Question

Consider a sphere of radius \(r\). What is the length of a side of a cube that has the same surface area as the sphere?

Short answer

Answer: The length of a side of a cube that has the same surface area as a sphere with radius \(r\) is given by the formula \(s = r \sqrt{\frac{4 \pi }{6}}\).

Step-by-step solution

Calculate the surface area of the sphere

Given, a sphere of radius \(r\). The surface area (A) of a sphere can be calculated using the formula: A = \(4 \pi r^2\)

Calculate the surface area of the cube

Let the length of a side of the cube be \(s\). The surface area (A) of a cube with each side having equal length can be calculated using the formula: A = \(6s^2\)

Equating the surface areas

Since the surface area of the cube should be the same as the surface area of the sphere, we can equate the two surface area formulas and solve for the length of a side (s) of the cube. \(4 \pi r^2\) = \(6s^2\)

Solve for the cube's side length

Now, we will solve for \(s\). By dividing both sides of the equation by 6, we get: \(s^2 = \frac{4 \pi r^2}{6}\) Taking the square root of both sides: \(s = \sqrt{\frac{4 \pi r^2}{6}}\) Simplify the expression: \(s = r \sqrt{\frac{4 \pi }{6}}\) So the length of a side of a cube that has the same surface area as the sphere is \(s = r \sqrt{\frac{4 \pi }{6}}\).

Key concepts

Understanding Geometry

Geometry is a branch of mathematics that deals with the properties and relations of points, lines, surfaces, and solids. In the context of our exercise, geometry explores the dimensions and relationships between shapes like spheres and cubes. To make sense of these geometrical figures, it's crucial to understand their surface areas and how they compare.

When we talk about the surface area of a 3-dimensional object, we mean the total area that the surface of the object occupies. Comparing the surface areas of different shapes is much like comparing the amount of paint you'd need to completely cover each shape without any gaps. It's an essential concept for various practical applications ranging from architecture to manufacturing.

Calculating the Surface Area of a Sphere

The surface area of a sphere is an important measure of its size and is derived using a mathematical formula. For a sphere of radius \(r\), the surface area \(A\) is given by \(4 \times \text{{pi}} \times r^2\), where \(\text{{pi}}\) is approximately 3.14159. This formula helps us understand how much material would cover the entire surface of a spherical object.

Imagine wrapping a ball in gift paper; the area of paper used is essentially what the formula yields. It's critical for students to grasp this formula since it lays the foundation for comparing surface areas of different shapes and understanding geometric properties.

Surface Area of a Cube

A cube is a special case of a geometric solid where all faces are square and of equal size. The surface area of a cube is given by the formula \(6s^2\), where \(s\) represents the length of one of its sides. Since a cube has 6 faces and each face is a square, we simply find the area of one square and multiply by six.

For students, visualizing a cube and understanding that surface area accounts for all of its six sides is essential. Each face contributes to the total coverage, and hence the multiplication by six. Remember, the length of each side of the cube plays a crucial role in determining how large its surface area is - much like determining how many tiles you'd need to cover all faces of a cubic block.

Understanding Volume and Surface Area

Volume and surface area are two fundamental concepts in geometry which describe different aspects of a figure. Volume measures how much space a 3-dimensional object occupies, while surface area measures the extent of the object's surface. They are distinct yet interrelated; knowing the surface area does not directly tell us the volume and vice versa.

For instance, a balloon being blown up gets larger in volume without changing the surface area significantly, whereas a piece of paper can have a large surface area but negligible volume. It's important for students to differentiate between the two and to use the correct formulas for each when solving geometry problems. These measures have practical implications in many fields, from packaging to construction, where both the material coverage (surface area) and capacity (volume) are significant factors.