# 5.8 - Perimeter, Area, Volume

The student will

a) solve practical problems that involve perimeter, area, and volume in standard units of measure; and

b) differentiate among perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.

### BIG IDEAS

• So that I know how much fencing to put around my garden
• So that I know the distance that I ran during the Virginia 10 Miler
• So that I know if furniture will be too large for a space in my apartment
• So that I can arrange furniture in a bedroom
• So that I can purchase the right amount of carpeting
• So that I know how many plants I can fit in my garden
• So that I know how much water I need to fill up my fish tank

### UNDERSTANDING THE STANDARD

• A plane figure is any closed, two-dimensional shape.
• Perimeter is the path or distance around any plane figure. It is a measure of length.
• Area is the surface included within a plane figure. Area is measured by the number of square units needed to cover a surface or plane figure.
• Volume of a three-dimensional figure is a measure of capacity and is measured in cubic units.
• A polygon is a closed plane figure composed of at least three line segments that do not cross.
• To determine the perimeter of any polygon, add the lengths of the sides.
• Students should label the perimeter, area, and volume with the appropriate unit of linear, square, or cubic measure.
• A right triangle has one right angle.
• Students should use manipulatives to discover the formulas for the area of a right triangle and volume of a rectangular solid.
• Area of a right triangle = 1/2 base x height
• Volume of a rectangular solid = length × width × height
• Students would benefit from opportunities that include the use of benchmark fractions (e.g., 1/2 ,1/4 ) in determining perimeter.
• The area of a rectangle can be determined by multiplying the length of the base by the length of the height.
• The diagonal of the rectangle shown divides the rectangle in half creating two right triangles. The legs of the right triangles are congruent to the side lengths of the rectangle. The representation illustrates that the area of each right triangle is half the area of the rectangle. Exploring the decomposition of shapes helps students develop algorithms for determining area of various shapes (e.g., area of a triangle is ½ × base × height). • The distance from the top of the right triangle to its base is called the height of the triangle. • Two congruent right triangles can always be arranged to form a square or a rectangle.
• To develop the formula for determining the volume of a rectangular prism, volume = length × width × height, students will benefit from experiences filling rectangular prisms (e.g., shoe boxes, cereal boxes) with cubes by first covering the bottom of the box and then building up the layers to fill the entire box.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Solve practical problems that involve perimeter, area, and volume in standard units of measure. (a)
• Determine the perimeter of a polygon, with or without diagrams, when
• the lengths of all sides of a polygon that is not a rectangle or a square are given;
• the length and width of a rectangle are given; or
• the length of a side of a square is given. (a)
• Estimate and determine the area of a square and rectangle using whole number measurements given in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 24 square inches). (a)
• Develop a procedure for determining the area of a right triangle using only whole number measurements given in metric or U.S. Customary units, and record the solution with the appropriate unit of measure (e.g., 12 square inches). (a)
• Estimate and determine the area of a right triangle, with diagrams, when the base and the height are given. (a)
• Develop a procedure for determining volume using manipulatives (e.g., cubes). (a)
• Estimate and determine the volume of a rectangular prism with diagrams, when the length, width, and height are given, using whole number measurements. Record the solution with the appropriate unit of measure (e.g., 12 cubic inches). (a)
• Describe practical situations where perimeter, area, and volume are appropriate measures to use, and justify orally or in writing. (b)
• Identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation. (b)

### KEY VOCABULARY

Updated: Mar 06, 2019