Unit 1 Lesson 4 Area of Regular Polygons
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1 Unit 1 Lesson 4 Area of Regular Polygons Objectives Find areas of regular polygons. Find areas of composite figures. Vertical Alignment Before Lesson 1-4 Use inscribed and circumscribed figures and find the areas of circles. Lesson 1-4 Find areas of regular polygons and composite figures. After Lesson 1-4 Find the areas of similar figures. Common Core State Standard G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices Make sense of problems and persevere in solving them. Attend to precision. Scaffolding Questions Have students read the Why? section of the lesson. Ask How many sides does the table have? Six If the area of one of the triangular sections of the table is 5 square feet, what is the area of the table? 30 ft 2 How can you find the area of a table that is composed of 10 triangular parts? Find the sum of the areas of the triangular sections. 1. Areas of Regular Polygons Example 1 shows how to identify segments and angles in a regular polygon. Examples 2 and 3 show how to find the area of a regular polygon.
2 Example 1: In the figure, pentagon PQRST is inscribed in üx. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. center: point X; radius: XR or XQ; apothem: XN; central angle: RXN; m RXN = 72 Example 2: FURNITURE The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth? Example 3:
3 Focus on Mathematical Content A regular polygon can be divided into congruent isosceles triangles by drawing a line from each vertex to the center of the polygon. The altitude of one of these triangles is called an apothem. The area of the polygon can be determined by adding the areas of the triangles. If a polygon has a side of s units and an apothem of a units, then the area of one of these triangles is. By multiplying this formula by the number of sides and substituting P for the formula for perimeter contained in the result, you will find that the formula for the area of a regular polygon is. A composite figure is a figure that cannot be classified into the specific shapes that the student has studied. A composite figure is also called an irregular figure or irregular polygon. To find the areas of composite figures, separate the figures into shapes for which you can find the area. The area of the composite figure is the sum of the areas of these separate shapes. The formula for the area of a regular polygon does not apply to an irregular polygon. To find the area of an irregular polygon, separate the polygon into figures that have areas that can be calculated easily. Area of Irregular Polygons Watch for students who think that the area formula for a regular polygon applies to any polygon. Stress that to find the area of a polygon that is not regular, you may need to divide the polygon into shapes with known area formulas. Teaching with Technology Video Recording Separate students into groups. Give each group the same composite figure and ask them to find the area. Have students explain the process they used to find the area. Record their explanations. Show the videos to the entire class to see if students had different approaches to solving the same problem.
4 Teaching the Math Practices Precision Mathematically proficient students use clear definitions in discussion with others and in their own reasoning. Encourage students to connect the altitude of a triangle to the altitude of a polygon. Sense-Making Mathematically proficient students start by explaining the meaning of a problem to themselves and looking for entry points to its solution. They plan a solution pathway rather than simply jumping into a solution attempt. Perseverance Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, does this make sense? Sense-Making Mathematically proficient students consider analogous problems and try simpler forms of the original problem in order to gain insight into its solution. Encourage students to divide the shape of the state into a triangle and rectangle. Differentiated Homework Assessments: Have students turn in an exit slip with an explanation on how to find the area of a composite figure. Differentiated Instruction: Extension Have students describe how to find the area of an inscribed polygon. First, find the apothem, or the distance from the center of the polygon perpendicular to the midpoint of the opposite side. Then, find half the product of the perimeter and the apothem.
5 Extension Have the students find the area of the white stars of the United States flag. The regular pentagon in the center of each star has an apothem of 0.69 centimeter. The five triangles of each star have bases of 1 centimeter and height of 1.5 centimeters. The area of each star is cm 2. So the area of all the stars is cm 2. Interpersonal Learners Have the students discuss how finding the area of a composite figure is similar to finding the area of a parallelogram, triangle, or trapezoid. Students should realize that throughout this chapter, they have separated figures into simpler regions to find area.
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