Ever wondered how much paint you need for a water tank or how many ice‑cream scoops fit in a round bowl?
In simple words, surface area tells you how much outer skin a 3‑D shape has, while volume tells you how much space is inside. Think of a chocolate bar: the wrapper covers the surface area, and the chocolate itself fills the volume.
Cylinders – Surface Area and Volume
A cylinder looks like a soup can – two identical circles stacked and joined by a curved side. The round parts are called bases and the distance between them is the height. The distance from the centre of a base to its edge is the radius.
Key formulas
- Surface area = 2πr(r + h). The first 2πr gives the area of the two circles, and the second (r + h) part adds the curved side (called the lateral surface).
- Volume = πr²h. Imagine stacking many thin discs of area πr² on top of each other; the total height h gives the volume.
Worked example
Find the surface area and volume of a cylinder with radius 5 cm and height 12 cm.
- Surface area: 2 × π × 5 × (5 + 12) = 10π × 17 = 170π ≈ 534 cm².
- Volume: π × 5² × 12 = π × 25 × 12 = 300π ≈ 942 cm³.
So you’d need about 534 cm² of paint and the can would hold roughly 942 cm³ of liquid.
Cones – Surface Area and Volume
A cone is like an ice‑cream scoop: a circular base and a pointy tip called the apex. The slant from the base edge to the apex is the slant height (usually denoted l). It’s longer than the straight height because it runs along the side.
Key formulas
- Surface area = πr(r + l). The πr part is the base area, and πr l is the curved side (lateral surface).
- Volume = (1/3)πr²h. A cone holds exactly one‑third the volume of a cylinder with the same base and height.
Worked example
Radius = 4 cm, height = 9 cm. First find the slant height using the Pythagoras theorem: l = √(r² + h²) = √(4² + 9²) = √(16 + 81) = √97 ≈ 9.85 cm.
- Surface area: π × 4 × (4 + 9.85) = 4π × 13.85 ≈ 174 cm².
- Volume: (1/3) × π × 4² × 9 = (1/3) × π × 16 × 9 = 48π ≈ 151 cm³.
Spheres – Surface Area and Volume
A sphere is a perfect ball, like a basketball. Every point on its surface is the same distance from the centre. That distance is the radius. Because there are no edges or corners, the formulas are a bit shorter.
Key formulas
- Surface area = 4πr². Imagine four circles stacked together – that’s the total skin of the ball.
- Volume = (4/3)πr³. The extra 4/3 factor comes from integrating tiny discs from the top to the bottom of the sphere.
Worked example
Radius = 6 cm.
- Surface area: 4 × π × 6² = 4π × 36 = 144π ≈ 452 cm².
- Volume: (4/3) × π × 6³ = (4/3)π × 216 = 288π ≈ 904 cm³.
Quick Comparison Table
| Shape | Surface Area Formula | Volume Formula |
|---|---|---|
| Cylinder | 2πr(r + h) | πr²h |
| Cone | πr(r + l) where l = √(r² + h²) | (1/3)πr²h |
| Sphere | 4πr² | (4/3)πr³ |
📝 Likely Exam Questions
- Find the total surface area of a closed cylinder with radius 3 cm and height 10 cm.
Answer: 2π·3·(3+10)=6π·13=78π ≈ 245 cm². - A cone has a base radius of 5 cm and a slant height of 13 cm. Calculate its lateral surface area.
Answer: π·5·13 = 65π ≈ 204 cm². - The volume of a sphere is 904 cm³. Determine its radius (use π≈3.14).
Answer: (4/3)πr³ = 904 → r³ = 904·3/(4·π) ≈ 216 → r ≈ 6 cm. - A cylindrical water tank of radius 7 m and height 4 m is to be painted on the outside only. How much paint is needed if 1 litre covers 10 m²? (Ignore waste.)
Answer: Surface area = 2π·7·(7+4)=14π·11=154π ≈ 484 m². Paint needed = 484/10 ≈ 48.4 litres. - Compare the volume of a cone with radius 3 cm and height 9 cm to a cylinder with the same base and height.
Answer: Cone volume = (1/3)π·3²·9 = 27π ≈ 85 cm³. Cylinder volume = π·3²·9 = 81π ≈ 254 cm³. The cone holds one‑third of the cylinder’s volume.