Question

Express each statement as an equation involving the indicated variables. Surface Area of a Sphere The surface area \(S\) of a sphere is 4 times \(\pi\) times the square of the radius \(r\).

Short answer

S = 4 \pi r^2

Step-by-step solution

- Understand the Problem

Identify the known quantities and the relationship between them. The surface area of a sphere, denoted by the variable S, is described as being 4 times \( \pi \) times the square of the radius, denoted by the variable r.

- Write the Mathematical Representation

Translate the statement into a mathematical equation. The phrase '4 times \( \pi \) times the square of the radius' can be written as \( 4 \pi r^2 \). Therefore, the equation representing the surface area will be \( S = 4 \pi r^2 \).

Key concepts

Sphere Studies

A sphere is a three-dimensional shape with all points on its surface equidistant from its center. This unique shape is highly symmetrical. Examples of spheres include balls, bubbles, and planets. Understanding spheres is crucial in fields like geometry, physics, and engineering.
The properties of a sphere make calculating surface areas slightly different from flat shapes. Dive deeper into the world of spheres to unlock more mathematical secrets!

Surface Area Simplified

The surface area of a sphere can be thought of as the total area that the surface of the sphere occupies. Imagine peeling an orange and laying the peel flat; the total area of the peel represents the surface area.
Knowing how to calculate the surface area is essential for applications like determining material quantities for production or understanding physical phenomena in science.
The formula for the surface area of a sphere is given by:
\( S = 4 \pi r^2 \).
Here, \( S \) represents the surface area, \( \pi \) is a constant (approximately 3.14159), and \( r \) stands for the radius of the sphere.

The Relevance of Radius

The radius is the distance from the center of the sphere to any point on its surface. It is a crucial measurement that fundamentally defines the size of the sphere.
In the surface area equation \( S = 4 \pi r^2 \), the radius directly influences the surface area. If the radius changes, the surface area changes accordingly. Doubling the radius, for example, increases the surface area by a factor of four.
Measurement of the radius can be simple for small spheres, just a matter of using a ruler or calipers. For larger objects, more advanced techniques might be used.

Understanding the Mathematical Equation

Mathematical equations are concise ways to represent relationships between quantities. They are essential tools in both theoretical and applied mathematics.
In the context of our problem, the equation \( S=4 \pi r^2 \) tells us that the surface area \( S \) depends on the square of the radius \( r \). The term \( 4 \pi \), a constant, shows how this relationship is scaled for spheres.
Understanding these relationships helps in problem-solving and practical applications. For instance, knowing how surface area scales with radius is useful in fields ranging from materials science to astrophysics!