Prism

Right-angled prism, whose base is a right triangle with leg a = 3 cm and hypotenuse c = 13 cm, has the same volume as a cube with an edge length of 3 dm.

a) Find the height of the prism
b) Calculate the surface of the prism
c) What percentage of the cube's surface is prism surface area?

Correct result:

a) h =  1423 cm
b) S =  40806.3 cm²
c) p =  755.7 %

Solution:

$$\begin{gathered} x=\sqrt{13^2-3^2}=12.65cm \\ {S}_1={\displaystyle \frac{12.65\cdot 3}{2}}=18.97c{m}^2 \\ V=(10\cdot 3{)}^3=27000=S_{1}h \\ h={\displaystyle \frac{27000}{18.97}}=1423\text{ cm} \end{gathered}$$

$S=2S_1+h(13+3+12.65)=40806.3{\text{cm}}^2$

$S_k = 6 \cdot (10 \cdot 3)^2 = 5400 \ cm^2 \ \\ \ \\ p = 100 \dfrac { S }{S_k} = 755.7 \%$

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