Volume Of Spheres In Terms Of Pi And 3.14

🔑 Key Concepts

• The volume of a sphere is found using \(\displaystyle V=\frac{4}{3}\pi r^3\).
• The radius must be raised to the third power because volume measures three-dimensional space.
• An exact answer can be written as a multiple of \(\pi\).
• An approximate answer can be found by substituting \(3.14\) or using a calculator value for \(\pi\).
• Volume is measured in cubic units, such as cubic centimeters or \(\text{cm}^3\).

✏️ Worked Examples

Example 1: Answer in Terms of \(\pi\)

Find the volume of a sphere with a radius of \(12\) centimeters. Write the answer in terms of \(\pi\).

\(\displaystyle V=\frac{4}{3}\pi r^3\)

Substitute \(12\) for the radius:

\(\displaystyle V=\frac{4}{3}\pi(12)^3\)

Evaluate the exponent:

\(\displaystyle V=\frac{4}{3}\pi(1728)\)

Divide \(1728\) by \(3\):

\(V=4\pi(576)\)

Multiply:

\(V=2304\pi\)

Exact volume:
\(\boxed{2304\pi\text{ cm}^3}\)

Example 2: Approximate and Round

Find the volume of a sphere with a radius of \(9\) centimeters. Use \(3.14\) for \(\pi\) and round to the nearest cubic centimeter.

\(\displaystyle V=\frac{4}{3}\pi r^3\)

Substitute \(9\) for the radius:

\(\displaystyle V=\frac{4}{3}\pi(9)^3\)

Evaluate the exponent:

\(\displaystyle V=\frac{4}{3}\pi(729)\)

Divide \(729\) by \(3\):

\(V=4\pi(243)\)

\(V=972\pi\)

Use \(3.14\) for \(\pi\):

\(V\approx972(3.14)\)

\(V\approx3052.08\)

Round to the nearest cubic centimeter:

Approximate volume:
\(\boxed{3052\text{ cm}^3}\)

Exact Form or Decimal Approximation?

Answer in terms of \(\pi\)

Leave \(\pi\) in the final answer. For example:

\(\boxed{2304\pi\text{ cm}^3}\)

Use \(3.14\) or a calculator

Replace \(\pi\) with a decimal and round as directed. For example:

\(\boxed{3052\text{ cm}^3}\)

Always read the directions carefully to determine which form the answer should take.

Why Is the Radius Cubed?

A sphere is a three-dimensional solid. As the radius grows, the sphere becomes larger in three directions: width, depth, and height.

That three-dimensional growth is represented by:

\(r\cdot r\cdot r=r^3\)

This is why the radius is raised to the third power in the sphere volume formula and why the final answer is measured in cubic units.

🧠 Math Vocabulary

• Sphere: A three-dimensional solid whose surface points are all the same distance from its center.
• Radius: The distance from the center of a sphere to any point on its surface.
• Diameter: A line segment passing through the center of a sphere with endpoints on its surface; it is twice the radius.
• Volume: The amount of three-dimensional space contained inside a solid.
• Cubic units: Units used to measure volume, such as cubic centimeters or \(\text{cm}^3\).
• Exact value: A value written without rounding, often expressed as a multiple of \(\pi\).
• Approximation: A value close to the exact answer, commonly found by replacing \(\pi\) with a decimal.
• Cube: To raise a number to the third power by multiplying it by itself three times.
• Nearest cubic centimeter: A volume rounded to the nearest whole cubic centimeter.

💡 Main Idea

The volume of a sphere is found by cubing its radius, multiplying by \(\pi\) and \(4\), and dividing by \(3\). Depending on the directions, the final answer may remain in exact form as a multiple of \(\pi\), or it may be converted to a decimal approximation and rounded to a specified place value.

📚 What You Should Already Know

Students should know how to evaluate exponents, multiply and divide whole numbers, substitute values into formulas, work with expressions containing \(\pi\), distinguish between radius and diameter, and round decimal values to a requested place.

🚀 What Comes Next

Students can extend this work by calculating the volume of hemispheres, solving for a missing radius from a known volume, comparing spheres with different radii, and investigating how multiplying the radius by a scale factor changes the sphere’s volume by the cube of that factor.

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