Geometric Formulas for Volume and Surface Area Calculations

This page presents a comprehensive collection of formulas for calculating the volume, surface area, and other key measurements of various three-dimensional shapes. The information is organized in a clear, visual format with accompanying diagrams for each shape.

Cube (Würfel)

The cube section provides formulas for:

• Volume (V): a³
• Base area (AG): a²
• Surface area (A₁): 6a²

Definition: A cube is a three-dimensional solid object with six square faces of equal size.

Square Pyramid (Quadratische Pyramide)

For the square pyramid, the following formulas are given:

• Volume (V): 1/3 · a² · h
• Base area (AG): a²
• Lateral surface area (An): 2a · ha
• Total surface area (A₁): a$a + 2ha$ = a² + 2aha

Vocabulary: "ha" likely refers to the slant height of the pyramid.

Cone (Kreiskegel)

The cone section includes:

• Volume (V): 1/3 · πr² · h
• Base area (AG): πr²
• Lateral surface area (AM): πrs
• Total surface area (A₁): πr$r + s$ = πr² + πrs

Highlight: The formula s² = r² + h² is provided, which relates the slant height $s$ to the radius $r$ and height $h$ of the cone.

Cuboid (Quader)

For the cuboid, we have:

• Volume (V): a · b · c
• Base area (AG): ab
• Surface area (A₁): 2(ab + ac + bc)

Example: A cuboid with dimensions 3 cm, 4 cm, and 5 cm would have a volume of 3 · 4 · 5 = 60 cm³.

Cylinder (Kreiszylinder)

The cylinder formulas include:

• Volume (V): πr² · h
• Base area (AG): πr²
• Lateral surface area (AM): 2πr · h
• Total surface area (A₁): 2πr$r + h$ = 2πr² + 2πrh

Vocabulary: "AM" stands for "Mantelfläche" which is the lateral surface area of the cylinder.

Sphere (Kugel)

For the sphere, the document provides:

• Volume (V): 4/3 · πr³
• Surface area (A₁): 4πr²

Highlight: The volumen kugel warum 4/3 is a common question, and this formula shows that the volume of a sphere is indeed 4/3 times π times the cube of its radius.

The page effectively combines Formeln Körper Volumen und Oberfläche for quick reference, making it an excellent resource for students studying geometry or anyone needing to perform Volumenberechnung or surface area calculations for these common shapes. The inclusion of diagrams helps visualize each shape and its dimensions, aiding in understanding and application of the formulas.