Quadrangular pyramid

The regular quadrangular pyramid has a base length of 6 cm and a side edge length of 9 centimeters. Calculate its volume and surface area.

Result

V =  95.247 cm3
S =  137.823 cm2

Solution:

a=6 cm s=9 cm  S1=a2=62=36 cm2  u=2⋅ a=2⋅ 6≐6 2 cm≐8.4853 cm  h=s2−(u/2)2=92−(8.4853/2)2≐3 7 cm≐7.9373 cm  V=13⋅ S1⋅ h=13⋅ 36⋅ 7.9373≐36 7≐95.247≐95.247 cm3a=6 \ \text{cm} \ \\ s=9 \ \text{cm} \ \\ \ \\ S_{1}=a^2=6^2=36 \ \text{cm}^2 \ \\ \ \\ u=\sqrt{ 2 } \cdot \ a=\sqrt{ 2 } \cdot \ 6 \doteq 6 \ \sqrt{ 2 } \ \text{cm} \doteq 8.4853 \ \text{cm} \ \\ \ \\ h=\sqrt{ s^2 - (u/2)^2 }=\sqrt{ 9^2 - (8.4853/2)^2 } \doteq 3 \ \sqrt{ 7 } \ \text{cm} \doteq 7.9373 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 36 \cdot \ 7.9373 \doteq 36 \ \sqrt{ 7 } \doteq 95.247 \doteq 95.247 \ \text{cm}^3
h2=s2−(a/2)2=92−(6/2)2≐6 2 cm≐8.4853 cm  S2=a⋅ h2/2=6⋅ 8.4853/2≐18 2 cm2≐25.4558 cm2  S=S1+4⋅ S2=36+4⋅ 25.4558≐137.8234≐137.823 cm2h_{2}=\sqrt{ s^2 - (a/2)^2 }=\sqrt{ 9^2 - (6/2)^2 } \doteq 6 \ \sqrt{ 2 } \ \text{cm} \doteq 8.4853 \ \text{cm} \ \\ \ \\ S_{2}=a \cdot \ h_{2}/2=6 \cdot \ 8.4853/2 \doteq 18 \ \sqrt{ 2 } \ \text{cm}^2 \doteq 25.4558 \ \text{cm}^2 \ \\ \ \\ S=S_{1} + 4 \cdot \ S_{2}=36 + 4 \cdot \ 25.4558 \doteq 137.8234 \doteq 137.823 \ \text{cm}^2

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