Surface Area of Pyramids and Cones

Surface Area of Pyramids

• Definition: The surface area of a pyramid is the total area of all its faces. It includes the base area and the lateral area.

• Types of Pyramids:

  • Triangular Pyramid: Consists of 4 triangular faces. The base is a triangle.
  • Square Pyramid: Has a square base and 4 isosceles triangular faces. A regular pyramid example because the base is a regular polygon.
  • Rectangular Pyramid: Features a rectangular base and four triangular faces. Has two sets of triangular faces.
  • Pentagonal Pyramid: Base is a regular pentagon. All sides and angles are equal.

Surface Area Calculation

• Lateral Area (LA): Can be calculated using the formula: $LA = \frac{1}{2} P l$ Where:
  • P = perimeter of the base
  • l = slant height
  • Total Surface Area (SA) = Base Area (B) + Lateral Area (LA):
    $SA = B + LA$

Example of Surface Area Calculation for a Cone

• Problem: Maribel wants to spray glitter on a conical shrub.

  • Height of the shrub (h): 9 ft
  • Diameter of the base: 48 inches (convert to feet: 48 in = 4 ft)
  • Radius (r) = 2 ft (from diameter)

• Find the slant height (l): Use Pythagorean theorem:

  • $l = \sqrt{h^2 + r^2}$
  • Calculation:
  • $l = \sqrt{9^2 + 2^2} = \sqrt{81 + 4} = \sqrt{85}$
  • Approx. $l \approx 9.22$ ft

• Lateral Area:

  • LA = \pi r l = \pi (2)(9.22) <br /> \approx 57.92 \text{ square feet}

• Area of the base:

  • B = \pi r^2 = \pi (2)^2 = 4\pi \approx 12.57 <br />
  • Total Surface Area of the Cone:
  • SA = B + LA \approx 12.57 + 57.92 \approx 70.49 <br />

Finding Slant Height of a Pyramid

• Pythagorean Theorem: To find slant height, if apothem (a) and height (h) are known:
  • $a^2 + h^2 = l^2$
  • Example calculation for slant height:
  • If apothem = 24 in and height = 12 in:
    • $12^2 + 24^2 = l^2$
    • 144 + 576 = l^2 <br /> = 720 <br /> l = \sqrt{720} \approx 26.83 in

Irregular Pyramids

• Definition: Base is not a regular polygon, leading to non-congruent lateral triangular faces.
• Surface Area Calculation: Sum of areas of individual triangle faces.
  • For example: $A<em>1 + A</em>2 + A_3 + … = SA$

Parts of a Pyramid

• Apex: The point where all lateral faces meet.
• Slant Height: Length along a lateral face from the apex to the midpoint of the opposite side.
• Base Height: Height from the apex perpendicularly to the base.
• Apothem: The distance from the center of a regular polygon base to the midpoint of a side.

Regular Pyramids

• Calculate total surface area by adding:
  • Base Area (calculated using the formula for the area of the polygon that forms the base) + Lateral Area.
• Example: A pyramid with surface area represented as:
  $SA = B + \frac{1}{2} P l$ where you calculate B based on the formula for the regular polygon.