Q6. - Geometry

Question

Consider the following prism.

Suppose the bases are regular hexagons with 4cm edges.

a.Find the lateral area.

b.Find the base area.

c.Find the total area.

d.Find the volume.

Step-by-Step Solution

Answer

1. The lateral area is$120{\text{\hspace{0.17em}cm}}^2$ .
2. Th base area is $24\sqrt{3}{\text{\hspace{0.17em}cm}}^2$.
3. The total area is$\left(\text{120+48}\sqrt{3}\right){\text{cm}}^2$ .
4. The volume is$120\sqrt{3}{\text{\hspace{0.17em}cm}}^3$ .

Step 1. Given Information.

The bases of the regular hexagon are 4cm.

Step 2. Write the concept.

A prism is a polyhedron that has two equal, parallel sides called “bases” and its lateral sides are “parallelograms”.

 A rectangular prism is a polyhedron with two congruent and parallel bases. It has 6 faces and all the faces are in rectangle shape and have 12 edges.

 A right rectangular prism is a prism that has 6 faces that are rectangles and all angles are right angles.

An oblique prism in which the bases are not perpendicular to each other.

Step 3. Find the area.

The lateral area of the prism is given by,

$\text{Lateral\hspace{0.17em}\hspace{0.17em}Area}=\text{Height}\times \text{Parameter\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}Base}$

Then,

$\begin{array}{c}\text{Lateral\hspace{0.17em}\hspace{0.17em}Area}=(6\times 4)\times 5\\ =24\times 5\\ =120{\text{\hspace{0.17em}cm}}^2\end{array}$

Step 1. Given Information.

The bases of the regular hexagon are 4cm.

Step 2. Write the concept.

A prism is a polyhedron that has two equal, parallel sides called “bases” and its lateral sides are “parallelograms”.

 A rectangular prism is a polyhedron with two congruent and parallel bases. It has 6 faces and all the faces are in rectangle shape and have 12 edges.

 A right rectangular prism is a prism that has 6 faces that are rectangles and all angles are right angles.

An oblique prism in which the bases are not perpendicular to each other.

Step 3. Find the area.

The base area is given by,

$\text{Base\hspace{0.17em}\hspace{0.17em}Area}=\frac{3\sqrt{3}{s}^2}{2}$

Then,

$\begin{array}{c}\text{Base\hspace{0.17em}\hspace{0.17em}Area}=\frac{3\sqrt{3}{\left(4\right)}^2}{2}\\ =24\sqrt{3}{\text{\hspace{0.17em}cm}}^2\end{array}$

Step 1. Given Information.

The bases of the regular hexagon are 4cm.

Step 2. Write the concept.

A prism is a polyhedron that has two equal, parallel sides called “bases” and its lateral sides are “parallelograms”.

 A rectangular prism is a polyhedron with two congruent and parallel bases. It has 6 faces and all the faces are in rectangle shape and have 12 edges.

 A right rectangular prism is a prism that has 6 faces that are rectangles and all angles are right angles.

An oblique prism in which the bases are not perpendicular to each other.

Step 3. Find the total area.

To find the total area, add the lateral area and double of base area.

 

Since,

$\text{Base\hspace{0.17em}\hspace{0.17em}Area}=24\sqrt{3}{\text{cm}}^2$

 

And

$\text{Lateral\hspace{0.17em}\hspace{0.17em}Area}=120{\text{\hspace{0.17em}cm}}^2$

 

Then, total area of the prism is given in below.

$120\text{+2}\left(24\sqrt{3}\right)=\left(\text{120+48}\sqrt{3}\right){\text{cm}}^2$

Step 1. Given Information.

The bases of the regular hexagon are 4cm.

Step 2. Write the concept.

A prism is a polyhedron that has two equal, parallel sides called “bases” and its lateral sides are “parallelograms”.

 A rectangular prism is a polyhedron with two congruent and parallel bases. It has 6 faces and all the faces are in rectangle shape and have 12 edges.

 A right rectangular prism is a prism that has 6 faces that are rectangles and all angles are right angles.

An oblique prism in which the bases are not perpendicular to each other.

Step 3. Find the volume.

To find the volume, multiply the height by the base area.

$\text{Volume}=\text{Base\hspace{0.17em}\hspace{0.17em}Area}\times \text{Height}$

Since,

$\text{Base\hspace{0.17em}\hspace{0.17em}Area}=24\sqrt{3}{\text{cm}}^2$

Then,

$\begin{array}{c}\text{Volume}=24\sqrt{3}\left(5\right)\\ =120\sqrt{3}{\text{\hspace{0.17em}cm}}^3\end{array}$