11-2Areas of Regular PolygonsHolt GeometryWarm UpLesson PresentationLesson Quiz

Warm UpFind the unknown side lengths in each special right triangle.1. a 30-60-90 triangle with hypotenuse 2 ft2. a 45-45-90 triangle with leg length 4 in.3. a 30-60-90 triangle with longer leg length 3m11.2 Areas of Regular Polygons

Develop and apply the formula for the area of a regular polygon.Objectives11.2 Areas of Regular Polygons

ApothemCenter of a PolygonRadius of a Polygoncentral angle of a regular polygonVocabulary11.2 Areas of Regular Polygons

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 11.2 Areas of Regular Polygons

Finding the Area of an Equilateral Triangle11.2 Areas of Regular Polygons

11.2 Areas of Regular Polygons

Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.11.2 Areas of Regular PolygonsFinding the Area of Regular Polygons

To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles.The perimeter is P = ns.11.2 Areas of Regular Polygons

11.2 Areas of Regular Polygons

Find the area of regular heptagon with side length 2 ft to the nearest tenth.Example 3A: Finding the Area of a Regular PolygonDraw a segment that bisects the central angle and the side of the polygon to form a right triangle. 11.2 Areas of Regular Polygons

Example 3A ContinuedSolve for a. Step 2 Use the tangent ratio to find the apothem.11.2 Areas of Regular Polygons

Example 3A ContinuedStep 3 Use the apothem and the given side length to find the area.Area of a regular polygon The perimeter is 2(7) = 14ft.Simplify. Round to the nearest tenth. A 14.5 ft211.2 Areas of Regular Polygons

11.2 Areas of Regular Polygons

Example 3B: Finding the Area of a Regular PolygonFind the area of a regular dodecagon with side length 5 cm to the nearest tenth.Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. 11.2 Areas of Regular Polygons

Example 3B ContinuedSolve for a. Step 2 Use the tangent ratio to find the apothem. 11.2 Areas of Regular Polygons

Example 3B ContinuedStep 3 Use the apothem and the given side length to find the area.Area of a regular polygon The perimeter is 5(12) = 60 ft.Simplify. Round to the nearest tenth. A 279.9 cm211.2 Areas of Regular Polygons

Check It Out! Example 3 Find the area of a regular octagon with a side length of 4 cm.Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. 11.2 Areas of Regular Polygons

Step 2 Use the tangent ratio to find the apothem Solve for a.Check It Out! Example 3 Continued 11.2 Areas of Regular Polygons

Step 3 Use the apothem and the given side length to find the area.Check It Out! Example 3 Continued Area of a regular polygon The perimeter is 4(8) = 32cm.Simplify. Round to the nearest tenth. A 77.3 cm211.2 Areas of Regular Polygons

Lesson Quiz: Find the area of each regular polygon to the nearest tenth.1. a regular nonagon with side length 8 cmA 395.6 cm22. a regular octagon with side length 9 ft A 391.1 ft211.2 Areas of Regular Polygons

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