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Theorem — what does that mean?
The curved half, 2πr², plus the flat circle it sits on, πr²: three pi r squared.
In Swedish: Halvklotets begränsningsarea
A
The curved half, 2πr², plus the flat circle it sits on, πr²: three pi r squared.
A≈84.823cm²
A equals 3 times (π times 3 squared), equals 3 times (π times 9), is approximately 3 times 28.2743, is approximately 84.823 cm²
You are given
Round to 1 decimal.
Pick the unit your lengths are in. Areas are then labelled in square units and volumes in cubic units; nothing is converted.
r is the radius. The sphere’s surface is 4πr², so the curved half of a hemisphere is 2πr². Cutting the sphere exposes a flat circle of area πr², and the total is the two together: 2πr² + πr² = 3πr².
A radius of 3 cm: A = 3π · 9 = 27π ≈ 84.8 cm², of which 56.5 cm² is the dome and 28.3 cm² the flat base. A dome with a radius of 6 m has four times as much, 108π ≈ 339 m², because area grows with the square.
A question about painting a dome wants the curved part alone, 2πr²; a question about the material in a closed bowl wants all of it, 3πr². The card above gives the total; two thirds of it is the dome.
Turned round, r = √(A / 3π). A closed hemisphere with a total surface of 300 cm² has a radius of √(300 / 3π) = √31.8 ≈ 5.6 cm.
Why is the curved half exactly 2πr²?
Archimedes showed that the sphere’s surface equals the curved surface of the cylinder that encloses
it, 2πr · 2r = 4πr²; half of each is the hemisphere’s dome.
What if the base is open?
Then it is a bowl and only the curved part counts, inside and out if both are painted: 2 · 2πr².
How does it relate to the volume?
Through the radius only. For r = 3 the volume is 56.5 cm³ and the area 84.8 cm²; the numbers happen to
be close at that size and far apart at any other.
| Solve for | Rearranged | With the numbers | Comes to |
|---|---|---|---|
| r | 3 cm | ||
| A | 3πr² | 3π·3² | 84.8 cm² |
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