Mean with Shapes Practice Test Worksheets
Mean with Shapes questions challenge you to work out the hidden numbers on a 3D shape based on the mean of all face values. This page shows how to count the faces, multiply to find the total sum, and solve for the missing values.
For students who understand basic average calculations, but need practice applying it to 3D spatial geometry puzzles.
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What Do Mean with Shapes Questions Look Like?
Mean with shapes questions combine 3D shape geometry with arithmetic averages. You are given the mean of the numbers on all faces of a solid and must deduce the missing face values. Here is a worked example.
Triangle-Based Pyramid
Question: The mean of the numbers on the faces of the triangle-based pyramid shown is 6. What are the two hidden numbers?
Options:
- A. 6 and 7
- B. 6 and 10
- C. 6 and 15
- D. 15 and 16
- E. 1 and 8
Worked Method
Step 1: Identify the shape. A triangle-based pyramid has 4 faces.
Step 2: Find the total sum. The mean is 6, so: 4 × 6 = 24.
Step 3: Add the visible numbers: 1 + 2 = 3.
Step 4: Find the hidden total: 24 − 3 = 21.
Step 5: Match to the options. Only 6 and 15 add to 21.
Answer: 6 and 15 → C.
Tip: Pay close attention to the shape's base type. A triangle-based pyramid has 1 base + 3 sides = 4 faces total.
Rules to Help You Solve Mean with Shapes Questions
To solve mean with shapes questions like the example above, connect the shape face count to the total sum. Keep these four rules in mind:
🔍 Rule 1: Count the total faces
Different shapes have different face counts. You must know the count to calculate the total sum.
Pyramids: 4 faces (triangle-based) or 5 faces (square-based). Cube: 6 faces.
📠 Rule 2: Find the total sum
Multiply the given mean by the number of faces to find the sum of all numbers on the shape.
Example: A 5-faced shape with a mean of 7 has a total sum of 35.
➕ Rule 3: Add all known face values
Add every given face value, including numbers supplied by labels for faces you cannot see.
Note: The base is hidden from view, but its value is known because the label gives it as 12. Include each known value once.
➖ Rule 4: Find the missing total
Subtract the known sum from the total sum. This gives the combined total of the missing numbers, not their individual values. Choose the answer option whose numbers add to that total.
Example: Total 35 − Known 23 = 12. Both 2 and 10 and 5 and 7 have this sum. The total alone cannot distinguish them; in these questions, only one listed option has the required sum.
Two More Worked Examples
Here are two more examples showing how to calculate hidden values on prisms and cubes.
Triangular Prism
Question: The mean of the numbers on the faces of the triangular prism shown is 5. What are the two hidden numbers?
Options:
- A. 1 and 9
- B. 5 and 8
- C. 2 and 6
- D. 6 and 9
- E. 3 and 6
Worked Method
Step 1: Identify the shape. A triangular prism has 2 triangular bases + 3 rectangular sides = 5 faces.
Step 2: Find the total sum. The mean is 5, so: 5 × 5 = 25.
Step 3: Add the visible numbers: 1 + 2 + 9 = 12.
Step 4: Find the hidden total: 25 − 12 = 13.
Step 5: Match to the options. Only 5 and 8 add to 13.
Answer: 5 and 8 → B.
Tip: Keep prism face counts clear. A triangular prism has 5 faces, whereas a rectangular prism (like a cuboid) has 6 faces.
Cube with Hidden Values
Question: The mean of the numbers on the faces of the cube shown is 5. What are the three hidden numbers?
Options:
- A. 1, 2 and 3
- B. 3, 6 and 7
- C. 4, 5 and 6
- D. 1, 4 and 7
- E. 2, 3 and 4
Worked Method
Step 1: Identify the shape. A cube has 6 faces.
Step 2: Find the total sum. The mean is 5, so: 6 × 5 = 30.
Step 3: Add the visible numbers: 6 + 8 + 10 = 24.
Step 4: Find the hidden total: 30 − 24 = 6.
Step 5: Match to the options. Only 1, 2 and 3 add to 6.
Answer: 1, 2 and 3 → A.
Tip: The missing total of 6 is shared across three faces. Add all three numbers in each option; only option A has the required total. The mean alone does not determine each missing value.
