Sphere

A sphere is a three-dimensional geometric shape that is perfectly round, like a ball. It is defined as the set of all points in space that are a given distance (called the radius) from a given point (called the center).

Formulas

The following are important formulas related to spheres:

Surface Area: 4πr², where r is the radius of the sphere.
Volume: (4/3)πr³, where r is the radius of the sphere.

Surface Area of a Sphere

To find the surface area of a sphere, you can use the formula 4πr², where r is the radius of the sphere.

Volume of a Sphere

The volume of a sphere can be calculated using the formula (4/3)πr³, where r is the radius of the sphere.

Example Problems

Let's solve a couple of example problems to understand how to apply the formulas for surface area and volume of a sphere.

Example 1

Find the surface area and volume of a sphere with a radius of 5 units.

Surface Area = 4π * 5² = 4π * 25 = 100π square units

Volume = (4/3)π * 5³ = (4/3)π * 125 = 500π cubic units

Example 2

If the volume of a sphere is 288π cubic units, find its radius.

Volume = (4/3)πr³ = 288π

Now, solve for r: r³ = (3/4) * 288 = 216

r = 6 units

Summary

In summary, a sphere is a three-dimensional shape with a perfectly round surface. The formulas for its surface area and volume are 4πr² and (4/3)πr³ respectively, where r represents the radius of the sphere.

◂Math Worksheets and Study Guides Sixth Grade. Volume

The resources above cover the following skills:

Geometry (NCTM)

Use visualization, spatial reasoning, and geometric modeling to solve problems.

Use geometric models to represent and explain numerical and algebraic relationships.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.

Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.

Connections to the Grade 6 Focal Points (NCTM)

Measurement and Geometry: Problems that involve areas and volumes, calling on students to find areas or volumes from lengths or to find lengths from volumes or areas and lengths, are especially appropriate. These problems extend the students' work in grade 5 on area and volume and provide a context for applying new work with equations.