Surface Area and Volume of Pyramids

Surface Area and Volume of Pyramids

Pyramid 1

  • Given dimensions:

    • Base Square: 6 in x 6 in
    • Slant Height: 10 in
    • Height: Unknown (not directly provided, but can be calculated if needed)

  • Surface Area (SA):

    • The surface area of a pyramid is the sum of the area of the base and the areas of all the triangular faces.
    • Base Area: 6 in×6 in=36 in26 \text{ in} \times 6 \text{ in} = 36 \text{ in}^2
    • Lateral Area (4 triangular faces): 4×(12×6 in×10 in)=120 in24 \times (\frac{1}{2} \times 6 \text{ in} \times 10 \text{ in}) = 120 \text{ in}^2
    • Total Surface Area: 36 in2+120 in2=156 in236 \text{ in}^2 + 120 \text{ in}^2 = 156 \text{ in}^2

  • Volume (V):

    • The volume of a pyramid is given by the formula: V=13×Base Area×HeightV = \frac{1}{3} \times \text{Base Area} \times \text{Height}
    • To find the height, we can use the Pythagorean theorem on a right triangle formed by half the base side (3 in), the height (h), and the slant height (10 in).
    • 32+h2=1023^2 + h^2 = 10^2
    • 9+h2=1009 + h^2 = 100
    • h2=91h^2 = 91
    • h=91≈9.54 inh = \sqrt{91} \approx 9.54 \text{ in}
    • Volume: V=13×36 in2×9.54 in=114.48 in3V = \frac{1}{3} \times 36 \text{ in}^2 \times 9.54 \text{ in} = 114.48 \text{ in}^3

Pyramid 2

  • Given dimensions:

    • Base Square: 14 in x 14 in
    • Slant Height: Unknown
    • Height: Unknown

  • Missing Information:

    • The slant height and the height of this pyramid are not provided, making it impossible to calculate surface area and volume without additional information.

Pyramid 3

  • Given dimensions:

    • Base Triangle: 5 cm x 5 cm x 5 cm (Equilateral Triangle)
    • Height of the Triangles: 13 cm

  • Surface Area:

    • Because it's a triangular pyramid and we can see the diagram resembles such a pyramid, we must use the area of an equilateral triangle for the base of this pyramid. The other three sides are equal and we are assuming each of those triangles have a height of 13 cm.
    • Area of Equilateral Triangle: A=34a2A = \frac{\sqrt{3}}{4} a^2 where aa is the side length.
    • A=34(5 cm)2≈10.83 cm2A = \frac{\sqrt{3}}{4} (5 \text{ cm})^2 \approx 10.83 \text{ cm}^2
    • Area of Lateral Triangle: A=12bhA = \frac{1}{2} b h
    • A=12(5 cm)(13 cm)=32.5 cm2A = \frac{1}{2} (5 \text{ cm}) (13 \text{ cm}) = 32.5 \text{ cm}^2
    • Total Surface Area (SA): 10.83+3(32.5)=108.33 cm210.83 + 3(32.5) = 108.33 \text{ cm}^2

  • Volume (V):

    • Height needs to be supplied in order to compute the Volume (V).

Pyramid 4

  • Given dimensions:

    • Base Square: 12 mm x 12 mm
    • Height of the Triangles: 4 mm

  • Surface Area (SA):

    • Area of the Square: A=a2A = a^2
    • A=(12 mm)2=144 mm2A = (12 \text{ mm})^2 = 144 \text{ mm}^2
    • Area of Lateral Triangle: A=12bhA = \frac{1}{2} b h
    • A=12(12 mm)(4 mm)=24 mm2A = \frac{1}{2} (12 \text{ mm}) (4 \text{ mm}) = 24 \text{ mm}^2
    • Total Surface Area (SA): 144+4(24)=240 mm2144 + 4(24) = 240 \text{ mm}^2

  • Volume (V):

    • Height needs to be supplied in order to compute the Volume (V).