side length: triangle: 11.3B Warm Up 1. Find the measure 2. Find the missing 3. Find the mising of c and b: side lengths: side lengths: 4. Find the missing 5. Find the height of the side length: triangle:

11.3 Areas of Regular Polygons Geometry 11.3 Areas of Regular Polygons mbhaub@mpsaz.org

11.3 Areas of Regular Polygons Essential Question How can you find the area of a regular polygon? December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Goals Find the angle measures in regular polygons Find areas of regular polygons December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Quick Review 30-60-90 Triangles Right Triangle Trigonometry Area of a triangle December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons 30-60-90 Triangle 30 2a 60 a December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Area of Regular Polygons December 2, 2018 11.3 Areas of Regular Polygons

Area of an Equilateral Triangle s 30° 30° ? h 60° 60° base Remember: An equilateral triangle is also equiangular. So… all angles are ___°. 60° December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Finding h. s We can solve for h by using the 30-60-90 Triangle. h a 3 1 2 𝑠 3 = 3 2 𝑠 60° 60° 𝑎= 1 2 𝑠 ? December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Solving for Area s s s December 2, 2018 11.3 Areas of Regular Polygons

Area of an Equilateral Triangle s s s December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area. Solution: 𝐴= 1 2 𝑏ℎ 𝐴= 1 2 ∙8∙4 3 𝐴=16 3 8 8 4 𝟒 𝟑 4 8 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Your Turn Find the area. 𝐴= 1 2 𝑏ℎ 𝐴= 1 2 ∙10∙5 3 𝐴=25 3 10 10 5 10 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example 2 𝐴= 3 4 𝑠 2 15= 3 4 𝑠 2 𝑠 2 = 60 3 ≈34.64 𝑠=5.89 The area of an equilateral triangle is 15. Find the length of the sides. 5.89 December 2, 2018 11.3 Areas of Regular Polygons

Segments in a regular polygon. A Central Angle is formed by two radii to consecutive vertices. Center Central angle Radius December 2, 2018 11.3 Areas of Regular Polygons

Segments in a regular polygon. Center Radius r Apothem a Side s December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Apothem The perpendicular distance from the center of a regular polygon to one of its sides is called the apothem or short radius. It is the same as the radius of a circle inscribed in the polygon. Apothem is pronounced with the emphasis on the first syllable with the a pronounced as in apple (A-puh-thum). December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Apothem Radius Short December 2, 2018 11.3 Areas of Regular Polygons

Segments in a regular polygon. Measure of the Central Angle of a regular polygon with n sides: 𝑚∠1= 360 𝑛 Center Radius Central angle December 2, 2018 11.3 Areas of Regular Polygons

Measure of a central angle in a regular polygon. Central Angle of a Hexagon: x= 360 𝑛 x= 360 6 x=60° x December 2, 2018 11.3 Areas of Regular Polygons

Segments in a regular polygon. The radii are congruent. What are the measures of the other two angles? ____ This is an equilateral Δ. 60° 60° 60° 60° December 2, 2018 11.3 Areas of Regular Polygons

Area of a Regular Hexagon The area of the hexagon is equal to the area of one triangle multiplied by the number of triangles, n. Area = (Area of one )  (Number of s) December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Area of one triangle Radius r Apothem a This is the Area of only one triangle. s December 2, 2018 11.3 Areas of Regular Polygons

Area of a regular polygon Remember, there are n triangles. The total area then is r a s December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Perimeter The perimeter of the hexagon is s  n. p = s  n s s s r s s a s December 2, 2018 11.3 Areas of Regular Polygons

Area of a Regular Polygon 𝐴= 1 2 𝑎𝑠𝑛 or 𝐴= 1 2 𝑎𝑝 a = apothem s = side length n= number of sides p = perimeter This formula works for all regular polygons regardless of the number of sides. December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area. Draw a radius and an apothem. What kind of triangle is formed? 30-60-90 What is the length of the segment marked x? 6 r a 12 60 x 6 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area. 4. So what is r? 12 5. And what is a? 6. The perimeter is? 72 (6  12) 12 r a 12 6 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area. The apothem is and the perimeter is 72. The area is 12 12 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area of a regular hexagon with side length of 8. 𝐴= 1 2 𝑎𝑠𝑛 𝐴= 1 2 ∙4 3 ∙8∙6 𝐴=96 3 2 4 3 4 8 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Your Turn Find the area. 𝐴= 1 2 𝑎𝑠𝑛 9 𝐴= 1 2 ∙9 3 ∙18∙6 9 3 𝐴=486 3 18 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Example Find the area. 𝐴= 1 2 𝑎𝑠𝑛 3 𝐴= 1 2 ∙6∙4 3 ∙6 x 3 =6 6 x= 6 3 ∙ 3 3 𝐴=72 3 4 3 x= 6 3 3 x=2 3 December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Summary The area of any regular polygon can be found be dividing the shape into congruent triangles, finding the area of one triangle, then multiplying by the number of triangles. Or, multiply the length of the apothem by the perimeter and divide that by 2. December 2, 2018 11.3 Areas of Regular Polygons

11.3 Areas of Regular Polygons Homework mbhaub@mpsaz.org December 2, 2018 11.3 Areas of Regular Polygons