Area Topic 13: Lesson 2 Regular Polygons Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS Focus: (11)(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure. (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (9)(B) Apply the relationships in special right triangles 30-60-90 and 45-45-90 and the Pythagorean Theorem, including Pythagorean triples, to solve problems.

The center of a regular polygon is equidistant from the vertices The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

Example: 1 Find the area of a regular octagon with side length 6 cm to the nearest tenth. Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The central angle is . Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

Example: 1continued Step 2 Use the tangent ratio to find the apothem. The tangent of an angle is . opp. leg adj. leg Solve for a.

Example: 1continued Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 6(8) = 48 ft. Simplify. Simplify. Round to the nearest tenth. A  173.8 cm2

Example: 2 Find the area of a regular 30-gon with side length 10 cm to the nearest tenth. (note: this drawing only represents the apothem and one side!) Find the Central Angle Find the Apothem Find the Area Find the Perimeter of the Polygon. P = 10(30) = 300

Example: 3 Each side of a regular hexagon is 4 feet. Find both area and perimeter of the hexagon. Round to the nearest tenth. Step 1 Draw the hexagon. Draw an isosceles triangle with its vertex at the center of the hexagon. The central angle is . Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

Example: 3 continued Step 2 Use the tangent ratio to find the apothem. The tangent of an angle is . opp. leg adj. leg Solve for a.

Example: 3 continued Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 4(6) = 24ft. Simplify. Round to the nearest tenth. A  41.6 ft2

Finding the Area of a Regular Polygon *Hexagon Shortcut* You can also use this formula when finding area of a regular hexagon Substitute 4 for s Square 4 Multiply 16 by ¼ Distribute the 6 Simplify. Round to the nearest tenth. A  41.6 ft2

Example: 4 Find the perimeter and area of a regular pentagon when the apothem is 20 Formula Apothem a = 20 This is given in the question. 36° 20 Find the Central Angle

Example: 4 continued Find the perimeter and area of a regular pentagon when the apothem is 20. length of x y equals side of pentagon 20 y = 2x x y Find the Perimeter of the Polygon.

Example: 4 continued Find the perimeter and area of a regular pentagon when the apothem is 20. Find the Area 20 x y