# The base of the prism is triangular in shape with sides of 3 cm, 4 cm, 5 cm. Find the volume of the prism if it’s height is 10 cm.

a) 45 cu.cm

b) 50 cu.cm

c) 60 cu.cm

d) 65 cu.cm

Last updated date: 24th Mar 2023

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Answer

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Hint: To solve the question, we have to calculate the area of the triangular base of the prism which when is multiplied with the height of the prism, will give you the volume of the prism.

Complete step-by-step answer:

We know that,

The volume of a prism = Area of the base of the prism \[\times \]the height of the prism ..… (1)

To calculate the area of the triangular base of the prism, we use the Heron's formula of area of a triangle which is equal to \[\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]

Where a, b, c are three sides of the triangle and s is the semi-perimeter of the triangle.

We know that the formula for semi-perimeter of triangle with sides a, b, c is given by \[s=\dfrac{a+b+c}{2}\]

The given sides of the triangular base of the prism are 3 cm, 4 cm, 5 cm. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{3+4+5}{2}=\dfrac{12}{2}\]

\[\Rightarrow s=6\]

By substituting the values of semi-perimeter and the sides of triangle in the above mentioned formula of area of a triangle, we get

\[A=\sqrt{6\left( 6-3 \right)\left( 6-4 \right)\left( 6-5 \right)}\]

Where A represents the area of a triangular base of the prism.

\[A=\sqrt{6\left( 3 \right)\left( 2 \right)\left( 1 \right)}\]

\[A=\sqrt{\left( 3\times 2 \right)\left( 3 \right)\left( 2 \right)}\]

\[A=\sqrt{{{2}^{2}}\times {{3}^{2}}}\]

\[\begin{align}

& A=\sqrt{{{\left( 2\times 3 \right)}^{2}}} \\

& =2\times 3 \\

& =6c{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular base of the prism of sides 3 cm, 4 cm, 5 cm is equal to 6\[c{{m}^{2}}\].

The given value of height of the prism is equal to 10 cm.

By substituting the given and calculated values in equation (1) we get

Thus, the volume of a prism \[=6\times 10=60cu.cm\]

Hence, option (c) is the right choice.

Note: The possibility of mistake can be, not able to apply the correct formula for volume of prism and for the area of triangles of given sides. The alternative way of calculating the area of the triangle is by using the formula \[\dfrac{1}{2}bh\] where b, h are the base and height of the triangle. Since the given sides of the triangle from the right-angle triangle, the other sides of the triangle excluding hypotenuse, form the base and height of the triangle. Thus, we can calculate the area of the triangular base of the prism.

Complete step-by-step answer:

We know that,

The volume of a prism = Area of the base of the prism \[\times \]the height of the prism ..… (1)

To calculate the area of the triangular base of the prism, we use the Heron's formula of area of a triangle which is equal to \[\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]

Where a, b, c are three sides of the triangle and s is the semi-perimeter of the triangle.

We know that the formula for semi-perimeter of triangle with sides a, b, c is given by \[s=\dfrac{a+b+c}{2}\]

The given sides of the triangular base of the prism are 3 cm, 4 cm, 5 cm. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{3+4+5}{2}=\dfrac{12}{2}\]

\[\Rightarrow s=6\]

By substituting the values of semi-perimeter and the sides of triangle in the above mentioned formula of area of a triangle, we get

\[A=\sqrt{6\left( 6-3 \right)\left( 6-4 \right)\left( 6-5 \right)}\]

Where A represents the area of a triangular base of the prism.

\[A=\sqrt{6\left( 3 \right)\left( 2 \right)\left( 1 \right)}\]

\[A=\sqrt{\left( 3\times 2 \right)\left( 3 \right)\left( 2 \right)}\]

\[A=\sqrt{{{2}^{2}}\times {{3}^{2}}}\]

\[\begin{align}

& A=\sqrt{{{\left( 2\times 3 \right)}^{2}}} \\

& =2\times 3 \\

& =6c{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular base of the prism of sides 3 cm, 4 cm, 5 cm is equal to 6\[c{{m}^{2}}\].

The given value of height of the prism is equal to 10 cm.

By substituting the given and calculated values in equation (1) we get

Thus, the volume of a prism \[=6\times 10=60cu.cm\]

Hence, option (c) is the right choice.

Note: The possibility of mistake can be, not able to apply the correct formula for volume of prism and for the area of triangles of given sides. The alternative way of calculating the area of the triangle is by using the formula \[\dfrac{1}{2}bh\] where b, h are the base and height of the triangle. Since the given sides of the triangle from the right-angle triangle, the other sides of the triangle excluding hypotenuse, form the base and height of the triangle. Thus, we can calculate the area of the triangular base of the prism.

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