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

The formulas for calculating the volume of common three-dimensional shapes are:

**Cube:**V = s^{3}(where s is the length of a side)**Rectangular Prism:**V = l × w × h (where l is the length, w is the width, and h is the height)**Cylinder:**V = πr^{2}h (where π is approximately 3.14, r is the radius of the base, and h is the height)**Sphere:**V = (4/3)πr^{3}(where π is approximately 3.14 and r is the radius)**Cone:**V = (1/3)πr^{2}h (where π is approximately 3.14, r is the radius of the base, and h is the height)

**Example 1:** Find the volume of a cube with a side length of 5 cm.

Using the formula V = s^{3}, we have V = 5^{3} = 125 cm^{3}.

**Example 2:** Find the volume of a rectangular prism with dimensions 4 cm by 3 cm by 6 cm.

Using the formula V = l × w × h, we have V = 4 cm × 3 cm × 6 cm = 72 cm^{3}.

To understand volume better, it's important to practice using the formulas and solving various problems. Here are some tips for studying volume:

- Memorize the formulas for calculating the volume of different shapes.
- Understand the concept of cubic units and how they relate to volume.
- Practice solving problems involving volume, using real-life examples whenever possible.
- Explore the relationship between volume and surface area, and how changes in one affect the other.
- Use visual aids and manipulatives to help visualize and understand volume in three-dimensional space.
- Review and reinforce your understanding through practice problems and quizzes.

By mastering the concept of volume and its calculations, you'll be better equipped to solve problems involving three-dimensional objects in mathematics and real-world scenarios.

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Measurement (NCTM)

Understand measurable attributes of objects and the units, systems, and processes of measurement.

Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute.

Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems.

Understand that measurements are approximations and how differences in units affect precision.

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

Select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.

Grade 5 Curriculum Focal Points (NCTM)

Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area

Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.