Calculating Perimeter and Area of 2-D Shapes

In this lesson, students will learn how to calculate perimeter and area, and then apply what they learn to build a stage for a concert.

Preinstructional Planning

Objectives

Students will:

• Understand the formulas that measure the perimeter and area of these basic two-dimensional shapes: rectangles, circles, and triangles

Materials

• Setting the Stage With Geometry Worksheet: Geometry Works! The Stage Takes Shape printable
• Setting the Stage With Geometry Reference Sheet: Perimeter, Area, Surface Area, and Volume printable
• Answer Key: Setting the Stage With Geometry printable
• Setting the Stage With Geometry Classroom Poster printable
• Paper for calculations
• Rulers
• Optional: Setting the Stage With Geometry Take-Home Activity: Poster-Crazy printable
• Optional: What’s the Angle? Bonus Worksheet printable

During Instruction

Set Up

1. Make class sets of the Setting the Stage With Geometry Worksheet: Geometry Works! The Stage Takes Shape printable and the Setting the Stage With Geometry Reference Sheet: Perimeter, Area, Surface Area, and Volume printable.
2. Print a copy of the Answer Key: Setting the Stage With Geometry printable for your use.
3. Hang a copy of the Setting the Stage With Geometry Classroom Poster printable in your classroom or project it using a computer and projector.
4. Optional: Make class sets of the Setting the Stage With Geometry Take-Home Activity: Poster-Crazy printable and the What’s the Angle? Bonus Worksheet printable for students to complete as part of the Lesson Extensions.

Lesson Directions

Introduction to Formulas for Perimeter

Step 1: On the board, draw a rectangle labeled with a length of 4 feet and width of 3 feet. Then draw a right triangle with a base of 4 feet, a height of 3 feet, and a hypotenuse (the side opposite the right angle) of 5 feet. Ask the class to determine the distance around each shape (14 feet for the rectangle, 12 feet for the triangle).

Explain that perimeter is the distance around a polygon (a closed figure made up of line segments). To measure the perimeter of any polygon (a closed figure made up of line segments), you add together the lengths of the sides. Show that a quicker way to calculate perimeter for rectangles is to add the lengths of two adjacent sides and multiply by 2, i.e., 2(l + w).

Guided Practice

Step 2: Either individually or in pairs, ask students to complete the following problems:

• What is the perimeter of a rectangle with length = 10 meters and width = 7 meters? 34 meters
• What is the perimeter of a triangle with side lengths of 3 feet, 7 feet, and 9 feet? 19 feet
• Using a ruler, draw and label a rectangle with a length of 5 inches and width of 7 inches. Calculate the perimeter. 24 inches
• Using a ruler, draw and label a right triangle with side lengths of 6 centimeters, 8 centimeters, and 10 centimeters. Calculate the perimeter. 24 centimeters
• Does the formula for calculating the perimeter of a rectangle apply to squares? Explain your thinking. The formula does apply, but it is quicker to simply multiply the length of one side by 4 to find the perimeter. Make sure students understand that a square is also a rectangle, but a special type of rectangle.

Step 3: Draw a circle on the board. Draw a line from the center of the circle to the edge and mark it as 3 feet. Tell students that this is the radius. Point out that any line from the center of the circle to the edge is also a radius and would measure 3 feet. Indicate that the diameter of a circle is the distance from one edge of a circle to the other, passing through the center. In this case, the diameter is 6 feet. Then explain that, while circles do not have a perimeter, the distance around a circle is called the circumference. The formula for calculating the circumference: C (circumference) = π • d (diameter). Explain that π is a number equal to 3.14159… The decimal continues on infinitely, but to solve most math problems, 3.14 (pi taken to two decimal places) is acceptable.

Step 4: In groups or in pairs, ask students to calculate:

• The circumference of the circle you have drawn. Ask them to provide the answer rounded to the nearest half foot. 3.14 • 6 = 18.84 feet; the answer is 19 feet.
• The diameter for a circle with a circumference of 15.7 centimeters. 5 centimeters = 15.7 centimeters/3.14.

Introduction to Formulas by Area

Step 5: Point out the definition of area on the poster: the measure of a bounded region of a two-dimensional shape expressed in square units, e.g., square inches or square feet. Show your students the formula for the area of a rectangle: A(area) =l • w. Explain that the area of the rectangle you had drawn earlier is 12 square feet (4 • 3 = 12 square feet).

Step 6: Now point out the formula for the area of a triangle on the poster: A= 1/2 • [b(base) h (height)]. Refer to your drawing of a right triangle with a base of 4 feet and height of 3 feet and demonstrate how to calculate the area of 6 square feet (1/2 (4 • 3) = 6 square feet).

Step 7: Finally, go over the area formula for circles. Again, refer to the poster: A= π • r2, where r2 means radius squared, or r • r. Show that the area of a circle with a radius of 3 feet = 28.26 square feet (32 x 3.14 = 28.26 square feet).

Guided Practice

Step 8: Either in pairs or individually, ask students to:

• Determine the area of a rectangle with a length of 7 feet and a width of 3 feet. 3 x 7 = 21 square feet
• Determine the area of a triangle with a base of 10 centimeters and a height of 7 centimeters. 1/2 x 7 x 10 = 35 square centimeters
• Determine the area of a circle with a diameter of 10 centimeters. The trick is that, to find the radius of 5 cm, divide the diameter by 2: 52 x 3.14 = 78.5 square centimeters
• Maria determines that the area of a triangle with a base of 3 inches and a height of 7 inches is 10.5 inches. Is she correct? Explain your thinking. She is not correct. 10.5 is the correct number (1/2 x 3 x 7), but the unit of measure that should be used is square inches, not inches.
• Consider the following: Bob is trying to calculate the area of a triangle with a base of 5 meters and a height of 7 meters. He mistakenly switches the height and the base. How will this affect his answer? Explain your thinking. No effect. 1/2 x 3 x 7 = 1/2 x 7 x 3

Independent Practice 

Step 9: Distribute Setting the Stage With Geometry Worksheet: Geometry Works! The Stage Takes Shape printable. Tell students they should complete all the questions.

Step 10: Check for Understanding: Go over correct answers as a class using the Answer Key: Setting the Stage With Geometry printable.

Supporting All Learners 

Provide the Setting the Stage With Geometry Reference Sheet: Perimeter, Area, Surface Area, and Volume printable for students who are having trouble remembering which formulas go with which polygon.

Lesson Extensions

Use the following printables as extensions to the lesson:

• Setting the Stage With Geometry Take-Home Activity: Poster-Crazy printable
• What’s the Angle? Bonus Worksheet printable

Post Instructional 

Standards

• Grade 7: Geometry (CCSS 7.G.B.4 and 6)
• Grades 6–8: Make Sense of Problems, Reason Abstractly and Quantitatively, Construct Viable Arguments, Model with Mathematics, and Attend to Precision (CCSS MP1-4 and 6); NCTM Geometry

For more information, download the comprehensive Standards Chart: Geometry printable.