Grade 8 Cylinder Volume Derivation: Link to Prism Volume - Unit Mastery Quiz with Full Solutions

Purpose

This is a Grade 8 unit mastery quiz for the idea that a cylinder follows the same volume structure as a prism: volume equals base area times perpendicular height.

Use this record to assess mastery of the link between prism volume and cylinder volume. For broader formula practice, see the linked records on prism volume, cylinder formulas, and formula selection rather than repeating those units here.

Linked records

• Prism rule focus: rea.m08.geometry-measurement.volume.basic-formulas.prism-bh.unit-mastery-quiz
• Cylinder formula fluency: rea.m08.geometry-measurement.volume.basic-formulas.cylinders.unit-mastery-quiz
• Mixed volume formula selection: rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.unit-mastery-quiz
• Broader unit overview quiz: rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz

Quiz structure

• 10 questions
• 20 marks total
• Suggested time: 25-35 minutes
• Calculator: optional unless your course requires mental estimation first

Success criteria

A learner shows mastery if they can:

• explain that a cylinder is handled by the prism structure V = Bh
• identify B as the area of the circular base, so B = pi r^2
• use perpendicular height correctly
• move flexibly between V = Bh and V = pi r^2 h
• solve direct and reverse problems with correct units

Unit Mastery Quiz

Question 1 (2 marks)

A right prism has volume V = Bh. Explain why a cylinder can also use V = Bh.

Question 2 (2 marks)

For a cylinder with radius 5 cm and height 9 cm:

1. write the base area B
2. find the volume in exact form

Question 3 (2 marks)

A cylinder has diameter 12 m and height 4 m. Find its volume in terms of pi.

Question 4 (2 marks)

A cylinder and a prism each have the same base area, 30 cm^2, and the same height, 8 cm.

1. Find each volume.
2. State what this shows about the prism-volume link.

Question 5 (2 marks)

A student says, "Cylinder volume is a different idea from prism volume because one has a circle and one does not." Is the student correct? Explain.

Question 6 (2 marks)

Find the volume of a cylinder with base area 49pi cm^2 and height 11 cm.

Question 7 (2 marks)

A cylinder has volume 200pi cm^3 and height 8 cm. Find the radius.

Question 8 (2 marks)

A can is shaped like a cylinder with radius 3.5 cm and perpendicular height 10 cm.

1. Write the volume formula using B.
2. Rewrite it using r.
3. Calculate the volume to the nearest tenth.

Question 9 (2 marks)

A cylinder has radius 2 cm and height 7 cm. Another cylinder has radius 4 cm and the same height. Without fully recalculating both volumes from scratch, determine how the larger volume compares to the smaller one. Explain.

Question 10 (2 marks)

A cylinder and a triangular prism both follow V = Bh. Explain what changes and what stays the same when moving from the triangular prism formula to the cylinder formula.

Full Marking Solutions

Question 1 solution

A cylinder can use V = Bh because volume for prism-type solids is found by taking the area of a cross-section/base and multiplying by the perpendicular height. A cylinder has the same circular base all the way through, so its volume is also base area times height.

Marking:

• 1 mark for stating volume is base area times perpendicular height
• 1 mark for connecting the cylinder's constant circular base to that rule

Question 2 solution

Given r = 5 cm, h = 9 cm

1. Base area: B = pi r^2 = pi(5)^2 = 25pi cm^2

2. Volume: V = Bh = 25pi x 9 = 225pi cm^3

Marking:

• 1 mark for correct base area
• 1 mark for correct volume with units

Question 3 solution

Diameter is 12 m, so radius is 6 m.

V = pi r^2 h V = pi(6)^2(4) V = pi(36)(4) V = 144pi m^3

Marking:

• 1 mark for converting diameter to radius correctly
• 1 mark for correct volume

Question 4 solution

Since both solids have B = 30 cm^2 and h = 8 cm, each volume is

V = Bh = 30 x 8 = 240 cm^3

So both volumes are 240 cm^3.

This shows that the shape of the base can differ, but if the base area and perpendicular height are the same, the volume is the same.

Marking:

• 1 mark for the correct volume
• 1 mark for the correct interpretation of the prism-volume link

Question 5 solution

The student is not correct.

It is true that the bases are different shapes, but the volume structure is the same. For both prisms and cylinders, V = Bh. The only change is how the base area is found. For a cylinder, B = pi r^2 because the base is a circle.

Marking:

• 1 mark for correctly disagreeing
• 1 mark for explaining that the structure stays V = Bh while the base-area formula changes

Question 6 solution

Given B = 49pi cm^2 and h = 11 cm

V = Bh V = 49pi x 11 V = 539pi cm^3

Marking:

• 1 mark for using V = Bh
• 1 mark for correct result with units

Question 7 solution

Given V = 200pi cm^3, h = 8 cm

Use V = pi r^2 h: 200pi = pi r^2 (8)

Divide by pi: 200 = 8r^2

Divide by 8: 25 = r^2

r = 5 cm

Marking:

• 1 mark for correct equation and rearrangement
• 1 mark for correct radius

Question 8 solution

Given r = 3.5 cm, h = 10 cm

1. Using B: V = Bh

2. Using r: V = pi r^2 h

3. Calculate: V = pi(3.5)^2(10) V = pi(12.25)(10) V = 122.5pi V approx 384.8 cm^3

Marking:

• 1 mark for both correct formula forms
• 1 mark for correct numerical value to the nearest tenth

Question 9 solution

Volume depends on r^2 when height stays the same.

The radius changes from 2 cm to 4 cm, which is doubling the radius. Since 4^2 = 16 and 2^2 = 4, the base area and volume are multiplied by 16/4 = 4.

So the larger cylinder has 4 times the volume of the smaller cylinder.

Marking:

• 1 mark for recognizing that squaring the radius matters
• 1 mark for the correct comparison factor of 4

Question 10 solution

What stays the same:

• the volume structure V = Bh
• h still means perpendicular height

What changes:

• the base shape changes
• therefore the formula for B changes

For a triangular prism, B is the area of a triangle. For a cylinder, B = pi r^2 because the base is a circle.

So the cylinder formula V = pi r^2 h is not a new volume idea; it is the prism rule with a circular base substituted in.

Marking:

• 1 mark for identifying what stays the same
• 1 mark for identifying what changes

Answer key summary

1. Cylinder also uses base area times perpendicular height because the circular cross-section stays constant.
2. B = 25pi cm^2, V = 225pi cm^3
3. 144pi m^3
4. Each volume 240 cm^3; same B and h give same volume.
5. No; the structure remains V = Bh, only the base-area formula changes.
6. 539pi cm^3
7. 5 cm
8. V = Bh, V = pi r^2 h, V approx 384.8 cm^3
9. The larger cylinder has 4 times the volume.
10. Same structure V = Bh; different base-area formula.

Mastery threshold recommendation

Recommended mastery threshold: 16/20 (80%) with no major conceptual error on the prism-to-cylinder link.

Interpretation:

• 18-20: secure mastery
• 16-17: acceptable mastery
• 13-15: developing; review the meaning of B and the role of perpendicular height
• 0-12: not yet secure; revisit the linked prism and cylinder records before reassessment

A major conceptual error includes any of the following:

• treating cylinder volume as unrelated to V = Bh
• using diameter as radius without halving it
• confusing base area with circumference
• omitting cubic units on volume

Common misconceptions to watch for

• Mistake: Thinking B means "bottom length" instead of base area. Repair: In volume, B is always an area measurement.
• Mistake: Using pi d^2 instead of pi r^2. Repair: Convert diameter to radius first.
• Mistake: Believing cylinders need a completely separate rule from prisms. Repair: Start from V = Bh, then substitute the circular base area.
• Mistake: Using slanted or non-perpendicular height in a diagram. Repair: Height for volume is perpendicular distance between bases.

Extension prompts

• Explain why a cylinder can be viewed as a prism-like solid with infinitely many sides in the base boundary model.
• Compare how changing radius and changing height affect volume differently.
• Create two different solids with the same base area and same height, then justify why their volumes match under V = Bh.