Frustrum - volume, area

Calculate the surface and volume of the truncated cone. The radius of the smaller figure is 4 cm, the height of the cone is 4 cm, and the side of the truncated cone is 5 cm.

Correct answer:

S =  376.9911 cm²
V =  389.5575 cm³

Step-by-step explanation:

$$\begin{gathered} s=5\text{cm} \\ {r}_2=4\text{cm} \\ h=4\text{cm} \\ {s}^2=h^2+(r_1-r_2{)}^2 \\ {r}_1=r_2+\sqrt{s^2-h^2}=4+\sqrt{5^2-4^2}=7\text{cm} \\ {S}_1=\pi \cdot r_1^2=3.1416\cdot 7^2\doteq 153.938{\text{cm}}^2 \\ {S}_2=\pi \cdot r_2^2=3.1416\cdot 4^2\doteq 50.2655{\text{cm}}^2 \\ {S}_3=\pi \cdot (r_1+r_2)\cdot \sqrt{h^2+(r_1-r_2{)}^2}=3.1416\cdot (7+4)\cdot \sqrt{4^2+(7-4{)}^2}\doteq 172.7876{\text{cm}}^2 \\ S=S_1+S_2+S_3=153.938+50.2655+172.7876=376.9911{\text{cm}}^2 \end{gathered}$$

$V=\frac{1}{3}\cdot (S_1+\pi \cdot r_1\cdot r_2+S_2)\cdot h=\frac{1}{3}\cdot (153.938+3.1416\cdot 7\cdot 4+50.2655)\cdot 4=389.5575{\text{cm}}^3$

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