Practice · Grade 9 Solid Geometry
Hemisphere Surface — Exercises
Practice the surface of a hemisphere in rising difficulty: curved area 2πr², total surface 3πr², plus a boss question. Hint and worked solution, free.
Q1 of 7
0 correct
Find the curved area (the dome only).
r = 3, A = 2πr²
Quick answer
What is the best way to practice hemisphere surface area?
For each problem, decide first whether only the curved dome (2πr²) or the total surface including the base (3πr²) is being asked. Then square the radius, plug into the right formula, and round sensibly. Example r = 6: dome ≈ 226.19, total ≈ 339.29. Work through several problems with changing values of r and the diameter so the distinction between 2πr² and 3πr² becomes second nature.
HowTo
Solving strategy in 4 steps
This order works for every hemisphere problem — curved area or total surface.
- 1Step 1 of 4
Read: dome or total surface?
If only the curved area (shell, open dome) is asked, you need 2πr². If the circular base should count too (closed, solid half-body), use 3πr². This question decides everything that follows.
- 2Step 2 of 4
Sort out the radius
If only the diameter d is given, halve it first: r = d/2. A diameter d = 12 therefore gives r = 6. Continue with r, never with d.
- 3Step 3 of 4
Square the radius
Compute r². Example: 6² = 36. The square appears in both formulas — 2πr without the square is a classic mistake.
- 4Step 4 of 4
Substitute, calculate, round
Put r² into 2πr² or 3πr². Total for r = 6: 3π · 36 ≈ 339.29. Round to two decimal places unless something else is required.
Examples
Worked examples with full calculation
Four typical problems from Grade 9 tests. Try each one yourself first, then compare.
Easy
Curved area for r = 3.
A = 2πr²
= 2π · 3²
= 2π · 9 ≈ 56.55
Check: 2 · 3.1416 · 9 ≈ 56.55 ✓
Just the dome — the base does not count here.
Medium
Total surface area for r = 5.
A = 3πr²
= 3π · 5²
= 3π · 25 ≈ 235.62
Check: dome 2π·25 ≈ 157.08 + base π·25 ≈ 78.54 ≈ 235.62 ✓
Total = dome plus base. The factor 3 combines both.
Medium
Total surface area from the diameter d = 12.
r = d/2 = 6
A = 3πr² = 3π · 36
≈ 339.29
Check: 3 · 3.1416 · 36 ≈ 339.29 ✓
Form the radius first, otherwise you compute with double the value.
Hard
Boss: base area for r = 6 (the base circle only).
base = πr²
= π · 6² = π · 36
≈ 113.10
Check: total 339.29 − dome 226.19 ≈ 113.10 ✓
The base is the difference between total surface (3πr²) and dome (2πr²).
Pitfalls
Common mistakes — and how to avoid them
These five traps show up in almost every hemisphere problem.
Confusing dome and total surface
2πr² is only the curved area. With the base it is 3πr². Mark in the problem whether the base circle counts before you choose the formula.
Not squaring the radius
Both formulas contain r². 2πr (without the square) or 3πr is wrong — square first, then multiply by 2π or 3π.
Computing with the full sphere
The sphere has 4πr². The dome of the hemisphere is only half, so 2πr² — not 4πr².
Using the diameter instead of the radius
If d is given, halve it first: r = d/2. Plugging d in directly gives four times the correct value.
Rounding too early or wrongly
Compute with π = 3.14159… and round only at the end to two decimal places. Early rounding distorts the result.
Study
Practice with a plan — three short tips
Pick the formula first, then calculate
Say out loud for each problem whether you need 2πr² or 3πr². This single decision prevents the most common source of error.
15 minutes at a time, spread over several days
Three short sessions on three days beat one long session the night before the test. Keyword: spaced repetition.
Question every wrong answer
Was it the formula choice, the missing square, or the diameter? Note the cause — next time you will spot the mistake instantly.
FAQ
Frequently asked questions about practicing
Glossary
Terms in one sentence
- Hemisphere
- Half of a sphere, cut through the center.
- Curved area
- The dome of the hemisphere, with area 2πr².
- Base circle
- The circular cut face at the bottom, with area πr².
- Total surface
- Dome plus base, that is 2πr² + πr² = 3πr².
- Sphere surface
- Surface of the full sphere, 4πr².
- Radius (r)
- Distance from the center to the surface; half the diameter.
- Boss question
- The last and hardest problem of a practice set, combining several steps.