Pyramids and Their Surface Area

Pyramids and Surface Area

Definition of a Pyramid

• A pyramid is a three-dimensional shape that comes to a peak.
• The base of a pyramid can be any polygon (e.g., equilateral triangle, regular hexagon, square).
• The pyramids in Egypt have a square base, but the base can be any polygon whatsoever.

Surface Area of a Pyramid

• Two types of surface area:
  • Total Surface Area: Includes the area of the sides and the base.
  • Lateral Surface Area: Just the area of the sides (excluding the base).
• For a square-based pyramid, the lateral surface area is the sum of the areas of the four triangles on the sides.
• The total surface area includes the area of the square base in addition to the lateral surface area.

Slant Height

• Slant height is the height of each triangular face and stretches along the slant of that triangle.
• To find the lateral surface area, it's important to know the slant height.

Example

• Square base: 10 units wide and 10 units long.
• Slant height: 14 units.
• Lateral Area Calculation:
  • Area of one triangle: $\frac{1}{2} \times base \times height = \frac{1}{2} \times 10 \times 14 = 70$
  • Since there are four triangles: $4 \times 70 = 280$
  • Lateral Area = 280 square units.
• Surface Area Calculation:
  • Area of the square base: $10 \times 10 = 100$
  • Total Surface Area: Lateral Area + Base Area = $280 + 100 = 380$
  • Surface Area = 380 square units.

Altitude of the Pyramid

• Altitude is the height from the peak straight down to the center of the base, rather than along the slant.
• If you were to bore a hole straight down through the pyramid down to the base, that's the altitude.

Example

• Square base: 8 by 8 units.
• Altitude: 10 units.
• Need to find the slant height to calculate the lateral area.

Finding Slant Height Using the Pythagorean Theorem

• A right triangle is formed inside the pyramid with the altitude, half the base length, and the slant height as its sides.
• Half the base length: $\frac{8}{2} = 4$
• Using the Pythagorean theorem: $c^2 = a^2 + b^2$ where $c$ is the slant height.
• $c^2 = 4^2 + 10^2 = 16 + 100 = 116$
• $c = \sqrt{116}$

Calculating Areas

• Area of one triangle: $\frac{1}{2} \times base \times height = \frac{1}{2} \times 8 \times \sqrt{116} = 4\sqrt{116}$
• Since there are four triangles: $4 \times 4\sqrt{116} = 16\sqrt{116}$
• Lateral Area = $16\sqrt{116}$

Simplifying the Square Root

• $\sqrt{116} = \sqrt{4 \times 29} = 2\sqrt{29}$
• Lateral Area = $16 \times 2\sqrt{29} = 32\sqrt{29}$

Calculating Surface Area

• Area of the square base: $8 \times 8 = 64$
• Total Surface Area: Lateral Area + Base Area = $32\sqrt{29} + 64$

Key Points

• If given the altitude, use the Pythagorean theorem to find the slant height.
• Focus on drawing the picture and understanding the components (triangles for the sides, a polygon for the base) instead of memorizing formulas.