**Introduction**

A cube’s surface area is an essential topic in geometry and mathematics as a whole. It allows us to understand how much material is needed to cover the cube or estimate its environmental interaction. This article shall give a comprehensive guide on the surface area of a cube, including grade appropriateness, mathematical domain, common core standards, definition, key concepts, illustrative examples, real-life applications, practice tests, and frequently asked questions.

**Grade Appropriateness**

The concept of surface area is typically introduced in middle school (grades 6-8), with students encountering the surface area of a cube, specifically around 6th or 7th grade.

**Math Domain**

This topic falls under geometry, a branch of mathematics concerned with sizes, shapes, properties of space, and relative positions of figures.

**Applicable Common Core Standards**

The following common core standards apply to the topic of the surface area of a cube:

*6.G.A.4:* Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.

*7.G.B.6:* Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

**Definition of the Topic**

The sum of the area of a cube’s six faces makes up its total surface area. A cube has six equal square faces, and its total surface area can be determined by measuring the area of one face and multiplying it by six.

**Key Concepts**

A ** cube** is a three-dimensional (3D) shape with six equally sized square faces.

The ** total surface area** (TSA) is the area of all the external faces of a three-dimensional object.

** To find a cube’s total surface area **(TSA), solve the area of one face and multiply it by six.

The ** lateral surface area **(LSA) is the area of all the external faces of a three-dimensional object, excluding the top and base surfaces.

** To find a cube’s lateral surface area **(LSA), solve the area of one face and multiply it by four.

**Discussion with Illustrative Examples**

**Cube as a Solid Figure**

Three-dimensional solid figures have length, width, and height. Cubes, pyramids, prisms, cones, spheres, etc., are a few of the most well-known solid shapes.

A cube has six equal square faces. A cube has equal-length sides on each side.

Cube is also a regular hexahedron and is part of platonic solids.

It has twelve edges — a common boundary shared by the faces of a cube.

It has eight vertices — the point where three edges meet.

**Surface Area Formulas**

Let us denote the length of each side as “s.” To find the area of one square face, we’ll use the formula for the area of a square: **A = s².** Since there are six faces, we can find the total surface area using the formula: **Total Surface Area (TSA) = 6s²**. The lateral surface area, on the other hand, is calculated using the formula: Lateral **Surface Area (LSA) = 4s²** since it refers to the area covered by the four side faces of a cube.

Let us consider a cube with a side length of 5 cm. Calculating its total surface area, we have,

Total Surface Area = 6s²

Total Surface Area = 6(5 cm)²

Total Surface Area = 6(25 cm²)

Total Surface Area = 150 cm²

And for its lateral surface area, we’ll have,

Lateral Surface Area = 4s²

Lateral Surface Area = 4(5 cm)²

Lateral Surface Area = 4(25 cm²)

Lateral Surface Area = 100 cm²

Hence, a cube with a side length of 5 cm has a total surface area of **150 cm**** ^{2}** while its lateral surface area is

**100 cm²**.

**Finding the side length of a cube using its surface area **

A cube’s total surface area is 96 cm². What is its side length?

Total surface Area of a Cube = 6s^{2}

96 cm^{2} = 6s^{2}

$\frac{96}{6}$ cm^{2} = $\frac{6}{6}$s^{2}

16 cm^{2} = s^{2}

$\sqrt{16 cm^2}$ = $\sqrt{s^2}$

4 cm = s

Therefore, the side length of a cube can be calculated using the formula: $\sqrt{\frac{Total\ Surface\ Area}{6}}$.

**Examples with Solution**

**Example 1**

Determine the lateral surface area and total surface area of each cube.

a. | b. | c. |

**Solution**

a. LSA = 4s² LSA = 4(3 cm)² LSA = 4(9 cm²) LSA = 36 cm²TSA = 6s² TSA = 6(3 cm)² TSA = 6(9 cm²) TSA = 54 cm² | b. LSA = 4s² LSA = 4(5.5 in)² LSA = 4(30.25 in²) LSA = 121 in²TSA = 6s² TSA = 6(5.5 in)² TSA = 6(30.25 cm²) TSA = 181.5 in² | c. LSA = 4s² LSA = 4(8 m)² LSA = 4(64 m²) LSA = 256 m²TSA = 6s² TSA = 6(8 m)² TSA = 6(64 m²) TSA = 384 m² |

**Example 2**

Solve the surface area of a cube with a side length of 4 cm.

**Solution**

Total Surface Area = 6s²

Total Surface Area = 6(4 cm)²

Total Surface Area = 6(16 cm²)

Total Surface Area = 96 cm²

Therefore, a cube’s total surface area with a side length of 4 cm is **96 cm².**

**Example 3**

Solve for the total surface area of the given cube.

**Solution**

To solve for the cube’s total surface area, use the formula TSA = 6s^{2},

TSA = 6(6 ft)^{2}

TSA = 6(36 ft^{2})

TSA = **216 ft**^{2}

**Real-life Application with Solution**

**Problem 1**

Consider that you need to wrap a present that has a cube-shaped shape and 10-inch sides. To determine how much wrapping paper is needed, you need to find the total surface area of the cube.

**Solution**

TSA = 6s²

TSA = 6(10 in)²

TSA = 6(100 in²)

TSA = 600 in²

Therefore, the total surface area is **600 in².**

**Problem 2**

Larry has a cube-shaped container with a total surface area of 9600 cm^{2}. Find the length of each side.

**Solution**

Since we know the total surface area, we can use the formula: Side length of a cube =$\sqrt{\frac{Total\ Surface\ Area}{6}}$. So,

Side Length = $\sqrt{\frac{9600}{6}}$

Side Length = $\sqrt{1600}$

Side Length = 40 cm

Hence, each side of the cube-shaped container is **40 cm long**.

**Practice Test**

1. Solve for the total surface area of a cube given its side length of 7 cm.

2. Calculate the lateral surface area of a cube with a side length of 5 m.

3. Determine the total surface area of a cube with a side length of 6.5 ft.

4. A cube has a total surface area of 54 m². What is the length of each side?

5. Suppose the total surface area of a cube is 150 cm²; what is the volume of the cube?

*Answers:*

1. TSA = 294 cm^{2}

2. LSA = 100 m^{2}

3. TSA = 252.5 ft^{2}

4. Side Length = 3 m

5. Volume of the Cube = 5^{3} = 125 cm^{3}

**Frequently Asked Questions (FAQs)**

**Can the surface area of a cube ever be negative?**

No, the surface area is always a positive value, representing a physical measurement.

**Is the surface area of a cube always a whole number?**

No, the surface area of a cube is not always a whole number. If the cube’s side length is a whole number, then the surface area will be a whole number. However, if the side length is a fraction or a decimal, the surface area may also be a non-whole number.

**Can the surface area of a cube be zero?**

No, the surface area of a cube cannot be zero, as each face has a non-zero area.

**Can two cubes with different side lengths have the same surface area?**

No, two cubes with different side lengths will always have different surface areas, as the surface area and the square of the side length are directly proportional.

**Is a cube’s surface area the same as its volume?**

No, the surface area and volume of a cube are distinct properties. The total surface area is the total area of all six faces, while the volume is the amount of space enclosed by the cube.

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