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Common Core (Geometry)

Common Core for Mathematics

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 cone.

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.

**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 cylinder.**Volume of a Pyramid, Deriving the Formula**

Animated proof that volume of a pyramid is equal to Bh/3.

**Cavalieri's Principle**

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**

(4/3)πr^{3} gives you the volume of a sphere, but where does
the formula come from? Here is a simple explanation using geometry
and algebra.
**Surface Area of a Sphere, deriving the formula**

Proof that Surface Area of a Sphere is equal to 4πr^{2}
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 4πr^{2} after having figured a
formula for its volume.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Common Core (Geometry)

Common Core for Mathematics

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 cone.

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

How to get the Formula?

Explanation of the origins of the formula for the surface area of a cylinder.

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.

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.

(4/3)πr

Proof that Surface Area of a Sphere is equal to 4πr

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.

This video shows how Archimede's deduced that the surface area of a sphere is 4πr

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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