Surface Area of Cylinders, Cones and Spheres
[PATHS extension] Extend surface area to curved solids. Unroll cylinders, unwrap cones, and cover spheres.
Printable Worksheets
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Worksheet
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Q1: Imagine peeling the label off a cylindrical can. What flat shape would it be? What would its dimensions be in terms of the can's radius and height?
Q2: A sphere and a cube both have volume 1000 cm$^3$. Which do you think has the larger surface area? Make a prediction and explain your reasoning.
Learning Intentions
Know
- The surface area formulas for cylinders, cones and spheres.
- How curved surfaces unfold into flat shapes.
Understand
- Why the curved surface area of a cylinder is a rectangle when unrolled.
- Why a sphere's surface area formula contains $r^2$.
Can Do
- Calculate total and curved surface area of cylinders, cones and spheres.
- Solve composite solid problems involving curved surfaces.
Success Criteria
- I can calculate the total surface area of a closed cylinder, open cylinder, cone and sphere.
- I can identify which formula applies to the curved part and which to the base(s).
- I can solve practical problems involving painting, wrapping or covering curved objects.
Key Terms
Common Mistakes to Avoid
Wrong: Using the vertical height of a cone instead of the slant height in the curved surface area formula.
Right: CSA of cone = $\pi r l$ where $l = \sqrt{r^2 + h^2}$. Always calculate slant height first.
Wrong: Adding two circular bases to the surface area of an open cylinder or cone.
Right: A closed cylinder has two bases; an open cylinder has one or none. A cone has one base (unless open at both ends).
When you unroll a cylinder, the curved surface becomes a rectangle. The rectangle's width is the cylinder's height, and its length is the circumference of the base.
What to write in your book
- Cylinder CSA: $2\pi rh$ (unrolls to a rectangle)
- Closed cylinder TSA: $2\pi rh + 2\pi r^2$
- Open-top cylinder TSA: $2\pi rh + \pi r^2$
- The rectangle's width = height, length = circumference
Correct! $CSA = 2\pi rh = 2\pi(5)(8) = 80\pi$ cm$^2$.
Not quite. $CSA = 2\pi rh = 2\pi(5)(8) = 80\pi$ cm$^2$.
A cone's curved surface unrolls into a sector of a circle. A sphere has no flat faces — every point is curved.
Remember: A sphere has the smallest surface area for a given volume of any 3D shape. Nature uses spheres (bubbles, planets, cells) because they are efficient.
What to write in your book
- Cone CSA: $\pi rl$ where $l = \sqrt{r^2+h^2}$
- Cone TSA: $\pi rl + \pi r^2$
- Sphere SA: $4\pi r^2$
- Always calculate slant height first for cones
Correct! $SA = 4\pi r^2 = 4\pi(3)^2 = 36\pi$ cm$^2$.
Not quite. $SA = 4\pi r^2 = 4\pi(3)^2 = 36\pi$ cm$^2$.
Interactive: Cylinder Surface Area Calculator
Your Turn
Question 1: Find the total surface area of a closed cylinder with radius 4 cm and height 10 cm.
Question 2: Find the surface area of a sphere with radius 6 cm.
Question 3: A cone has radius 3 cm and height 4 cm. Find its total surface area.
Revisit Your Thinking
Look back at your Think First prediction about unrolling a cylinder's label. Was the flat shape a rectangle? What were its dimensions in terms of the can's radius and height? If a can has radius 3.5 cm and height 10.5 cm, what is the exact area of the label (curved surface only)?
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
What is the curved surface area of a cylinder with radius 5 cm and height 8 cm?
A sphere has radius 3 cm. What is its surface area?
A cone has radius 6 cm and perpendicular height 8 cm. What is its slant height?
An open cylinder (no top) has radius 4 cm and height 9 cm. What is its total surface area?
Which solid has the largest surface area for a given volume?
A closed cylindrical water tank has diameter 1.4 m and height 2.1 m.
(a) Find the total surface area of the tank. (2 marks)
(b) If the tank is painted and 1 litre of paint covers 8 m$^2$, how many litres are needed? (1 mark)
A solid is formed by placing a hemisphere of radius 5 cm on top of a cylinder of radius 5 cm and height 8 cm.
(a) Find the curved surface area of the cylinder. (1 mark)
(b) Find the curved surface area of the hemisphere. (1 mark)
(c) Find the total surface area of the composite solid. (2 marks)
A conical tent has a base diameter of 3 m and a perpendicular height of 2 m.
(a) Find the slant height of the tent. (1 mark)
(b) Find the curved surface area of the tent. (2 marks)
(c) Waterproof fabric costs $\$28$ per m$^2$. Calculate the cost of the fabric for the tent. (2 marks)
Cylinder CSA
$2\pi rh$
Cylinder TSA
$2\pi rh + 2\pi r^2$
Cone CSA
$\pi rl$
Cone TSA
$\pi rl + \pi r^2$
Sphere
$4\pi r^2$
Slant height
$l = \sqrt{r^2+h^2}$
Real-Life Link
Hot water cylinders in Australian homes are typically insulated to reduce heat loss. The amount of insulation needed depends directly on the curved surface area. A standard 250-litre cylinder has radius about 27 cm and height 145 cm, giving a curved surface area of about 2.5 m$^2$. When designing water tanks for rural Australian properties, engineers calculate the total surface area to determine the amount of corrugated steel or concrete needed. For spherical storage tanks used in industrial settings, the spherical shape minimises surface area (and therefore heat loss and material cost) for a given volume. This is why LNG storage tanks are often spherical.
Game Time!
Test your curved surface area skills in an interactive challenge.
Play Curved SA Challenge