Understanding the Surface Area of a Cube and Its Applications
POSTED ON MARCH 27, 2023

It’s time to level up your knowledge about cubes. Now that you know this special three-dimensional shape, it’s time to uncover new interesting things about the cube, its surfaces, and its areas. This is where the surface area of the cube and its applications enter!

Have you ever wondered how stores charge for gift-wrapping services or how your parents estimate the cost of painting your house? Don’t worry, let’s learn about them today. Time to learn about the cube’s surface area and its application!

## What Is the Surface Area of a Cube?

The surface area of a cube is the sum of the area of the cube’s faces. Imagine wrapping a cube (or a cube-shaped gift) perfectly. The total amount of surface or wrapping paper needed to cover the cube represents the cube’s total surface area.

When finding the total amount of wrapping paper needed, consider the amount of wrapping paper that you need to cover all six square faces of the cube. Another great way of understanding how surface areas work is to think of painting a cube-shaped object.

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When finding the total area of the surface that needs to be painted, take a look at each of the areas of all six faces. Use the sum of the areas to know the total amount of paint needed to paint the wooden’s surfaces. Anything related to the cube’s surfaces, don’t be surprised if the total surface area will be mentioned as well!

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#### Example 1

Suppose that the wooden cube’s square surfaces each have an area of 12 in². What is the total surface area of the wooden cube?

The total area that needs to be painted is equal to the sum of all six square surfaces.

Area = 6 x 12 = 72 in²

This means that by knowing the area of one surface, it’s now easy to know the cube’s total surface area.

#### Example 2

Now, what if it costs 60 cents to paint one square inch? How much would it cost to paint all surfaces?

From the previous example, the total surface area of the wooden cube is 72 in². Since it costs 60 cents (or \$0.60) to paint 1 in², the total cost to paint the wooden cube is equal to:

From this, it costs \$43.20 to paint the wooden cube. Who knew painting and geometry can sound this fun, right? This time, let’s simplify the steps we need when finding the total surface area of any cubes.

## How To Find the Cube’s Total Surface Area?

The surface area of a cube is equal to 6s² square units, where s represents the side length of each square making up the cube. The video perfectly sums up the surface area of a cube’s definition. It is simply the total area of the six squares that make up the cube.

Total Surface Area = 6 s² = 6s²

Time to break down this concept and understand the formula behind the cube’s surface area.Take a look at the net making up one cube - it contains six faces.

Let’s say one face has a side length of s units, the area of one face will be equal to s². A cube is made up of six equal squares, so its total surface area equals 6 times the area of the square, s².

This means that when given the length of the side of the cube, follow the steps below to find the surface area of the cube:

STEP 1: Find out if you’re given the cube’s side length or one surface’s area.

STEP 2: If you have the area, skip to Step 3. If you have the length of the cube, find the square surface’s area first.

STEP 3: Multiply the resulting area by 6.

Remember these and you’ll be fine! Why don’t you test your understanding by working on the examples shown below? When you’re ready, head over to the section that follows!

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#### Example 3

One face of a cube has an area of 70 ft². What is it the surface area of the cube?

The area of one of the cube’s faces is already given, so we’re down to two steps! Remember that the surface area of the cube is simply the sum of the areas of the cube’s square surfaces.

To find the surface area of the cube, find the sum of the areas of the squares (70 ft²) or multiply the square’s area by 6.

Total Surface Area = 6 x 70 = 420 ft²

This means that the surface of the cube is equal to 420 ft².

#### Example 4

Now, take a look at the wooden block shown below. Use what you know so far to calculate the surface area of the object.

When working with cubes, remember that all the edges (or sides) are equal. All six surfaces are going to be identical squares each with an edge length of 8 cm. When finding the surface area of the bock, begin by finding the area of one square surface.

One square surface has an area of 8 x 8 = 64 cm², so find the surface area of the cube by multiplying the square’s area by 6.

Total Surface Area = 6 x 64 = 384 cm²

Hence, the surface area of the wooden block is 384 cm².

## How To Solve Problems Involving the Cube’s Total Surface Area?

When solving word problems and working with applications that involve the cube and its surface area, be mindful of the given edge length. Use this to your advantage when calculating the surface area. Keep in mind that __the surface area will always be measured in squared units. __

Here are some common types of surface area problems you might encounter in your math class or even in real life:

• The total cost of painting a surface.
• Knowing the area capacity of a cube-shaped figure.
• Find the area and cost of packing cubes or blocks.

When working with cubes and their surface areas, immediately identify the length of the cube’s side or one of its surfaces’ areas. Of course, the best way to master this is through examples, so work on the problem below to test your knowledge!

#### Example 5

Suppose that it costs \$0.45 per square inch to wrap a gift in a special craft paper. If Mary wants to have a care package be packed into a box (see image below) and wrapped with the craft paper, how much would it cost her?

By taking a look at the figure, it’s clear that the box is a cube, so begin by calculating its total surface area. Use the fact that s = 14 inches and implement this into the surface area’s formula.

Total Surface Area = 6 x 142 = 1176 in²

Now, it costs \$0.45 per square inch to wrap the box into the premium craft paper that Mary wants. Since the box has a surface area of 1176 in², multiply this by the cost per square inch to find the total cost.

Total Cost= 0.45 x 1176 = \$529.20

Hence, it’ll cost Mary a total of \$529.20 to wrap her special care package.