Volume of Rectangular Prisms (Formula & Examples)

Welcome to the Area and Volume of Shapes!

Back in 4th grade, you learned how to find the area (the amount of flat surface) of rectangles and squares. In 5th grade, we are going to expand on that knowledge to learn how to find the areas of parallelograms, triangles, trapezoids, and rhombuses, as well as the volume of 3D solids to measure how much space they take up.
You might think, "There are so many formulas, it sounds tough...", but don't worry! Every single formula can be figured out by transforming these shapes into the rectangles and squares you already know. Once you understand the mechanism behind each one, you'll be solving them like a breeze!

1. Measuring 3D Space: "Volume"

What is Volume?

The amount of three-dimensional space that an object, like a box or a classroom, takes up is called its volume.
Just like we measured area by counting "how many \(1\text{ cm}\) squares fit inside," we measure volume by counting "how many cubes with \(1\text{ cm}\) sides fit inside."

The volume of a cube with \(1\text{ cm}\) sides is written as \(1\text{ cm}^3\) and is read as "one cubic centimeter." The small "3" indicates that three measurements of length (length, width, and height) have been multiplied together.

Formulas for the Volume of Rectangular Prisms and Cubes

By thinking about how many \(1\text{ cm}^3\) cubes line up inside a 3D solid, we get the following formulas:

[Volume of a Rectangular Prism]
The number of cubes lining the base layer is "length \(\times\) width," and these are stacked up to the "height."
Volume of a rectangular prism \(=\) length \(\times\) width \(\times\) height

[Volume of a Cube]
For a cube, the length, width, and height are all the exact same length.
Volume of a cube \(=\) edge \(\times\) edge \(\times\) edge

(Example) What is the volume of a rectangular prism with a length of \(4\text{ cm}\), a width of \(5\text{ cm}\), and a height of \(3\text{ cm}\)?
Equation: \(4 \times 5 \times 3 = 60\)
Answer: \(60\text{ cm}^3\)

A Unit for Larger Volumes: \(1\text{ m}^3\) (One Cubic Meter)

When measuring the volume of air inside a room or the space inside a large shipping container, using \(1\text{ cm}^3\) makes the numbers way too huge to work with. That's why we use the volume of a cube with \(1\text{ m}\) sides as a benchmark. We write this as \(1\text{ m}^3\) (one cubic meter).

[Key Point: How many \(\text{cm}^3\) are in \(1\text{ m}^3\)?]
Remember, \(1\text{ m} = 100\text{ cm}\).
Because \(100\text{ cm}\) cubes line up along the length, \(100\text{ cm}\) along the width, and \(100\text{ cm}\) along the height:
\(100 \times 100 \times 100 = 1000000\)
This means that \(1\text{ m}^3 = 1000000\text{ cm}^3\) (one million \(\text{cm}^3\))! It has 6 zeros, so be very careful with place value.

[Common Mistake]
Just because \(1\text{ m} = 100\text{ cm}\), many students accidentally write "\(1\text{ m}^3 = 100\text{ cm}^3\)"! For volume, you multiply by \(100\) three times—for length, width, and height—so remember that the number of zeros increases by \(2 \times 3 = 6\).

[Fun Fact: The Deep Connection Between Volume and Liquid Capacity (\(\text{L}\) and \(\text{mL}\))]
The units of liquid capacity you learned in 2nd and 3rd grade, like "\(\text{L}\) (liters)" and "\(\text{mL}\) (milliliters)," actually match up perfectly with volume!
・The amount of water that fills \(1\text{ cm}^3 = 1\text{ mL}\)
・The amount of water that fills a cube with \(10\text{ cm}\) edges (\(10 \times 10 \times 10 = 1000\text{ cm}^3\)) \(= 1\text{ L}\)
・A box of \(1\text{ m}^3\) holds an amazing \(1000\text{ L}\) of water (about 5 full bathtubs!).

2. Area of a Parallelogram

Transforming a Parallelogram into a Rectangle!

The secret to finding the area of a parallelogram is to cut off a piece, move it over, and "transform it into a familiar rectangle."
If you cut off the slanted corner straight down and attach it to the other side, it turns into a neat rectangle!
When you do this, the side that corresponds to the rectangle's width is called the base, and the straight vertical line is called the height.

[Formula for the Area of a Parallelogram]
Area of a parallelogram \(=\) base \(\times\) height

[Key Point: The Relationship Between Base and Height]
The height is "the length that extends perpendicular (at a right angle) to the base."
The length of the slanted side is NOT the height! Searching for the right-angle mark (perpendicular relationship) is always your shortcut to the correct answer.

[Common Mistake]
When the length of the slanted side is written on a test question, it can be tempting to multiply by it. Watch out—that slanted side is a trap! Always make sure to look for the height that meets the base at a right angle.

3. Area of a Triangle

A Triangle is "Half" of a Parallelogram!

To find the area of a triangle, take an identical copy of the triangle, flip it upside down, and join them together. When you do... it turns into a parallelogram!
The area of one triangle is exactly half (\(\frac{1}{2}\)) of that parallelogram.

[Formula for the Area of a Triangle]
Area of a triangle \(=\) base \(\times\) height \(\div 2\)

(Example) What is the area of a triangle with a base of \(6\text{ cm}\) and a height of \(4\text{ cm}\)?
Equation: \(6 \times 4 \div 2 = 12\)
Answer: \(12\text{ cm}^2\)

[Key Point: Triangles Where the Height Falls Outside the Shape]
In an obtuse triangle (a triangle with one wide angle), when you try to drop a perpendicular line straight down from the top vertex to the base, it can fall outside of the base.
This might seem confusing at first, but don't worry! The perpendicular distance between the vertex and the "line extending straight out from the base" is the height. Just calculate using the original length of the base itself without extending it.

[Common Mistake]
Getting so focused on the multiplication that you forget the final \(\div 2\) is a very common slip-up! Get into the habit of writing down the full equation, including "\(\div 2\)," right from the start.

4. Area of a Trapezoid

What Happens When You Combine Two Trapezoids...?

A four-sided shape (quadrilateral) with one pair of opposite parallel sides is called a trapezoid. Of the two parallel sides, the top one is called the top base, and the bottom one is called the bottom base.
Just like with triangles, let's take an identical copy of the trapezoid, turn it upside down, and fit them together snugly.
Look at that: it transforms into a large parallelogram whose base length is "top base \(+\) bottom base"!
A single trapezoid is half the area of this large parallelogram.

[Formula for the Area of a Trapezoid]
Area of a trapezoid \(=\) (top base \(+\) bottom base) \(\times\) height \(\div 2\)

(Example) What is the area of a trapezoid with a top base of \(3\text{ cm}\), a bottom base of \(7\text{ cm}\), and a height of \(4\text{ cm}\)?
Equation: \((3 + 7) \times 4 \div 2 = 10 \times 4 \div 2 = 20\)
Answer: \(20\text{ cm}^2\)

[Key Point: Always Use Parentheses ( ) !]
Following the order of operations, you must "add the top base and bottom base" first. If you forget the parentheses and write \(3 + 7 \times 4 \div 2\), the multiplication will be calculated first, giving you the wrong answer. Always be sure to put parentheses around "(top base \(+\) bottom base)."

5. Area of a Rhombus

Let's Enclose the Rhombus in a Rectangle!

A four-sided shape where all four sides are equal in length is called a rhombus. A rhombus has a special property: its two diagonals (lines connecting opposite corners) cross each other at right angles.
If you draw a rectangle that fits snugly around the outside of the rhombus, you'll see that the area of the rhombus is exactly half of that rectangle.

[Formula for the Area of a Rhombus]
The length and width of the enclosing rectangle are the exact same lengths as the two diagonals of the rhombus.
Area of a rhombus \(=\) diagonal \(\times\) diagonal \(\div 2\)

(Example) What is the area of a rhombus with diagonals of \(6\text{ cm}\) and \(8\text{ cm}\)?
Equation: \(6 \times 8 \div 2 = 24\)
Answer: \(24\text{ cm}^2\)

Study Summary (Formula Quick Reference)

Let's organize the important formulas we learned today! Review them while picturing how the shapes transform in your mind.

[Volume Formulas]
・Volume of a rectangular prism \(=\) length \(\times\) width \(\times\) height
・Volume of a cube \(=\) edge \(\times\) edge \(\times\) edge
・Unit conversion: \(1\text{ m}^3 = 1000000\text{ cm}^3\)

[Area Formulas]
・Area of a parallelogram \(=\) base \(\times\) height
・Area of a triangle \(=\) base \(\times\) height \(\div 2\)
・Area of a trapezoid \(=\) (top base \(+\) bottom base) \(\times\) height \(\div 2\)
・Area of a rhombus \(=\) diagonal \(\times\) diagonal \(\div 2\)

[A Tip from Your Teacher]
Notice how the formulas for the triangle, trapezoid, and rhombus all have a \(\div 2\) at the end! If you remember the reason—"we divide by 2 because we are finding half of a rectangle or parallelogram that we reshaped"—you'll never forget it on a test. Look closely at each shape, double-check where the right angles are, and have fun working through practice problems!