Површина тростране призме

Текст задатака објашњених у видео лекцији:

Пр.1)   Израчунати површину правилне тростране призме

           чија је оесновна ивица 12cm, а висина призме 10cm.

Пр.2)   Ако је површина основне правилне тростране призме

           $36\sqrt 3 c{m^2}$, а висина 8cm, израчунати површину

           те призме.

Пр.3)   Ако је површина павилне тростране пизме $8\sqrt 3  + 60c{m^2}$

           а основна ивица 4cm, израчунати висину те призме.

Пр.4)   Ако је полупречник уписаног круга у основу правилне

           тростране јенакоивичне призме 2cm, израчунати пиршину

           те пизме.

Пр.5)   Основа паве призме је павоугли троугао чије су катете

           12cm  и 16cm. Израчунати површину ове призме, ако је

            највећа бочна страна квадрат.

Пр.1)  

\[\begin{gathered}
a = 12cm \hfill \\
\underline {H = 10cm} \hfill \\
P = ? \hfill \\
\hfill \\
P = 2B + M \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = \frac{{{a^2}\sqrt 3 }}{4} \hfill \\
B = \frac{{{{12}^2}\sqrt 3 }}{4} \hfill \\
B = \frac{{144\sqrt 3 }}{4} \hfill \\
B = 36\sqrt 3 c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = 3aH \hfill \\
M = 3 \cdot 12 \cdot 10 \hfill \\
M = 360c{m^2} \hfill \\
\hfill \\
\hfill \\
\hfill \\
\end{gathered}
\end{array} \hfill \\
\hfill \\
P = 2 \cdot 36\sqrt 3 + 360 \hfill \\
P = \left( {72\sqrt 3 + 360} \right)c{m^2} \hfill \\
\end{gathered} \]

Пр.2)   

\[\begin{gathered}
B = 36\sqrt 3 c{m^2} \hfill \\
\underline {H = 8cm} \hfill \\
P = ? \hfill \\
\hfill \\
P = 2B + M \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = \frac{{{a^2}\sqrt 3 }}{4} \hfill \\
36\sqrt 3 = \frac{{{a^2}\sqrt 3 }}{4} \hfill \\
{a^2}\sqrt 3 = 144\sqrt 3 \hfill \\
{a^2} = \frac{{144\sqrt 3 }}{{\sqrt 3 }} \hfill \\
{a^2} = 144 \hfill \\
a = 12cm \hfill \\
\end{gathered} &\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = 3aH \hfill \\
M = 3 \cdot 12 \cdot 8 \hfill \\
M = 288c{m^2} \hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\hfill \\
\end{gathered}
\end{array} \hfill \\
\hfill \\
P = 2 \cdot 36\sqrt 3 + 288 \hfill \\
P = \left( {72\sqrt 3 + 288} \right)c{m^2} \hfill \\
\end{gathered} \]

Пр.3)   

\[\begin{gathered}
P = \left( {8\sqrt 3 + 60} \right)c{m^2} \hfill \\
\underline {a = 4cm} \hfill \\
H = ? \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = \frac{{{a^2}\sqrt 3 }}{4} \hfill \\
B = \frac{{{4^2}\sqrt 3 }}{4} \hfill \\
B = 4\sqrt 3 c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
P = 2B + M \hfill \\
8\sqrt 3 + 60 = 2 \cdot 4\sqrt 3 + M \hfill \\
M = 8\sqrt 3 + 60 - 8\sqrt 3 \hfill \\
M = 60c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
M = 3aH \hfill \\
60 = 3 \cdot 4 \cdot H \hfill \\
60 = 12 \cdot H \hfill \\
H = 5cm \hfill \\
\hfill \\
\end{gathered}
\end{array} \hfill \\
\end{gathered} \]

Пр.4)   

\[\begin{gathered}
H = a \hfill \\
\underline {{r_u} = 2cm} \hfill \\
P = ? \hfill \\
\hfill \\
{r_u} = \frac{1}{3}h \hfill \\
2 = \frac{1}{3}h \hfill \\
h = 6cm \hfill \\
h = \frac{{a\sqrt 3 }}{2} \hfill \\
6 = \frac{{a\sqrt 3 }}{2} \hfill \\
a = \frac{{12}}{{\sqrt 3 }} = 4\sqrt 3 cm \hfill \\
H = a = 4\sqrt 3 cm \hfill \\
B = \frac{{{a^2}\sqrt 3 }}{4} \hfill \\
B = \frac{{{{\left( {4\sqrt 3 } \right)}^2}\sqrt 3 }}{4} \hfill \\
B = 12\sqrt 3 c{m^2} \hfill \\
M = 3aH \hfill \\
M = 3 \cdot 4\sqrt 3 \cdot 4\sqrt 3 \hfill \\
M = 144c{m^2} \hfill \\
P = 2B + M \hfill \\
P = 2 \cdot 12\sqrt 3 + 144 \hfill \\
P = \left( {24\sqrt 3 + 144} \right)c{m^2} \hfill \\
\end{gathered} \]

Пр.5)   

\[\begin{gathered}
a = 12cm \hfill \\
b = 16cm \hfill \\
\underline {H = c} \hfill \\
P = ? \hfill \\
\end{gathered} \]

\[\begin{gathered}
{c^2} = {a^2} + {b^2} \hfill \\
{c^2} = {12^2} + {16^2} \hfill \\
{c^2} = 144 + 256 \hfill \\
{c^2} = 400 \hfill \\
c = 20cm \hfill \\
H = c = 20cm \hfill \\
B = \frac{{ab}}{2} \hfill \\
B = \frac{{12 \cdot 16}}{2} \hfill \\
B = 96c{m^2} \hfill \\
M = aH + bH + cH \hfill \\
M = 12 \cdot 20 + 16 \cdot 20 + 20 \cdot 20 \hfill \\
M = 960c{m^2} \hfill \\
P = 2B + M \hfill \\
P = 2 \cdot 96 + 960 \hfill \\
P = 1152c{m^2} \hfill \\
\end{gathered} \]