Why Care About Cylinders, Cones, and Spheres?
Ever wondered how much paint you need for a water tank, or how many ice‑cream scoops fit in a cone? Those are real‑world uses of surface area and volume.
💡 In Simple Words: Surface area tells you how much skin a shape has. Volume tells you how much space it can hold. For cylinders, cones and spheres, we just plug the right numbers into a few neat formulas.
Surface Area of a Cylinder
A cylinder looks like a soup can. It has two circular ends (called bases) and a curved side (called the lateral surface). The radius is the distance from the centre of a base to its edge, and the height is how tall the can is.
Formula: Surface Area = 2πr (r + h)
- 2πr is the circumference of the base (think of walking around the rim).
- Multiply that by the height to get the curved part.
- Then add 2πr² for the two circular ends.
Worked Example: A cylindrical tank has radius 5 cm and height 12 cm. Find its total surface area (ignore the top if it’s open).
Step 1: Find curved area = 2πr × h = 2 × 3.14 × 5 × 12 = 376.8 cm².
Step 2: Find base area = πr² = 3.14 × 5² = 78.5 cm².
Step 3: If the top is open, surface area = curved + one base = 376.8 + 78.5 = 455.3 cm².
Volume of a Cylinder
Volume is the amount of space inside. For a cylinder it’s simply the area of the base times the height.
Formula: Volume = πr²h
Example: Same tank, radius 5 cm, height 12 cm.
Volume = 3.14 × 5² × 12 = 3.14 × 25 × 12 = 942 cm³.
Surface Area of a Cone
A cone is like an ice‑cream scoop on a wafer. It has a circular base and a slanting side called the lateral surface. The slant height (ℓ) is the distance from the tip to any point on the rim, measured along the side.
Formula: Surface Area = πr (r + ℓ)
- πr² is the area of the base.
- πrℓ is the curved part.
Worked Example: A cone has radius 4 cm and slant height 9 cm.
Surface Area = 3.14 × 4 × (4 + 9) = 3.14 × 4 × 13 = 163.28 cm².
Volume of a Cone
Think of a cone as a cylinder that’s been sliced down to a point. Its volume is one‑third of a cylinder with the same base and height.
Formula: Volume = (1/3)πr²h
Example: Same cone, radius 4 cm, height 12 cm.
Volume = (1/3) × 3.14 × 4² × 12 = (1/3) × 3.14 × 16 × 12 ≈ 200.96 cm³.
Surface Area of a Sphere
A sphere is a perfect ball, like a basketball. It has only one continuous surface.
Formula: Surface Area = 4πr²
Example: A sphere with radius 7 cm.
Surface Area = 4 × 3.14 × 7² = 4 × 3.14 × 49 = 615.44 cm².
Volume of a Sphere
Volume tells you how much air fits inside a ball.
Formula: Volume = (4/3)πr³
Example: Same sphere, radius 7 cm.
Volume = (4/3) × 3.14 × 7³ = (4/3) × 3.14 × 343 ≈ 1436.03 cm³.
Quick Comparison of Formulas
| Shape | Surface Area | Volume |
|---|---|---|
| Cylinder | 2πr(r + h) | πr²h |
| Cone | πr(r + ℓ) | (1/3)πr²h |
| Sphere | 4πr² | (4/3)πr³ |
📝 Likely Exam Questions
- Find the total surface area of a closed cylindrical tank with radius 6 cm and height 15 cm.
Answer: SA = 2πr(r + h) = 2 × 3.14 × 6 × (6 + 15) = 2 × 3.14 × 6 × 21 = 791.88 cm². - A cone has a base radius of 5 cm and a height of 12 cm. Calculate its volume.
Answer: V = (1/3)πr²h = (1/3) × 3.14 × 25 × 12 ≈ 314 cm³. - Determine the surface area of a sphere whose diameter is 10 cm.
Answer: r = 5 cm, SA = 4πr² = 4 × 3.14 × 25 = 314 cm². - A cylindrical pipe 8 cm in radius is 30 cm long. How much water can it hold?
Answer: V = πr²h = 3.14 × 64 × 30 = 6031.2 cm³. - Find the curved surface area of a cone with slant height 13 cm and base radius 3 cm.
Answer: CSA = πrℓ = 3.14 × 3 × 13 = 122.46 cm².