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How to find the surface area of a hemisphere — step by step (Grade 9)

Q: When do you use 2πr² and when 3πr²?

Use 2πr² for the curved surface alone, such as a bowl or an open dome. Use 3πr² when the flat circular base is part of the surface too, that is, for a closed, solid half-body.

Q: Why is the dome 2πr²?

The dome of a hemisphere is exactly half of the full sphere's surface. Since a sphere has 4πr², half of it is 4πr² ÷ 2 = 2πr².

Q: What is the area of the base?

The base is a circle with the same radius r as the hemisphere, so its area is πr². This is exactly the part that separates the total surface from the curved surface alone.

Q: How do I work with the diameter instead of the radius?

First find the radius: r = d/2. Then substitute r into 2πr² or 3πr² as usual. A diameter d = 12 therefore gives r = 6.

Q: What is the difference from a sphere's surface area?

A full sphere has surface area 4πr². The closed hemisphere has only 3πr², because half a dome (2πr²) plus one base (πr²) together add up to less than the whole sphere.

A hemisphere has two surfaces: the curved dome 2πr² and the flat circular base πr². Together they give the total surface 3πr². Worked example: r = 6 → total ≈ 339.29, dome only ≈ 226.19. This topic belongs to the solid geometry of Grade 9 / Year 10.

By Valerij Hofmann · Fullstack Developer

Updated May 29, 2026Intermediate

Quick answer

A hemisphere has two surface formulas: the curved surface (the dome only) is 2πr², and the total surface including the flat base is 3πr². For r = 6, the dome is ≈ 226.19 and the total surface is ≈ 339.29.

At a glance

Summary of this tutorial
Formula	3πr²
Method	Dome 2πr² + base πr²
Steps	4
Result (r = 6)	≈ 339.29
Dome only (r = 6)	≈ 226.19
Grade level	Grade 9 (ages 14–15)

Worked example: 3πr² for r = 6

EXAMPLE

3π · 6²

We substitute r = 6 into the total formula 3πr² to get the complete outer surface.

The steps to a hemisphere's surface area

These steps work for any hemisphere — you just decide whether the base counts.

Step 1 · Start
3x + 7 = 22
Identify the unknown x and the target 22.
Step 2 · Subtract 7
3x = 22 − 7
Undo the “+ 7”. Same operation on both sides keeps the balance.
Step 3 · Simplify
3x = 15
Tidy up the right-hand side.
Step 4 · Divide by 3
x = 15 ÷ 3
Undo the “× 3”. Now x is alone.
Step 5 · Check
x = 5
Substitute back: 3 · 5 + 7 = 22 ✓

Why 3πr² is correct

The dome is exactly half of the full sphere's surface 4πr², so it is 2πr². The flat cut is a circle with area πr². Adding both parts gives 2πr² + πr² = 3πr². The factor 3 is no coincidence — it is "two half-sphere parts plus one base part".

Practice it yourself

Open the hemisphere calculator

Type your own radius. See the dome and total surface with every step.

Start practice set

Hemisphere problems

Frequently asked questions

How do you find the surface area of a hemisphere?

Use two formulas: the curved surface (the dome only) is 2πr², and the total surface including the circular base is 3πr². Substitute the radius and square it. Example r = 6: dome = 2π · 36 ≈ 226.19; total = 3π · 36 ≈ 339.29.

When do you use 2πr² and when 3πr²?

Why is the dome 2πr²?

What is the area of the base?

How do I work with the diameter instead of the radius?

What is the difference from a sphere's surface area?

Quick answer

At a glance

Worked example: 3πr² for r = 6

The steps to a hemisphere's surface area

Step 1 · Start

Step 2 · Subtract 7

Step 3 · Simplify

Step 4 · Divide by 3

Step 5 · Check

Why 3πr² is correct

Practice it yourself

Frequently asked questions