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Understanding Pyramids: Shapes, Properties, and Formulas

Learn about different types of pyramids, including regular and irregular, their properties, formulas for lateral area and volume, and how to calculate them. Explore the concept of slant height, lateral faces, bases, and vertices in pyramid geometry. Discover the essence of mathematics and collaborative learning through classic quotes and practical examples from geometry problems.

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Understanding Pyramids: Shapes, Properties, and Formulas

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  1. 12.2pyramids

  2. Names of pyramids • Names are based on the shape of the pyramids base. • Reference whether it is regular or not • To label a pyramid you start with the vertex, and then a hyphen, and then list the vertices of the base.

  3. This pyramid is a regular pentagonal pyramid • Referred to as V-ABCDE • V is the vertex • ABCDE is the base

  4. The segment from the vertex perpendicular to the base is the altitude.

  5. The 5 faces that have V in common are the lateral faces. The edges of the lateral faces are called the lateral edges.

  6. Most pyramids that you will encounter are regular pyramids, with the following properties • Base is a regular polygon • All lateral edges are congruent • All lateral faces are congruent isosceles triangles. The height of a lateral face is called the SLANT HEIGHT of the pyramid. • The altitude meets the base at its center.

  7. Slant height

  8. 2 methods for finding the lateral area of a regular pyramid • Method 1 • Find the area of one lateral face and multiply by n (number of faces)

  9. Method 2 • Theorem – the lateral area of a regular pyramid equals half the perimeter of the base times the slant height (l). • L.A. = ½ Pl

  10. Find the L.A.

  11. Find the L.A. • Hint, need to find slant height • Probably going to need to find the apothem of the base.

  12. Volume • The volume of a pyramid equals one third the area of the base times the height of the pyramid. (V=1/3 Bh) • Tip – if the base is a regular polygon then you need to use the formula A= ½ aP to find the base area.

  13. Find the volume • Suppose this figure has base edges of 8 and height of 12

  14. Again the formulas may seem basic but there are values that we need to determine using things like special right triangles, apothems, trigonometry, isosceles triangles, etc. • “Mathematicians stand on each other's shoulders.” • Gauss • What one person has accomplished in mathematics is due to the many that came before them, this idea reverberates through concepts as well.

  15. Pg. 485 1-10, 16, 18, 21, 23, 27

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