Volumes and Surface Areas of Pyramids, Cones and Spheres

GeoGebra Institute of Hong Kong, Anthony OR 柯志明, Dec 5, 2014

GeoGebraBook for elementary proofs of the formulas for volumes and surface areas of pyramids, cones and spheres.

Table of Contents

  1. Cavalieri's Principle
    1. Cavalieri's Principle
  2. Volume of Pyramids
    1. Trisecting the Cube into 3 Pyramids "Yangma"（陽馬）
    2. Volume of Pyramids
    3. Volume of Pyramids - Chinese Proof
  3. Curved Surface Area of Cones
    1. Curved Surface Area of Cones (Combined Version)
  4. Volume and Surface Area of Spheres
    1. Volume of Spheres
    2. Surface Area of Spheres
    3. Volume of Spheres with Proof
    4. Surface Area of Spheres with Proof
  5. Volume of Spheres and the Double Box-lid (牟合方蓋)
    1. Volume of Spheres in Nine Chapters on the Mathematical Art《九章算術》的球體體積公式
    2. Tracing Double Box-lid (Mouhefanggai 牟合方蓋)
    3. Double Box-lid (Mouhefanggai) 牟合方蓋面面觀
    4. Double Box-lid (Mouhefanggai 牟合方蓋) as Intersection of 2 Cylinders
    5. Volumes of Mouhefanggais and Spheres 牟合方蓋和球體的體積
  6. Relationships between Volumes and Surface Areas of Similar Figures
    1. Volumes and Surface Areas of Similar Cuboids

1.

Cavalieri's Principle

  • 1. Cavalieri's Principle

1.1.

Cavalieri's Principle

Acknowledgement: Inspired by Steve Phelps' applet "[url=http://www-beta.geogebra.org/m/3137147]Figure 6.7 Cavalieri's principle[/url]".
– Anthony OR 柯志明

2.

Volume of Pyramids

  • 1. Trisecting the Cube into 3 Pyramids "Yangma"（陽馬）
  • 2. Volume of Pyramids
  • 3. Volume of Pyramids - Chinese Proof

2.1.

Trisecting the Cube into 3 Pyramids "Yangma"（陽馬）

– Anthony OR 柯志明

2.2.

–

2.3.

–

3.

Curved Surface Area of Cones

  • 1. Curved Surface Area of Cones (Combined Version)

3.1.

Curved Surface Area of Cones (Combined Version)

– Anthony OR 柯志明

4.

Volume and Surface Area of Spheres

  • 1. Volume of Spheres
  • 2. Surface Area of Spheres
  • 3. Volume of Spheres with Proof
  • 4. Surface Area of Spheres with Proof

4.1.

Volume of Spheres

The figure shows a hemisphere of radius [i]r[/i] and a cylinder of base radius and height [i]r[/i] with an inverted cone of the same height and base radius removed. Drag the red point to see the cross-sections of the two solids at a height [i]h[/i].[br](a) Express [i]x[/i] and [i]y[/i] in terms of [i]r[/i] and [i]h[/i].[br](b) Are the cross-sections equal in area?[br](c) Hence show that the volume of the sphere of radius [i]r[/i] is 4/3 π [i]r[/i]³.
– Anthony OR 柯志明

4.2.

–

4.3.

–

4.4.

–

5.

Volume of Spheres and the Double Box-lid (牟合方蓋)

  • 1. Volume of Spheres in Nine Chapters on the Mathematical Art《九章算術》的球體體積公式
  • 2. Tracing Double Box-lid (Mouhefanggai 牟合方蓋)
  • 3. Double Box-lid (Mouhefanggai) 牟合方蓋面面觀
  • 4. Double Box-lid (Mouhefanggai 牟合方蓋) as Intersection of 2 Cylinders
  • 5. Volumes of Mouhefanggais and Spheres 牟合方蓋和球體的體積

5.1.

Volume of Spheres in Nine Chapters on the Mathematical Art《九章算術》的球體體積公式

– Anthony OR 柯志明

5.2.

–

5.3.

–

5.4.

–

5.5.

–

6.

Relationships between Volumes and Surface Areas of Similar Figures

  • 1. Volumes and Surface Areas of Similar Cuboids

6.1.

Volumes and Surface Areas of Similar Cuboids

– Anthony OR 柯志明

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