How to find the volume of a prism - Intermediate Geometry

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Question

A rectangular box has two sides with the following lengths:

and

If it possesses a volume of $84$ cm$^{3}$ , what is the area of its largest side?

Answer

The volume of a rectangular prism is found using the following formula:

$V=l\times w\times h$

If we substitute our known values, then we can solve for the missing side.

$84=3\times 4\times x$

Divide both sides of the equation by 12.

$\frac{84}{12}=\frac{12x}{12}$

We now know that the missing length equals 7 centimeters.

This means that the box can have sides with the following dimensions: 3cm by 4cm; 7cm by 3cm; or 7cm by 4cm. The greatest area of one side belongs to the one that is 7cm by 4cm.

$A=l\times w$

$A=4\times 7$

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Question

A prism with a square base has a height of feet.

If the edge of the base is feet, what is the volume of the prism?

Answer

The volume of a prism is given as

where

B = Area of the base

and

h = height of the prism.

Because the base is a square, we have

So plugging in the value of B that we found and h that was given in the problem we get the volume to be the following.

$V = 4\cdot4 = 16,ft^3$

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Question

Find the surface area of the rectangular prism:

Answer

The surface area of a rectangular prism is

Substituting in the given information for this particular rectangular prism.

$\SA=2(13\cdot 9)+2(13\cdot 4)+2(9\cdot 4) \SA=410\ \text{units}^2$

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Question

Find the volume of a rectangular prism with a width of , height of and length of .

Answer

The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

Because all the necessary information has been provided to solve for the volume, all that needs to be done is substituting in the values for the variables.

Therefore:

$V = 2:cm \cdot 3:cm \cdot 8:cm$

$V = 6:cm^2 \cdot 8:cm$

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Question

A rectangular prism has the dimensions of , , and . What is the volume of the prism?

Answer

The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

The given information does not explicitly state which side each dimension measurement correlates to on the prism. Volume simply requires the multiplication of the dimensions together.

Volume can be solved for in the following way:

$V = 2:in \cdot 5:in \cdot 7:in$

$V = 10:in^2 \cdot 7:in$

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Question

A small rectangular fish tank has sides that are wide, long, and high. Which formula would not work to find the correct volume of the fish tank?

Answer

In this question the formula that uses addition will not yield the correct volume of the fish tank:

This is the correct answer because in order to find the volume of any rectangular prism one needs to multiply the prism's length, width, and height together.

The volume of a rectangular prism is given by the following equation:

$V= l \cdot w \cdot h$

In this equation, is length, is width, and is height.

To restate, is the correct answer because it will NOT yield the correct volume, you would need to multiply by the product of and , not add.

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Question

Find the volume of the prism.

Answer

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.

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

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(3)(9)=13.5$

Now, find the volume of the prism.

$\text{Volume of Prism}=13.5\times 20 =270$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(5)(9)=22.5$

Now, find the volume of the prism.

$\text{Volume of Prism}=22.5\times 14=315$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(6)(12)=36$

Now, find the volume of the prism.

$\text{Volume of Prism}=36\times 19=684$

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Question

Find the volume of a prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(5)(12)=30$

Now, find the volume of the prism.

$\text{Volume of Prism}=30\times 17=510$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(13)(17)=110.5$

Now, find the volume of the prism.

$\text{Volume of Prism}=110.5\times 20 =2210$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(13)(19)=123.5$

Now, find the volume of the prism.

$\text{Volume of Prism}=123.5\times 28 =3458$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(1)(4)=2$

Now, find the volume of the prism.

$\text{Volume of Prism}=2\times 4=8$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(3)(8)=12$

Now, find the volume of the prism.

$\text{Volume of Prism}=12\times 10=120$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(9)(10)=45$

Now, find the volume of the prism.

$\text{Volume of Prism}=45\times 14=630$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(7)(10)=35$

Now, find the volume of the prism.

$\text{Volume of Prism}=35\times 12=420$

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Question

Find the volume of the prism.

Answer

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.

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

$\text{Area of Right Triangle Base}=\frac{1}{2}(11)(18)=99$

Now, find the volume of the prism.

$\text{Volume of Prism}=99\times 29 =2871$

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Question

Find the volume of the prism.

Answer

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.

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

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Question

Find the volume of the prism.

Answer

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.

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

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