Overview
This lecture covers how to calculate the lateral area, total surface area, and volume of a regular quadrangular (square-based) pyramid, using worked examples and Pythagoras’ Theorem.
Properties of a Regular Quadrangular Pyramid
- A regular quadrangular pyramid has a square base with all sides equal.
- The term "regular" refers to the base being a regular polygon (square).
Area and Volume Calculations
- The volume of a pyramid is given by: ( V = \frac{\text{Base Area} \times \text{Height}}{3} ).
- To find the base area for a square with side ( 6\sqrt{2} ), use ( (6\sqrt{2})^2 = 72 ) cm².
- The height of the pyramid can be found using the Pythagorean theorem in a right triangle formed by the height, half the diagonal of the base, and the slant edge.
- The diagonal of a square with side ( 6\sqrt{2} ) is ( 6\sqrt{2} \times \sqrt{2} = 12 ) cm, so half-diagonal = 6 cm.
- Using Pythagoras: ( h^2 + 6^2 = 10^2 ) gives height ( h = 8 ) cm.
Worked Example 1: Volume Calculation
- Base area = 72 cm², height = 8 cm.
- Volume = ( \frac{72 \times 8}{3} = 192 ) cm³.
Lateral Area Calculation
- Each lateral face is a triangle with base equal to a base edge of the square.
- The height of each face (apothem) is found using Pythagoras: forms a right triangle with leg 8 cm and half the base side 6 cm, hypotenuse 10 cm.
- Area of one face = ( \frac{6 \times 10}{2} = 30 ) cm².
- Total lateral area = 4 faces × 30 cm² = 120 cm² (review example for specific values).
Worked Example 2: Area and Volume with Different Base
- For a square base with side 12 cm and height 8 cm.
- Volume = ( \frac{12^2 \times 8}{3} = 384 ) cm³.
- Lateral area for one face = ( \frac{12 \times 10}{2} = 60 ) cm², so total lateral area = 4 × 60 = 240 cm².
- Total surface area = lateral area + base area = 240 + 144 = 384 cm².
Key Terms & Definitions
- Lateral Area — the sum of the areas of all the triangular faces (not including the base).
- Total Surface Area — lateral area plus base area.
- Volume — the measure of the space inside the pyramid, ( \frac{\text{Base Area} \times \text{Height}}{3} ).
- Apothem of Face — height of a triangular face from vertex to midpoint of base edge.
- Regular Pyramid — a pyramid with a regular polygon as its base and all lateral edges equal.
Action Items / Next Steps
- Practice applying Pythagoras’ theorem to find pyramid heights and face apothems.
- Solve similar problems for pyramids with different base sizes or types.
- Review homework on lateral and total area calculations for other pyramid shapes.