Geometry - How to find the volume of a prism

Find the volume of the prism.

Explanation

Recall how to find the volume of a prism:

$\text{Volume of Prism}=\text{Area of base}\times\text{height of prism}$

Find the area of the base, which is a right triangle.

$\text{Area of Right Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Right Triangle Base}=\frac{1}{2}(12)(15)=90$

Now, find the volume of the prism.

$\text{Volume of Prism}=90 \times 20 =1800$

Find the volume of the prism.

Explanation

To find the volume of the prism, use the following formula:

$\text{Volume of Prism}=\text{Area of Base}\times \text{height}$

Since the base of this prism is a triangle, find the area of a triangle.

$\text{Area of Triangle}=\frac{\text{base}\times \text{height}}{2}$

$\text{Area of Triangle}=\frac{18 \times 4}{2}=36$

Now, plug in the area of the base and the height of the prism to find the volume.

$\text{Volume of Prism}=36 \times 12 =432$

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base, as shown by the figure below.

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(8)(8)(20)=1280$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$

Plug in the given side to find the area of the triangle.

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(8^2)=16\sqrt3$

Next, notice that the width of the rectangular prism is also the height of the triangular prism.

Now, find the volume of the triangular prism.

$\text{Volume of Triangular Prism}=16\sqrt3(10)=160\sqrt3$

To find the volume of the entire figure, add the individual volumes together.

$\text{Volume of Figure}=\text{Volume of Rectangular Prism}+\text{Volume of Triangular Prism}$

$\text{Volume of Figure}=1280+160\sqrt3=1834.26$

Make sure to round to places after the decimal.

Find the volume of the prism.

Explanation

Recall how to find the volume of any prism.

$\text{Volume of Prism}=\text{Area of Base}\times \text{Height}$

Since the base is a trapezoid, recall how to find the area of a trapezoid.

$\text{Area of Trapezoid}=\frac{base_1+base_2}{2}(\text{height})$

Plug in the given bases and height to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{3+8}{2}(12)=66$

Now, plug this in to find the volume of the prism.

$\text{Volume of Prism}=66 \times 6 =396$

Find the volume of the prism.

Explanation

To find the volume of the prism, use the following formula:

$\text{Volume of Prism}=\text{Area of Base}\times \text{height}$

Since the base of this prism is a triangle, find the area of a triangle.

$\text{Area of Triangle}=\frac{\text{base}\times \text{height}}{2}$

$\text{Area of Triangle}=\frac{12 \times 3}{2}=18$

Now, plug in the area of the base and the height of the prism to find the volume.

$\text{Volume of Prism}=18\times 10 =180$

Find the volume of the regular hexagonal prism.

Explanation

Recall the formula to find the volume of any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

Since the base is a regular hexagon, recall how to find the area of a regular hexagon.

$\text{Area of Regular Hexagon}=\frac{3\sqrt3}{2}(\text{side})^2$

Plug in the given side to find the area of the base.

$\text{Area of Base}=\frac{3\sqrt3}{2}(2)^2=6\sqrt3$

Next, plug in the area of the base and the height to find the volume of the prism.

$\text{Volume of Prism}=6\sqrt3 \times 10=60\sqrt3=103.92$

Remember to round to places after the decimal.

Find the volume of the prism.

Explanation

Recall how to find the volume of any prism.

$\text{Volume of Prism}=\text{Area of Base}\times \text{Height}$

Since the base is a trapezoid, recall how to find the area of a trapezoid.

$\text{Area of Trapezoid}=\frac{base_1+base_2}{2}(\text{height})$

Plug in the given bases and height to find the area of the trapezoid.

$\text{Area of Trapezoid}=\frac{4+8}{2}(11)=66$

Now, plug this in to find the volume of the prism.

$\text{Volume of Prism}=66 \times 10 =660$

A triangular prism with an equilateral triangle base is placed on top of a rectangular prism with a square base as shown by the figure below.

Find the volume of the figure.

Explanation

In order to find the volume of the figure, we will first need to find the volume of the rectangular and triangular prisms.

Recall how to find the volume for any prism:

$\text{Volume of Prism}=\text{Area of base}\times \text{height}$

First find the volume of the rectangular prism:

$\text{Volume of Rectangular Prism}=(7)(7)(13)=637$

Next, find the volume of the triangular prism.

Notice that the length of the rectangular prism is the same as the side of the equilateral triangle that makes up one of the bases of the triangular prism.

Recall how to find the area of an equilateral triangle:

$\text{Area of Equilateral Triangle}=\frac{\sqrt3}{4}(\text{side})^2$