Intermediate Geometry

Intermediate Geometry explores the properties and relationships of shapes, sizes, and theorems in two and three dimensions.

Properties of TrianglesPolygons and Their PropertiesCircles and Their Elements
Similarity and CongruenceThe Pythagorean Theorem and DistanceVolume and Surface Area of 3D Shapes
Geometry in ArchitectureNavigation and MappingDesign and Art
Breaking Down Complex ProblemsChecking Your Answers
Properties of Triang...Polygons and Their P...Circles and Their El...Similarity and Congr...The Pythagorean Theo...Volume and Surface A...Geometry in Architec...Navigation and Mappi...Design and ArtBreaking Down Comple...Checking Your Answer...
Advanced Topics

Volume and Surface Area of 3D Shapes

3D Geometry: Measuring Space

Three-dimensional (3D) shapes like cubes, cylinders, and spheres have two important measurements: volume (how much space it takes up) and surface area (total area of all surfaces).

Key Formulas

  • Cube: \(V = s^3\), \(A = 6s^2\)
  • Rectangular Prism: \(V = lwh\), \(A = 2(lw + lh + wh)\)
  • Cylinder: \(V = \pi r^2 h\), \(A = 2\pi r (r + h)\)

Everyday Uses

These calculations help in construction (how much concrete for a pillar?), packaging (how much material for a can?), and baking (volume of a cake pan).

Examples

  • Determining the amount of paint needed to cover a box.

  • Calculating the water a swimming pool can hold.

In a Nutshell

Volume tells us how much space is inside; surface area tells us how much material covers a shape.

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Geometry in Architecture