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: A1 + A2 + 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.