Depth · one concept from first principles · step 2 of 4
Same height, radius times ●, volume times ● squared
A cylinder's base radius is doubled but its height is not. We need how many times bigger the new volume is.
Geometry
Your answer
How to solve
Strategy Solve an Easier Related Problem — A cylinder's volume is its base area times its height. The height is untouched, so the answer is simply how much the base grew -- and a circle grows in two directions when its radius does.
1STEP 1
See what the volume depends on
A cylinder's volume is its base area multiplied by its height, and the height does not change.
Whatever the base does, the volume does.
7.G.B.6Look For A PatternWith the height fixed, the volume changes exactly as the base area does.
Why?
A cylinder stacks into layers that are all copies of its base, so its volume is the base's area taken as many times as the height says.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Why?
Multiplying a bigger base by the same height simply repeats the bigger amount the same number of times.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
2STEP 2
Work out both base areas
The radius goes from 10 to 20 cm.
10 × 10 × 3.14 = 314, 20 × 20 × 3.14 = 1256
The base grows from 314 to 1256 cm².
8.G.C.9Draw A Diagram3STEP 3
Compare them
Divide the new base area by the old one.
1256 ÷ 314 = 4
The base is 4 times as big, so the volume is too.
7.G.A.1Solve An Easier Related ProblemAnswer
4 times
Check with the volumes themselves: 314 × 13 = 4082 and 1256 × 13 = 16328, and 16328 ÷ 4082 = 4.
Takeaway
A circle grows in two directions when its radius does, so doubling the radius makes the base four times as big.
- See what the volume depends on
- Work out both base areas
- Compare them
Where next?
Another one like thissuggested
Step back and solidifysuggested
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