In Exercises 15–20, write the specified quantity as a...

In Exercises 15–20, write the specified quantity as a function of the specified variable. It will help in each case to draw a picture.
A sphere is contained in a cube, tangent to all six faces. Find the surface area of the cube as a function of the radius of the sphere.

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Key Concepts

Function Notation in Geometry

Expressing a geometric quantity as a function of a given variable involves establishing a relationship where the output (e.g., surface area) is defined based on the input (e.g., the sphere's radius). This functional representation is a common mathematical technique that simplifies complex geometric relationships.

Surface Area Calculation

The surface area of a cube is calculated using the formula 6 times the square of the side length. By expressing the side length in terms of another variable, such as the radius of an inscribed sphere, the surface area can be represented as a function of that variable.

Relationship Between Dimensions

When a sphere is inscribed in a cube, the sphere's diameter is equal to the side length of the cube. This fundamental relationship allows one to translate the sphere's radius directly into the cube’s dimension, forming the basis for further calculations.

Inscribed Shapes

In geometry, one shape is said to be inscribed in another when every point on the first shape touches the boundary of the second shape. For example, when a sphere is inscribed in a cube, it touches all six faces, establishing a clear relationship between their dimensions.

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