Area and Volume
The Skill
Area Formulas
Rectangle: $A = l \times w$
Triangle: $A = \frac{1}{2} \times b \times h$
Parallelogram: $A = b \times h$
Trapezium: $A = \frac{1}{2}(a + b) \times h$
Circle: $A = \pi r^2$
Volume Formulas
Cuboid: $V = l \times w \times h$
Prism: $V = \text{Area of cross-section} \times \text{length}$
Cylinder: $V = \pi r^2 h$
Cone: $V = \frac{1}{3}\pi r^2 h$ (Higher)
Sphere: $V = \frac{4}{3}\pi r^3$ (Higher)
Pyramid: $V = \frac{1}{3} \times \text{base area} \times h$ (Higher)
Surface Area
Cuboid: $SA = 2lw + 2lh + 2wh$
Cylinder: $SA = 2\pi r^2 + 2\pi rh$
Sphere: $SA = 4\pi r^2$ (Higher)
Units
- Area: cm², m², km²
- Volume: cm³, m³, litres (1 litre = 1000 cm³)
To convert: 1 m² = 10,000 cm² (100²), 1 m³ = 1,000,000 cm³ (100³)
The Traps
Common misconceptions and how to avoid them.
Forgetting the ½ in triangle area "The Missing Half"
The Mistake in Action
Find the area of a triangle with base 12cm and height 8cm.
Wrong: $A = 12 \times 8 = 96\text{cm}^2$
Why It Happens
Students remember to multiply base by height but forget the crucial ½. They're calculating the area of the rectangle the triangle fits inside.
The Fix
A triangle is half of a rectangle/parallelogram with the same base and height.
$$A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 12 \times 8 = 48\text{cm}^2$$
Memory aid: Triangle has half the area of a rectangle → half × base × height
Spot the Mistake
Triangle: base = 12cm, height = 8cm. Find the area.
$A = 12 \times 8 = 96\text{cm}^2$
Click on the line that contains the error.
Using diameter instead of radius in circle formulas "The D-R Disaster"
The Mistake in Action
A circle has diameter 10cm. Find its area.
Wrong: $A = \pi r^2 = \pi \times 10^2 = 100\pi = 314.16\text{cm}^2$
Why It Happens
Students see "10cm" and plug it straight into the formula without checking if it's the radius or diameter.
The Fix
Always check: Does the question give you radius or diameter?
- Radius = distance from centre to edge
- Diameter = distance across the whole circle = 2 × radius
If given diameter 10cm, then radius = 5cm.
$$A = \pi r^2 = \pi \times 5^2 = 25\pi = 78.54\text{cm}^2$$
The wrong answer (100π) is 4 times too big!
Spot the Mistake
A circle has diameter 10cm. Find its area.
$A = \pi r^2 = \pi \times 10^2$
$= 100\pi = 314.16\text{cm}^2$
Click on the line that contains the error.
Using slant height instead of perpendicular height "The Slant Slip"
The Mistake in Action
Find the area of a triangle with base 8cm and slant side 10cm. The perpendicular height is 6cm.
Wrong: $A = \frac{1}{2} \times 8 \times 10 = 40\text{cm}^2$
Why It Happens
Students use the slant height (a side of the triangle) instead of the perpendicular height.
The Fix
The height in the area formula must be perpendicular (at 90°) to the base.
Look for:
- A line with a small square (indicating 90°)
- The word "perpendicular" or "vertical"
$$A = \frac{1}{2} \times 8 \times 6 = 24\text{cm}^2$$
Not: $\frac{1}{2} \times 8 \times 10 = 40$
Spot the Mistake
Triangle: base = 8cm, slant side = 10cm, perpendicular height = 6cm
$A = \frac{1}{2} \times 8 \times 10$
$= 40\text{cm}^2$
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
A circle has radius 7cm. Find its area. Give your answer to 1 decimal place.
Solution
Formula: $A = \pi r^2$
Substitute: $A = \pi \times 7^2$
Calculate: $$A = \pi \times 49$$ $$A = 153.938...$$ $$A = 153.9\text{cm}^2 \text{ (1 d.p.)}$$
Answer: 153.9 cm²
Question
A cylinder has radius 5cm and height 12cm. Find its volume. Give your answer in terms of π.
Solution
Formula: $V = \pi r^2 h$
Substitute: $$V = \pi \times 5^2 \times 12$$ $$V = \pi \times 25 \times 12$$ $$V = 300\pi \text{ cm}^3$$
Answer: 300π cm³
Question
A triangular prism has a cross-section that is a right-angled triangle with base 6cm and height 4cm. The length of the prism is 15cm. Find the volume.
Solution
Step 1: Find the area of the cross-section (triangle) $$A = \frac{1}{2} \times 6 \times 4 = 12\text{cm}^2$$
Step 2: Use the prism formula $$V = \text{Area of cross-section} \times \text{length}$$ $$V = 12 \times 15 = 180\text{cm}^3$$
Answer: 180 cm³
Level 2: Scaffolded
Fill in the key steps.
Question
A trapezium has parallel sides of 8cm and 12cm. The perpendicular height is 5cm. Find the area.
Level 3: Solo
Try it yourself!
Question
A circle has area 78.54 cm². Find the radius. Give your answer to 1 decimal place.
Show Solution
$$A = \pi r^2$$ $$78.54 = \pi r^2$$ $$r^2 = \frac{78.54}{\pi} = 25$$ $$r = \sqrt{25} = 5\text{cm}$$
Answer: 5.0 cm
Question
A shape consists of a rectangle 8cm by 5cm with a semicircle attached to one of the shorter sides. Find the total area to 1 decimal place.
Show Solution
Rectangle area: $8 \times 5 = 40\text{cm}^2$
Semicircle: Diameter = 5cm, so radius = 2.5cm $$\text{Area} = \frac{1}{2} \times \pi \times 2.5^2 = \frac{1}{2} \times \pi \times 6.25 = 9.817...\text{cm}^2$$
Total: $40 + 9.817... = 49.8\text{cm}^2$ (1 d.p.)
Answer: 49.8 cm²
Examiner's View
Mark allocation: Basic area/volume is 2-3 marks. Composite shapes or reverse problems are 3-4 marks.
Common errors examiners see:
- Using diameter instead of radius
- Using wrong height (slant vs perpendicular)
- Unit conversion errors
- Forgetting to halve for triangles
What gains marks:
- Showing the formula you're using
- Stating units in your answer
- Breaking composite shapes into parts clearly
AQA Notes
AQA provides cone and sphere formulas on the formula sheet.