Geometry & Measures
Grade 5-6Volume & Surface Area of Cylinders
A cylinder can be treated as a circular prism, so its volume uses the same cross-sectional area times length idea as any other prism, while its curved surface area comes from unrolling the curved face into a rectangle. This lesson covers the volume and curved surface area formulae, working backwards from a given volume or surface area, and applying these formulae to hollow pipes and melting and recasting problems.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Volume of a cylinder
The volume of a cylinder is the area of its circular cross-section multiplied by its height: π × radius² × height. This follows the same pattern as any prism, using the circle area formula for the cross-section.
A cylinder has radius 4 cm and height 10 cm. Its volume is π × 4² × 10 = 160π cm³.
Curved surface area of a cylinder
The curved surface of a cylinder, if unrolled, forms a rectangle with one side equal to the circumference of the circular base and the other equal to the height. This gives a curved surface area of 2 × π × radius × height. The total surface area, if needed, also includes the two circular ends, each with area π × radius².
A cylinder has radius 3 cm and height 9 cm. Its curved surface area is 2 × π × 3 × 9 = 54π cm². Its total surface area also adds two circular ends: 2 × π × 3² = 18π cm², giving 54π + 18π = 72π cm² in total.
Working backwards from volume or surface area
If the volume or curved surface area is given along with one dimension, set up an equation using the appropriate formula and solve for the missing radius or height.
A cylinder has a volume of 250π cm³ and a radius of 5 cm. Since 250π = π × 5² × h, dividing both sides by 25π gives h = 10 cm.
Hollow pipes and melting and recasting
A hollow pipe is the difference between two cylinders of the same length: subtract the cross-sectional area of the inner circle from the outer circle before multiplying by the length. Melting and recasting problems keep the total volume of material the same before and after, even though the shape changes.
A hollow pipe has outer radius 6 cm, inner radius 4 cm and length 15 cm. Its cross-sectional area is π × (6² - 4²) = 20π cm², so its volume is 20π × 15 = 300π cm³. A cylindrical rod is melted down and recast into smaller discs of equal volume, so the volume of the original rod equals the total volume of all the discs combined.
Worked Examples
Three exam-style questions, fully solved.
Find the volume of a cylinder with radius 4 cm and height 10 cm, leaving your answer in terms of π.
Easy- 1.Use the volume formula: π × radius² × height with radius 4 and height 10
Answer: 160π cm³
A cylinder has a volume of 942 cm³ and a height of 12 cm. Find the radius of the cylinder. Use π = 3.14.
Medium- 1.Set up an equation using the volume formula: 942 = 3.14 × r² × 12
- 2.Divide both sides by 3.14 × 12: r² = 942 ÷ 37.68
- 3.Take the square root to find the radius
Answer: r = 5 cm
A cylinder has a volume of 150π cm³ and a curved surface area of 60π cm². Find the radius and the height of the cylinder.
Hard- 1.Write the volume equation: r² × h = 150
- 2.Write the curved surface area equation: 2 × r × h = 60, so r × h = 30, meaning h = 30 ÷ r
- 3.Substitute into the volume equation: r² × (30 ÷ r) = 150, so 30r = 150, giving r = 5
- 4.Find the height using r × h = 30: h = 30 ÷ 5
Answer: r = 5 cm, h = 6 cm
Avoid These
The most common mistakes students make.
Confusing the volume formula (π × radius² × height) with the curved surface area formula (2 × π × radius × height), especially mixing up which one squares the radius.
Using the diameter instead of the radius in either formula without halving it first.
When finding total surface area, forgetting to add the two circular ends to the curved surface area.
When a hollow pipe is involved, using the area of only the outer or inner circle instead of subtracting one from the other.
In melting and recasting problems, forgetting that the total volume stays the same, and instead trying to match a different measurement such as surface area.
FAQ
Questions parents and students ask.
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