Quadrangular pyramid

Given is a regular quadrangular pyramid with a square base. The body height is 30 cm and volume V = 1000 cm³. Calculate its side a and its surface area.

Result

a =  10 cm
S =  708.276 cm2

Solution:

h=30 cm V=1000 cm3  V=13hS1  S1=3⋅ V/h=3⋅ 1000/30=100 cm2  S1=a2  a=S1=100=10 cmh=30 \ \text{cm} \ \\ V=1000 \ \text{cm}^3 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } h S_{1} \ \\ \ \\ S_{1}=3 \cdot \ V / h=3 \cdot \ 1000 / 30=100 \ \text{cm}^2 \ \\ \ \\ S_{1}=a^2 \ \\ \ \\ a=\sqrt{ S_{1} }=\sqrt{ 100 }=10 \ \text{cm}
h2=h2+(a/2)2=302+(10/2)2≐5 37 cm≐30.4138 cm S2=a⋅ h2/2=10⋅ 30.4138/2≐25 37 cm2≐152.0691 cm2  S=S1+4⋅ S2=100+4⋅ 152.0691≐708.2763≐708.276 cm2h_{2}=\sqrt{ h^2 + (a/2)^2 }=\sqrt{ 30^2 + (10/2)^2 } \doteq 5 \ \sqrt{ 37 } \ \text{cm} \doteq 30.4138 \ \text{cm} \ \\ S_{2}=a \cdot \ h_{2}/2=10 \cdot \ 30.4138/2 \doteq 25 \ \sqrt{ 37 } \ \text{cm}^2 \doteq 152.0691 \ \text{cm}^2 \ \\ \ \\ S=S_{1} + 4 \cdot \ S_{2}=100 + 4 \cdot \ 152.0691 \doteq 708.2763 \doteq 708.276 \ \text{cm}^2

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