# Volume word problems on the SAT test

### Additional topics in math

**Volume word problems** require making calculations of volumes of three-dimensional shapes using volume formulas. To calculate the volume, plug the given dimension into the relevant volume formula.

**Volume formulas of five basic shapes:**

The volume formulas of five basic shapes are given at the beginning of the math sections of the SAT exam: right rectangular prism, right circular cylinder, sphere, right circular cone and rectangular pyramid.**Right rectangular prism** **volume** formula is V= lwh.**Right circular cylinder** **volume** formula is V=πr^{2}h.**A sphere volume** formula is V= ^{4}/_{3} πr^{3}.**A right circular cone volume** formula is V= ^{1}/_{3} πr^{2}h.**A rectangular pyramid volume** formula is V=^{1}/_{3} lwh.

**Calculating the effect of changes in dimensions on volume: **The power of the dimension determines the size of the change in the volume value.

- If a dimension in the volume formula is raised to a first power, the volume changes by the same factor as the shape.
- If a dimension in the volume formula is raised to a second power, when the shape changes by a factor the volume changes by a square of the factor.
- If a dimension in the volume formula is raised to a third power, when the shape changes by a factor the volume changes by a third degree of the factor.

**Comparing volumes of two shapes:** In these questions we are given ratios between the dimensions of two shapes and we are required to compare their volumes. We need to calculate the total effect on the volume of all the ratios between the shapes.

__Continue reading this page for detailed explanations and examples.__

### Volume formulas worksheet

**In the formulas below: **

l=length, w=width, h=height, V=volume, A=area

Circle measures: π=3.14159, diameter=2*radius

### The volume of a right rectangular prism

**A right rectangular prism** is a three-dimensional object with 6 faces, where all the 6 faces are rectangles. In a right rectangular prism, the angles between the base and sides are right angles.

**The volume of a right rectangular prism** is defined as the product of the area of one face (length*width) multiplied by its height.

Right rectangular prism volume formula is **V= lwh.**

Consider the following example:

Calculate the volume of a rectangular prism that has a height of 6 centimeters, a length of 3 centimeters and a width of 4 centimeters.

Right rectangular prism volume formula is V= lwh.

V=lwh=3*4*6=72 cubic centimeters.

### The volume of a right circular cylinder

**A right circular cylinder** has 2 identical and parallel circular bases at the ends. The elements are perpendicular to the bases, therefore the cylinder is called right.

**The volume of a right circular cylinder** is defined as the product of the area of the circular base multiplied by the height of the cylinder.

Circular base area formula is A=πr^{2}.

Right circular cylinder volume formula is **V=πr ^{2}h**.

Consider the following example:

Calculate the volume of a circular cylinder that has a radius of 5 centimeters and a height of 10 centimeters.

Right circular cylinder volume formula is V=πr^{2}h.

V=πr^{2}h=π*5^{2}*10=250π=785 cubic centimeters.

### The volume of a sphere

**A sphere** is a three-dimensional object that has a surface of a ball, all the point on the surface of the sphere are lying at the same distance (the radius) from the center.

A sphere volume formula is **V= ^{4}/_{3} πr^{3}**.

Consider the following example:

Calculate the volume of a sphere that has a radius of 5 centimeters.

A sphere volume formula is ^{4}/_{3} πr^{3}.

V=^{4}/_{3} πr^{3}=^{4}/_{3}*π*5^{3}=^{ 4}/_{3}*5^{3}*π=167π =524 cubic centimeters.

### The volume of a right circular cone

**A right circular cone** is a cone in which the center point of the circular base is joined with the vertex of the cone and forms a right angle (the height is perpendicular to the radius or the circle).

A right circular cone volume formula is **V= ^{1}/_{3} πr^{2}h**.

Consider the following example:

A cone has a height of 10 centimeters and a circular base with a radius of 3 centimeters. What is the volume of the cone?

A right circular cone volume formula is V= ^{1}/_{3} πr^{2}h.

V= ^{1}/_{3} πr^{2}h=^{1}/_{3}*π*3^{2}*10=30π=30*3.14159=94 cubic centimeters.

### The volume of a rectangular pyramid

**A rectangular pyramid** is a pyramid that has four-sided base and a vertex.

**A rectangular pyramid** **volume** is defined as the product of the area of the base multiplied by the height of the pyramid (the height is the distance from the center point of the base to the vertex) divided by 3.

A rectangular pyramid volume formula is **V= ^{1}/_{3} lwh**.

Consider the following example:

A pyramid has a height of 10 centimeters a length of 3 centimeters and a width of 5 centimeters. What is the volume of the pyramid?

A rectangular pyramid volume formula is V=^{1}/_{3} lwh.

V=^{1}/_{3} lwh= ^{1}/_{3}*3*5*10=50 cubic centimeters.

### Calculating dimensions given the volumes values of the shapes

To calculate an unknown dimension, plug the given dimensions and the volume values into the volume formula and solve.

Consider the following example:

A sphere has a volume of 33.5 cubic centimeters.

What is the radius of the sphere?

A sphere volume formula is V= ^{4}/_{3} πr^{3}.^{4}/_{3} πr^{3}=33.5

πr^{3}=33.5*3/4

πr^{3}=25.125

3.14159r^{3}=25.125

r^{3}=25.125/3.14159

r^{3}=8

r=^{3}√8

r=2

The radius of the sphere is 2 centimeters.

Consider the following example:

A pyramid has a square base with a height that is 3 timed bigger than the length of the base.

If the volume of the pyramid is 27 cubic centimeters, what is its height?

x= the length and the width of the square pyramid

3x= the height of the pyramid

A rectangular pyramid volume formula is V=^{1}/_{3} lwh.

V=^{1}/_{3} lwh= ^{1}/_{3} x*x*3x^{1}/_{3} x*x*3x=27

x^{3}=27

x=^{3}√27

x=3

### Calculating the effect of changes in dimensions on volume

**The power of the dimension determines the size of the change in the volume value:**

**If a dimension in the volume formula is** **raised to a** **first power**, the volume changes by the same factor as the shape.__For example:__ A rectangular pyramid volume formula is V=^{1}/_{3} lwh. The length, width and height in the formula are raised to a first power. If we double the length or the width or the height (one of them), then the volume will also be doubled.

**If a dimension in the volume formula is** **raised to a** **second power**, when the shape changes by a factor the volume changes by a square of the factor.__For example:__ A right circular cone volume formula is V= ^{1}/_{3} πr^{2}h. The radius of the circle is raised to a second power. If we double the radius, the volume will change by 2^{2}=4. The height in the formula is raised to a first power. If we double the height, then the volume will also be doubled.

**If a dimension in the volume formula is** **raised to a third power**, when the shape changes by a factor the volume changes by a third degree of the factor.

For example: A sphere volume formula is ^{4}/_{3} πr^{3}. The radius of the sphere is raised to a third power. If we double the radius, the volume will change by 2^{3}=8.

Consider the following example:

A circular cylinder has a radius of 5 centimeters and a height of 10 centimeters. What is the change in the volume if we double the radius and halve the height?

Right circular cylinder volume formula is V=πr^{2}h.

We see in the formula that the radius is raised to a second power and the height is raised to a first power.

If we doble the radius, the volume will be multiplied by 2^{2}=4.

If we halve the height, the volume will be multiplied by 0.5^{1}.

The total change in the volume is 4*0.5=2.

We can check the answer by calculating the volumes in the two scenarios:

V=πr^{2}h=π*5^{2}*10= 250π cubic centimeters.

V=πr^{2}h=π*10^{2}*5= 500π cubic centimeters.

The volume value is doubled.

### Comparing volumes of two shapes

In these questions we are given ratios between the dimensions of two shapes and we are required to compare their volumes.

We need to calculate the total effect on the volume of all the ratios between the shapes.

Consider the following example:

A right circular cone has a volume of 200 cubic centimeters.

A right circular cone volume formula is V= ^{1}/_{3} πr^{2}h.

Which of the following cones has a higher volume than the given cone?

A. A cone with 3 times the radius and 1/10 times the height.

B. A cone with 2 times the radius and 1/4 times the height.

C. A cone with 1/2 times the radius and 3 times the height.

D. A cone with 1/3 times the radius and 10 times the height.

A right circular cone volume formula is V= ^{1}/_{3} πr^{2}h.

We see in the formula that the radius is raised to a second power and the height is raised to a first power.

**Answer A:** A cone with 3 times the radius and 1/10 times the height:

The total change is (3^{2})*^{1}/_{10}=^{9}/_{10 }<1.

This cone has a smaller volume than the given cone.

**Answer B:** A cone with 2 times the radius and 1/4 times the height.

The total change is (2^{2})*^{1}/_{4}=^{4}/_{4}=1

This cone has the same volume as the given cone.

**Answer C:** A cone with 1/2 times the radius and 3 times the height.

The total change is (^{1}/_{2})^{2}*3=(^{1}/_{4})*3=^{2}/_{4} <1.

This cone has a smaller volume than the given cone.

**Answer D:** A cone with 1/3 times the radius and 10 times the height.

The total change is (^{1}/_{3})^{2}*10=(^{1}/_{9})*10=^{10}/_{9} >1

This cone has a bigger volume than the given cone.

**The correct answer is D**– A cone with 1/3 times the radius and 10 times the height has a bigger volume than the given cone.