Скачать или смотреть mathBeats: Surface Area and Volume (Open the Wormhole)

Why Divide by 2? Why Divide by 3?
Don't Memorize — Travel Through Dimensions!

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A triangle is a two-dimensional cone
Area is base times height, then divide by two
A circular cone is a three-dimensional cone
Volume is base times height, then divide by three
(The magic of dimension!)

(A circle): circumference is two pi r
Area is pi r squared
Take the circumference, (multiply by r, divide by two)
(A sphere): surface area is four pi r squared
Volume is four pi r cubed, divided by three
(Multiply by r, then divide by three)
(A trace of dimension!)

Cones and shapes that fill the space inside
Universes built with r and h, one shared law
Area comes from two factors
Volume comes from three
Divide by two, divide by three
The wormhole key, across dimensions!

Surface area of a sphere? Imagine this
Four circles laid out flat
That's why the total surface area is
Exactly four times a circle's area
(Four pi r squared)

Cones and shapes that fill the space inside
Universes built with r and h, one shared law
Area comes from two factors
Volume comes from three
Divide by two, divide by three
The wormhole key, across dimensions!

Wait, watch out for traps!
The surface area of a cylinder is
Two pi r squared plus two pi r h
(Top, bottom, and the side!)
(The side of a cone?) Unfold it, it's a sector!
Radius is the slant height
Arc is the base circumference
(Scaling trap!) If length doubles
Area becomes four times, volume eight times
Don't trust length alone!
(Dimension tells the truth!)

Travelers! Final checkpoint before the jump! (Quiz time!)
1: A cylinder and a cone with the same radius and height
What fraction of the cylinder's volume is the cone?
2: If a sphere's radius doubles
By what factor does its volume grow?
3: Unfold the side of a cone, what shape appears?
And name the formula for its glowing area!