What is the area of the curved surface of a cylinder?
Let the radius of the circular base of the cylinder be r; let its height be h. Suppose the cylinder is cut as shown in the figure below. We get a rectangle.
Step-by-step solution
Idea: When the curved surface is cut and flattened it becomes a rectangle. One side of the rectangle was the circle round the base, the other was the height.
- Cut the curved surface straight down and open it out. It lies flat as a rectangle, with no overlap and no gap.½ mark
- The rectangle’s length is the edge that went round the base circle: circumference = 2πr. Its width is the height h.1 mark
- Area of rectangle = length × width = 2πr × h, so CSA = 2πrh.1 mark
- If both ends are closed, add two circles: TSA = 2πrh + 2πr2 = 2πr(h + r). If one end is open (like a tumbler): 2πrh + πr2 = πr(2h + r).½ mark
Check: Wrap a sheet of paper 22 cm × 10 cm round into a tube of height 10 cm: its radius is 22 ÷ (2π) ≈ 3.5 cm, and 2π × 3.5 × 10 ≈ 220 cm2 = 22 × 10 ✓.
Answer to write in the exam
Curved surface opened out = rectangle
Length = circumference of base = 2πr; width = h
CSA = 2πr × h
∴ CSA = 2πrh; TSA = 2πrh + 2πr2 = 2πr(h + r)
Common mistakes that cost marks
- Using the diameter in place of the radius: 2πdh gives double the area.
- Writing πr2h (the volume) for the curved surface area.
- Adding the two circular ends when only the curved surface is asked for.
How this can come in the exam
The curved surface area of a cylinder of radius 7 cm and height 10 cm is (π = 227)
- 220 cm2
- 440 cm2
- 1540 cm2
- 748 cm2
Show answer
(B) 440 cm2
2 × 227 × 7 × 10 = 440 cm2. (748 cm2 would be the total surface area.)
A closed cylindrical tin has radius 14 cm and height 20 cm. Find the area of tin sheet used. (π = 227)
Show answer
TSA = 2πr(h + r) = 2 × 227 × 14 × 34 = 88 × 34 = 2992 cm2.Try one yourself
Find the curved surface area of a pipe of radius 3.5 cm and length 2 m. (π = 227)
Show answer
2 m = 200 cm; CSA = 2 × 227 × 3.5 × 200 = 4400 cm2.
More questions like this
- Savitri had to make a model of a cylindrical kaleidoscope for her science project. She wanted to use chart paper to make the curved surface of the kaleidoscope (see the figure). What would be the area of chart paper required by her, if she wanted to make a kaleidoscope of length 25 cm with a 3.5 cm radius? You may take π = 227.
- Two cylinders, A and B, are given. The radius of cylinder B is twice that of cylinder A, and the height of cylinder B is half that of cylinder A. Find the ratio of the curved surface area of A to the curved surface area of B. Also find the ratio of the volume of A to the volume of B.
- The radii of two cylinders are in the ratio 2:3, and their heights are in the ratio 3:2. Find
- The edge of a cube measures r cm. The largest possible right circular cylinder is cut out of the cube. What do you think is the volume of the cylinder (in cm3)?
- The radius of the base of a cylinder is increased by 10%. At the same time, the height of the cylinder is decreased by x%. Given that the volume of the cylinder remains unchanged, find the value of x.