Hey everyone! Mohi here from Singapore Maths Academy. I’m excited to walk you through a classic maths problem that comes up in both GCSE exams and our Year 8 class (which, by the way, is on Mondays at 6pm – feel free to join if you’re keen!).
Today, instead of working out the volume of a cylinder, we’re going to focus on finding its surface area.
Ready? Let’s jump in and break down the method, why it works, and how you can use it to ace questions on this topic.
What Exactly Is Surface Area?
Before we start crunching any numbers, it’s important to understand what surface area means for 3D shapes like cylinders.
Definition: Surface area is the sum of the areas of all the faces that make up a 3D shape.
Think of everyday items. The surface of your desk, the screen of your phone, or even the wrapping paper covering a gift – those are all surfaces.
Visualising the Cylinder
First, imagine you’re looking at a standard cylinder, like a can of beans or a Pringles tube. A cylinder consists of three main surfaces:
- The top circle
- The bottom circle (identical to the top one)
- The curved side (the bit that wraps around the tube, like the wall of the can)
Quick Visual: The “Net”
If we cut the cylinder open and lay it flat, we create what is called a net.
- The top is a circle.
- The bottom is another circle.
- The main body, if you were to cut it lengthwise and unroll it, turns into a rectangle!
The Three Surfaces of a Cylinder
To find the total surface area, we need to calculate the area of all three parts and then add them up:
- The Top Circle
- The Bottom Circle
- The Curved Surface (the rectangle)
Step 1: The Area of the Circles
You probably remember the formula for the area of a circle from your earlier maths lessons. If not, don’t worry – here it is:
Area of a circle = πr²
Where r is the radius of the circle.
Since our cylinder has two identical circles (top and bottom), we multiply this area by 2:
Area of both circles = 2πr²
Step 2: The Curved Surface (The Rectangle)
Here’s where it gets interesting. As we saw in the visual above, the curved part of the cylinder is actually a rectangle when you open it out.
But how big is this rectangle?
- Height: This is just the height of the cylinder (h).
- Width: The width of the rectangle is actually the distance around the cylinder. In math terms, that is the circumference of the circle!
Recap: Circumference of a Circle
The circumference (C) is the distance around the edge of a circle:
C = 2πr
So, the area of our rectangle is:
Area of rectangle = width × height Area of rectangle = (2πr) × h Area of rectangle = 2πrh
Step 3: Putting It All Together
Now, we add the area of the circles and the area of the rectangle together.
Total Surface Area = Area of both circles + Area of rectangle Total Surface Area = 2πr² + 2πrh
Let’s Factorise for Simplicity
Check this out: both terms have 2πr in them. We can factorise to make the formula easier to type into a calculator:
Surface Area = 2πr(r + h)
And that’s the neatest formula for the total surface area of a cylinder!
Full Example: Step-by-Step Calculation
Let’s do a full worked example using numbers so you can see how it all fits together.
Problem: Suppose you have a cylinder with a radius of 4 cm and a height of 10 cm. What is its surface area?
Step 1: Write Down the Formula Surface Area = 2πr(r + h)
Step 2: Plug in the Values
- r = 4
- h = 10
= 2π × 4(4 + 10) = 8π(14)
Step 3: Calculate 8 × 14 = 112
So, the exact answer is: Surface Area = 112π cm²
If you need a decimal answer, multiply by π (approx 3.14): Surface Area ≈ 351.86 cm²
Why Not Just Memorise the Formula?
Of course, at GCSE (and even A-level), you’ll often get the formula handed to you in your exam paper. But understanding why the rectangle width equals the circle’s circumference helps you remember it naturally.
Plus, if you ever need to work out a variation—like a hollow cylinder or a cylinder missing a lid—you’ll know exactly how to adapt the formula.
Common Mistakes to Avoid
- Missing the rectangle: Remember, the curved part is essential for the total surface area.
- Forgetting a circle: Don’t forget there are usually two circles (top and bottom).
- Mixing up formulas: Area of a circle is πr², circumference is 2πr. Don’t confuse them!
- Unit mismatch: Make sure your radius and height are in the same units (e.g., both in cm or both in m).
Practice Questions
Ready for a challenge? Try these:
- A cylinder has a radius of 7 cm and a height of 15 cm. Find its total surface area.
- If the height of a cylinder is 20 cm, the total surface area is 1884 cm², and the radius is 6 cm, check if these values make sense by plugging them into the formula.
Summary: Quick Reference Table
| Surface | Formula | Notes |
| Top circle | πr² | |
| Bottom circle | πr² | Multiply by 2 for both circles |
| Rectangle (side) | 2πrh | The unrolled “label” |
| Total | 2πr² + 2πrh | Expanded form |
| Factorised | 2πr(r + h) | Best for calculators |
FAQ
Does it matter which units I use? No, but radius and height must be in the same unit! The answer will be in units-squared (e.g., cm², m²).
What if the cylinder is “open top” (no lid)? Then just use one circle on the bottom, not two. Surface Area = πr² + 2πrh
Join Our Classes!
If this made sense and you want to learn more cool maths topics (or get ahead for your exams), join our Singapore Maths Academy classes.
The Year 8 class runs every Monday at 6pm, and we cover questions just like this—step by step, with plenty of time for questions.
Did this help you get a clearer understanding of surface area? Let us know in the comments!
Keep practising, and keep visualising!