courses › Geometry

Volumes, and why doubling a size more than doubles its weight.

Pen and paper is fine · no calculator needed why?

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skills · practice stats

From rounds of this course only: box, review and trial answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

• Cylinders & cones par 17 s not tried yet

  worked example

  A cylinder has radius 1 and height 9. Its volume is kπ. What is k?

  Answer: 9

  1. V = πr²h = π × 1 × 9 = 9π, so k = 9.

• Area scale factor par 20 s not tried yet

  worked example

  Two shapes are similar: every length of the big one is 3 times the matching length of the small one. The small one has an area of 10 cm². What is the big one's area, in cm²?

  Answer: 90

  1. Areas scale by the square of the length factor: 3² = 9.
  2. 10 × 9 = 90 cm².

• Spheres par 19 s not tried yet

  worked example

  A sphere has diameter 4. Its volume is kπ. What is k? Give k as a whole number or a fraction.

  Answer: 32/3

  1. The radius is half the diameter: 2.
  2. V = (4/3)πr³ = (4/3) × 8 × π = (32/3)π, so k = 32/3.

• Volume scale factor par 18 s not tried yet

  worked example

  A bell weighs 5,632 kg. A model of it is built at 1/8 of its size in every length, from the same material. How many kilograms does the model weigh?

  Answer: 11

  1. Weight scales like volume, by the cube of the length factor: 8³ = 512.
  2. 5,632 ÷ 512 = 11 kg.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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