Edexcel Summer 2024 3H Question 9

Original exam question

The diagram shows a cube and a square-based pyramid.

The volume of the cube is equal to the volume of the pyramid.
Work out the perpendicular height, h cm, of the pyramid.

Exam Alert: Questions combining 3D shapes and equating their volumes are common in GCSE Maths, especially on the AQA, Edexcel, and OCR papers. They test your ability to recall volume formulas, substitute values correctly, and solve the resulting equation. Let's break down a classic example.

The Question

You are given:

A cube with side length 6 cm.

A square-based pyramid with a base measuring 8 cm by 8 cm and an unknown perpendicular height h cm.

A key statement: The volume of the cube is equal to the volume of the pyramid.

Your task: Work out the perpendicular height, h , of the pyramid.

Step 1: Visualising the Problem

It always helps to sketch the shapes, even if the exam paper provides a diagram. Here’s a simple representation of what we’re dealing with:

flowchart TD
    A["Start: Two 3D Shapes"] --> B["Cube"]
    A --> C["Square-Based Pyramid"]
    B --> D["Side = 6 cm"]
    C --> E["Base: 8 cm × 8 cm, Height = h cm"]
    D --> F["Volume of Cube"]
    E --> G["Volume of Pyramid"]
    F --> H["Key Condition:, Volume Cube = Volume Pyramid"]
    G --> H
    H --> I["Form an equation"]
    I --> J["Solve for h"]

Step 2: Recall the Essential Formulas

You must know these volume formulas for your GCSE:

Volume of a cube (or any cuboid):

V_cube = side × side × side = side³

Volume of a pyramid:

V_pyramid = ¹⁄₃ × area of base × perpendicular height

Crucial note: The height must be the perpendicular height from the apex to the base. For our square-based pyramid, the base area is simply side × side.

Step 3: Calculate the Volume of the Cube

Substitute the given side length into the cube formula:

V_cube = 6³

V_cube = 6 × 6 × 6 = 216

So, the volume of the cube is 216 cm³.

Step 4: Write an Expression for the Volume of the Pyramid

First, find the area of the square base:

Base Area = 8 × 8 = 64 cm²

Now, apply the pyramid volume formula. The volume is:

V_pyramid = ¹⁄₃ × 64 × h

V_pyramid = ^64h⁄₃

Step 5: Apply the "Equal Volumes" Condition and Solve

The question states: V_cube = V_pyramid

Therefore:
216 = ^64h⁄₃

This is now an equation in h . Let's solve it step-by-step.

Step 5.1: Multiply both sides by 3 to eliminate the fraction.
216 × 3 = 64h
648 = 64h

Step 5.2: Divide both sides by 64 to isolate h .
h = ⁶⁴⁸⁄₆₄

Step 5.3: Simplify the fraction. Both numerator and denominator are divisible by 8.
h = ^{648 ÷ 8}⁄_{64 ÷ 8} = ⁸¹⁄₈

h = 10.125

Step 6: State the Final Answer with Units

The perpendicular height of the pyramid is:

h = 10.125 cm

You could also leave the answer as the exact fraction ⁸¹⁄₈ cm, which is often preferable unless the question specifies a decimal.

Common Pitfalls and Exam Tips

Using the wrong height: For the pyramid volume, you must use the perpendicular height, not the slant height.

Forgetting the ¹⁄₃: The factor of ¹⁄₃ in the pyramid formula is easily missed. Remember: Pyramid volume is one-third of the corresponding cuboid's volume.

Incorrect base area: For a square-based pyramid, ensure you calculate the area (side²) correctly.

Algebraic errors: When solving 216 = ^64h⁄₃ , a common mistake is to incorrectly multiply or ÷. Take it slowly.

Units: Always include units (cm for height, cm³ for volume) in your working where appropriate, and definitely in your final answer.

Why This Question is Important

This problem is a great example of a multi-topic question at GCSE. It tests:

Geometry: Knowledge of 3D shapes and their properties.

Number: Arithmetic with fractions and decimals.

Algebra: Forming and solving an equation from a worded condition.

Mastering this type of question builds confidence for the longer, more complex problems found in Papers 2 and 3.

Try a Similar Question (Self-Test)

> A cube has a side length of 5 cm. A square-based pyramid has a base of 7.5 cm by 7.5 cm. If their volumes are equal, what is the perpendicular height of the pyramid?

(Answer at the bottom of the post.)

Final Thought: Always break 3D problems down into clear steps:

1. Sketch,
2. Write formulas,
3. Calculate known parts,
4. Form an equation,
5. Solve.

This structured approach turns a daunting 3D problem into a manageable sequence of calculations.