Examples, solutions, videos, and lessons to help High School students learn how to give
an informal argument for the formulas for the circumference of a
circle, area of a circle, volume of a cylinder, pyramid, and
Use dissection arguments, Cavalieri's principle,
and informal limit arguments.
Give an informal argument using Cavalieri's principle for the
formulas for the volume of a sphere and other solid figures.
Common Core: HSG-GMD.A.1 and HSG-GMD.A.2
Circumference of a circle, 2πr
How to get the Formula?
Area of a circle, how to get the formula.
Surface Area of a Cylindrical Solid
Explanation of the origins of the formula for the surface area of a
Volume of a Pyramid, Deriving the Formula
Animated proof that volume of a pyramid is equal to Bh/3.
If two solids lying between parallel planes have equal heights and
all cross sections at equal distance from their bases have equal
areas, then the solids have equal volumes.
Formula for the Volume of a Cone
This video explain why the volume of a cone is 1/3 the volume of the
cylinder that surrounds it. It uses Cavalieri's Principle.
Volume of a Sphere, How to get the formula animation
gives you the volume of a sphere, but where does
the formula come from? Here is a simple explanation using geometry
Surface Area of a Sphere, deriving the formula
Proof that Surface Area of a Sphere is equal to 4πr2
using geometry and algebra.
Formula for the Volume of a Sphere
This video shows how Archimede's deduced found the volume of a
sphere. To do so he had to use a formula for the volume of a cone.
Formula for Surface Area of a Sphere
This video shows how Archimede's deduced that the surface area of a
sphere is 4πr2
after having figured a
formula for its volume.
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