Volume is the amount of space occupied by a three-dimensional object. It is measured in cubic units, such as cubic centimeters (cm^{3}) or cubic meters (m^{3}).

There are different formulas for calculating the volume of different shapes:

**Cube:**Volume = s^{3}, where s is the length of a side.**Rectangular Prism:**Volume = length × width × height.**Cylinder:**Volume = πr^{2}h, where r is the radius of the base and h is the height.**Sphere:**Volume = (4/3)πr^{3}, where r is the radius.**Cone:**Volume = (1/3)πr^{2}h, where r is the radius of the base and h is the height.

Let's solve some example problems to practice calculating volumes.

If the side length of a cube is 5 cm, what is its volume?

Volume = 5^{3} = 125 cm^{3}.

A rectangular prism has a length of 7 cm, a width of 4 cm, and a height of 3 cm. What is its volume?

Volume = 7 × 4 × 3 = 84 cm^{3}.

If the radius of a cylinder is 2 m and the height is 6 m, what is its volume?

Volume = π × 2^{2} × 6 = 24π m^{3}.

Here are some tips for studying volume:

- Practice using the volume formulas for different shapes.
- Understand the concept of volume as the amount of space occupied by an object.
- Work on visualizing the shapes and how they occupy space in three dimensions.
- Review and memorize the formulas for calculating volume.
- Solve plenty of practice problems to reinforce your understanding.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

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

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.