi r^2 = 2\pi r^2 = 2 \times \pi \times 6^2 = 72\pi\text{ cm}^2$
2. Curved surface area of cylinder: $2\pi rh = 2 \times \pi \times 6 \times 10 = 120\pi\text{ cm}^2$
3. Area of the base of the cylinder: $\pi r^2 = \pi \times 6^2 = 36\pi\text{ cm}^2$
4. Total surface area: $72\pi + 120\pi + 36\pi = 228\pi\text{ cm}^2$
5. Final answer: $\bf{228\pi\text{ cm}^2}$

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Key Equations

Shape	Volume Formula	Surface Area Formula	Notes
Cuboid	$\bf{V = l \times w \times h}$	$\bf{SA = 2(lw + lh + wh)}$	Must memorize
Cylinder	$\bf{V = \pi r^2 h}$	$\bf{SA = 2\pi r^2 + 2\pi rh}$	Must memorize
Sphere	$\bf{V = \frac{4}{3}\pi r^3}$	$\bf{SA = 4\pi r^2}$	Formula usually provided
Cone	$\bf{V = \frac{1}{3}\pi r^2 h}$	$\bf{CSA = \pi rl}$	$l$ is the slant height
Pyramid	$\bf{V = \frac{1}{3}Ah}$	Sum of all faces	$A$ = Area of base

Units Note:

• Area: $mm^2, cm^2, m^2$
• Volume: $mm^3, cm^3, m^3$
• Capacity: $1\text{ litre} = 1000\text{ cm}^3$

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Common Mistakes to Avoid

❌ Wrong: Using the diameter when the formula requires the radius, or vice versa. ✅ Right: Always check if the question gives you the diameter ($d$) or the radius ($r$). Remember that $r = \frac{d}{2}$. Underline the words "radius" or "diameter" in the question to remind yourself.

❌ Wrong: Forgetting that a closed cylinder has two circular ends when calculating total surface area. ✅ Right: The total surface area of a closed cylinder is the curved surface area ($2πrh$) plus the area of both circular ends ($2πr^2$).

❌ Wrong: Including the area of the circular base when calculating the surface area of a solid hemisphere. ✅ Right: The surface area of a solid hemisphere is half the surface area of a sphere ($2πr^2$) plus the area of the circular base ($πr^2$), giving a total of $3πr^2$.

❌ Wrong: Not converting all measurements to the same units before calculating volume or surface area. ✅ Right: Circle the units given in the question. If you have mixed units (e.g., cm and m), convert before you start calculating.

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Exam Tips

• Command Words: If the question says "Calculate," show every step. If it says "Show that," you must start from the formula and reach the given answer clearly.
• Calculator Tip: Use the $\pi$ button on your calculator rather than $3.14$ or $22/7$ for better accuracy, unless the question specifies otherwise.
• Rounding: Do not round your numbers in the middle of a calculation. Keep the full decimal on your calculator and round only at the very end (usually to 3 significant figures).
• Units: Always check if the dimensions are in the same units. If length is in $m$ and width is in $cm$, convert them to be the same before calculating.
• Real-world Context: Be prepared for "Rate of flow" questions. If water flows into a cylinder, the Volume divided by Time gives the Flow Rate.
• Formula Sheet: Check your exam paper's front page immediately to see which formulas are provided so you don't panic. Cuboids, Prisms, and Cylinders are usually NOT provided.