Површина четворостране призме

Текст задатака објашњених у видео лекцији:

Пр.1)   Израчунати површину правилане четвораостране призме

           чија је основна ивица 10cm, а висина призме 15cm.

Пр.2)   Ако је површина основе правилне четворостране призме

           256$c{m^2}$, а висина 11cm, израчунати површину те призме.

Пр.3)   Ако је површина правилне четворостране призме 190$c{m^2}$,

           а основна ивица 5cm, израчунати висину те призме.

Пр.4)   Ако дијагонала павилне четворостране призме на равни основе

           гради угао од ${60^ \circ }$, а дијагонала основе је $6\sqrt 2 cm$,

           израчунати површину те призме.

Пр.5)   Основа праве пизме је ромб чије су дијагонале 12cm  и 16cm.

            Ако су висина призме и основна ивица у размери 3:2, изачунати

            поршину призме.

Пр.6)   Основа праве призме је једнакокраки троугао чије су основене

            ивице 27cm и 11cm, а крак трапеза је 17cm. Израчунати поврпину

            те призме ако је висина призме за 5cm дужа од висине основе.

Пр.1)  

\[\begin{gathered}
a = 10cm \hfill \\
\underline {H = 15cm} \hfill \\
P = ? \hfill \\
P = 2B + M \hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = {a^2} \hfill \\
B = {10^2} \hfill \\
B = 100c{m^2} \hfill \\
\end{gathered} &{}&\begin{gathered}
M = 4aH \hfill \\
M = 4 \cdot 10 \cdot 15 \hfill \\
M = 600c{m^2} \hfill \\
\end{gathered}
\end{array} \hfill \\
P = 2 \cdot 100 + 600 \hfill \\
P = 800c{m^2} \hfill \\
\end{gathered} \]

Пр.2)   

\[\begin{gathered}
B = 256c{m^2} \hfill \\
\underline {H = 11cm} \hfill \\
P = ? \hfill \\
P = 2B + M \hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = {a^2} \hfill \\
256 = {a^2} \hfill \\
a = 16cm \hfill \\
\end{gathered} &{}&\begin{gathered}
M = 4aH \hfill \\
M = 4 \cdot 16 \cdot 11 \hfill \\
M = 704c{m^2} \hfill \\
\end{gathered}
\end{array} \hfill \\
P = 2 \cdot 256 + 704 \hfill \\
P = 1216c{m^2} \hfill \\
\end{gathered} \]

Пр.3)  

\[\begin{gathered}
P = 190c{m^2} \hfill \\
\underline {a = 5cm} \hfill \\
H = ? \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = {a^2} \hfill \\
B = {5^2} \hfill \\
B = 25c{m^2} \hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
P = 2B + M \hfill \\
190 = 2 \cdot 25 + M \hfill \\
M = 190 - 50 \hfill \\
M = 140c{m^2} \hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = 4aH \hfill \\
140 = 4 \cdot 5 \cdot H \hfill \\
140 = 20H \hfill \\
H = \frac{{140}}{{20}} \hfill \\
H = 7cm \hfill \\
\end{gathered}
\end{array} \hfill \\
\end{gathered} \]

Пр.4)   

\[\begin{gathered}
d = 6\sqrt 2 cm \hfill \\
\underline {\alpha = {{60}^ \circ }} \hfill \\
P = ? \hfill \\
\hfill \\
d = a\sqrt 2 \hfill \\
6\sqrt 2 = a\sqrt 2 \hfill \\
a = 6cm \hfill \\
H = \frac{{D\sqrt 3 }}{2} \hfill \\
\end{gathered} \]

\[\begin{gathered}
D = 2d = 2 \cdot 6\sqrt 2 = 12\sqrt 2 cm \hfill \\
H = \frac{{12\sqrt 2 \cdot \sqrt 3 }}{2} = 6\sqrt 6 cm \hfill \\
\hfill \\
P = 2B + M \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = {a^2} \hfill \\
B = {6^2} \hfill \\
B = 36c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = 4aH \hfill \\
M = 4 \cdot 6 \cdot 6\sqrt 6 \hfill \\
M = 144\sqrt 6 c{m^2} \hfill \\
\end{gathered}
\end{array} \hfill \\
\hfill \\
P = 2 \cdot 36 + 144\sqrt 6 \hfill \\
P = \left( {72 + 144\sqrt 6 } \right)c{m^2} \hfill \\
\end{gathered} \]

Пр.5)   

\[\begin{gathered}
{d_1} = 12cm \hfill \\
{d_2} = 16cm \hfill \\
\underline {H:a = 3:2} \hfill \\
P = ? \hfill \\
\hfill \\
{a^2} = {\left( {\frac{{{d_1}}}{2}} \right)^2} + {\left( {\frac{{{d_2}}}{2}} \right)^2} \hfill \\
{a^2} = {\left( {\frac{{12}}{2}} \right)^2} + {\left( {\frac{{16}}{2}} \right)^2} \hfill \\
{a^2} = 36 + 64 \hfill \\
{a^2} = 100 \hfill \\
a = 10cm \hfill \\
H:a = 3:2 \hfill \\
H:10 = 3:2 \hfill \\
H = \frac{{3 \cdot 10}}{2} \hfill \\
H = 15cm \hfill \\
\hfill \\
P = 2B + M \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = \frac{{{d_1} \cdot {d_2}}}{2} \hfill \\
B = \frac{{12 \cdot 16}}{2} \hfill \\
B = 96c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = 4aH \hfill \\
M = 4 \cdot 10 \cdot 15 \hfill \\
M = 600c{m^2} \hfill \\
\end{gathered}
\end{array} \hfill \\
\hfill \\
P = 2 \cdot 96 + 600 \hfill \\
P = 192 + 600 \hfill \\
P = 792c{m^2} \hfill \\
\end{gathered} \]

Пр.6)   

\[\begin{gathered}
a = 27cm \hfill \\
b = 11cm \hfill \\
c = 17cm \hfill \\
\underline {H = h + 5} \hfill \\
P = ? \hfill \\
\hfill \\
B = \frac{{a + b}}{2} \cdot h \hfill \\
\end{gathered} \]

\[\begin{gathered}
\frac{{a - b}}{2} = \frac{{27 - 11}}{2} = \frac{{16}}{2} = 8 \hfill \\
{c^2} = {h^2} + {\left( {\frac{{a - b}}{2}} \right)^2} \hfill \\
{17^2} = {h^2} + {8^2} \hfill \\
289 = {h^2} + 64 \hfill \\
{h^2} = 225 \hfill \\
h = 15cm \hfill \\
H = 15 + 5 = 20cm \hfill \\
\hfill \\
P = 2B + M \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
\begin{gathered}
B = \frac{{a + b}}{2} \cdot h \hfill \\
B = \frac{{27 + 11}}{2} \cdot 15 \hfill \\
B = 285c{m^2} \hfill \\
\end{gathered} &\begin{gathered}
\hfill \\
\hfill \\
\hfill \\
\end{gathered} &\begin{gathered}
M = aH + bH + 2cH \hfill \\
M = 27 \cdot 20 + 11 \cdot 20 + 2 \cdot 17 \cdot 20 \hfill \\
M = 1440c{m^2} \hfill \\
\end{gathered}
\end{array} \hfill \\
\hfill \\
P = 2 \cdot 285 + 1440 \hfill \\
P = 2010c{m^2} \hfill \\
\end{gathered} \]