Exploring Area and Perimeter of Similar Polygons

Objective:

Students will be able to calculate the area and perimeter of similar polygons using the properties of similar figures.

Assessment:

Students will be assessed through a worksheet containing various problems involving the calculation of the area and perimeter of similar polygons. The problems will vary in complexity to gauge students' understanding of the topic thoroughly.

Key Points:

• Similar Polygons: Polygons are similar if they have the same shape but not necessarily the same size.

• Ratio of Perimeters: The ratio of the perimeters of similar polygons is the same as the scale factor. This applies to all corresponding parts of similar shapes.

• Area of Similar Polygons Theorem: If two polygons are similar, then the ratio of their areas is equal to the square of the scale factor.

Definitions and Formulas:

• Perimeters: The ratio of the perimeters is the same as the scale factor. In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor.

• Area of Similar Polygons Theorem: If two polygons are similar, the ratio of their areas is equal to the square of the scale factor.

Opening:

• Engage students by showing two different shapes and asking them to identify if they are similar or not.

• Discuss with students what makes two polygons similar and how they can identify this.

Introduction to New Material:

• Define similar polygons and explain the properties they share.

• Show examples of similar polygons and demonstrate how to calculate the ratio of perimeters and areas.

• Address the misconception that polygons need to be the same size to be similar.

Guided Practice:

• Provide examples for students to practice calculating the ratios of perimeters and areas of similar polygons.

• Scaffold questioning starting from basic problems to more challenging ones.

• Monitor student performance by circulating the room and providing support as needed.

Independent Practice:

• Assign a worksheet with problems involving the area and perimeter of similar polygons.

• Encourage students to show their work and explain their calculations clearly.

• Monitor students as they work independently, providing guidance where necessary.

Closing:

• Have students share their answers to a few problems and explain their reasoning.

• Summarize the key concepts learned during the lesson about similar polygons.

Extension Activity:

• For early finishers, provide them with a set of more complex problems involving real-life scenarios that require the application of area and perimeter concepts to similar polygons.

Homework:

• Create an assignment for students to find and draw two similar polygons in their home environment. Calculate the area and perimeter of both polygons and compare their ratios.

Standards Addressed:

• G-SRT.A.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for two similar figures.

• G-SRT.A.2: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Assessment Worksheet - Area and Perimeter of Similar Polygons

1. Problem 1:

   • Given two similar rectangles with a scale factor of 2. If the perimeter of the smaller rectangle is 24 cm, what is the perimeter of the larger rectangle?

2. Problem 2:

   • Two similar triangles have a scale factor of 3. If the area of the smaller triangle is 18 cm², what is the area of the larger triangle?

3. Problem 3:

   • A square and a rectangle are similar with a scale factor of 4. If the side length of the square is 5 cm, what is the perimeter of the rectangle?

4. Problem 4:

   • Two similar polygons have a scale factor of 2.5. If the area of the smaller polygon is 32 in², what is the area of the larger polygon?

5. Problem 5:

   • Given two similar pentagons with a scale factor of 1.5. If the perimeter of the smaller pentagon is 30 ft, what is the perimeter of the larger pentagon?

Real-Life Scenario for Guided Practice

Scenario: Urban Planning Project

Description: Students will be presented with a scenario where they are urban planners designing two similar parks for a city. The scale model of Park A has a scale factor of 2 compared to the actual Park B.

Task:

1. Calculate the perimeter of Park A if its scale model perimeter is 24 meters.

2. Determine the area of Park B if the area of Park A is 36 square meters.

3. Discuss the implications of scaling up the parks in terms of resources required and overall design considerations.

Discussion Points:

• How does scaling affect the perimeter and area of the parks?

• What real-world factors need to be considered when designing similar spaces on different scales?

• Why is it important for urban planners to understand the concept of similar polygons in their work?

By incorporating real-life scenarios like this urban planning project, students can see the practical applications of similar polygons in a context that is relevant to their lives and future career possibilities.

Step-by-Step Solutions:

Problem 1:

1. The scale factor is 2, and the perimeter of the smaller rectangle is 24 cm.

2. To find the perimeter of the larger rectangle, multiply the scale factor by the perimeter of the smaller rectangle:
   Perimeter of larger rectangle = 2 * 24 = 48 cm

Problem 2:

1. The scale factor is 3, and the area of the smaller triangle is 18 cm².

2. To find the area of the larger triangle, square the scale factor and multiply by the area of the smaller triangle:
   Area of larger triangle = 3² * 18 = 162 cm²

Problem 3:

1. The scale factor is 4, and the side length of the square is 5 cm.

2. To find the perimeter of the rectangle, multiply the scale factor by the side length of the square:
   Side length of rectangle = 4 * 5 = 20 cm
   Perimeter of rectangle = 2 * (length + width) = 2 * (20 + 5) = 50 cm

Problem 4:

1. The scale factor is 2.5, and the area of the smaller polygon is 32 in².

2. To find the area of the larger polygon, square the scale factor and multiply by the area of the smaller polygon:
   Area of larger polygon = 2.5² * 32 = 200 in²

Problem 5:

1. The scale factor is 1.5, and the perimeter of the smaller pentagon is 30 ft.

2. To find the perimeter of the larger pentagon, multiply the scale factor by the perimeter of the smaller pentagon:
   Perimeter of larger pentagon = 1.5 * 30 = 45 ft

These step-by-step solutions provide a clear and straightforward approach for students to follow when solving problems involving the area and perimeter of similar polygons.