A mirror is in the shape of a regular nonagon with 6 inch sides (nonagon = 9-gon). What is the area of the mirror? 3 ͕ͨ͢20° Ɣ ͕
3 360 ͕ Ɣ Ɣ Ɣ 40° ͕ͨ͢20° 20° 20° 9
͕ Ɣ 8.24 ͢͝
1 ̻ Ɣ ͕͊ 2 1 ̻ Ɣ 8.24 ʚ9 · 6ʛ 2 70° 70° ̻ Ɩ 222.5 ͢͝ ͦ
3 6 3 Big Ideas Geometry Section 11.3 Tri = 3 Octa = 8 Tetra = 4 Nona = 9 Penta = 5 Deca = 10 Hexa = 6 Dodeca = 12 Hepta (Septa) = 7 You are decorating the top of the table with small ceramic tiles. The tabletop is a regular octagon with a radius of 19.6 inches. What is the area you are covering? Whenever we have a word problem, we should always draw it out. Since it’s about a regular polygon, let’s draw a triangle slice
Since the radius of the polygon is 19.6 in, the congruent sides of the 19.6 in 19.6 in isosceles triangle are this length a Time to identify what we need to solve for!
ͥ The area equation is ͕͊ ͦ **Recall ͊ Ɣ ͢ · ͧ**
s To solve for both the apothem and side length, we need to get the central angle 360 Ɣ Ɣ 45° 22.5° 22.5° 8
19.6 in 19.6 in The apothem will bisect this angle and the side it’s intersecting a Time to figure out which trigonometric ratios to use! If it helps, separate the right triangle from inside the isosceles triangle 67.5° 67.5°
ͧ s ͧ 2 2 We have two targets to solve for and the only side we have is hypotenuse. So we must either use sin or cos! 22.5° Apothem Octagon Side 19.6 in ͕ ͧ ͧ͢͝67.5° Ɣ 19.6 ͗ͣͧ67.5° Ɣ 2 a 19.6 or or ͕ ͧ ͗ͣͧ22.5° Ɣ ͧ͢͝22.5° Ɣ 2 19.6 19.6 67.5° ͕ Ɩ 18.11 ͢͝ ͧ Ɩ 15 ͢͝ ͧ 2 You are decorating the top of the table with small ceramic tiles. The tabletop is a regular octagon with a radius of 19.6 inches. What is the area you are covering?
1 ̻ Ɣ ͕͊ 2 1 ̻ Ɣ 18.11 ʚ8 · 15ʛ 2 ̻ Ɩ 1086.5 ͦ͢͝ Since we have the radius of the regular octagon, all we need is the central angle to solve the area 360 Ɣ Ɣ 45° 8
1 19.6 in 19.6 in ̻ Ɣ ͦͦ͢ · ͧ͢͝ 2 1 ̻ Ɣ 19.6 ͦ 8 · ͧ͢͝45° 2 ̻ Ɩ 1086.5 ͦ͢͝ A mathematician is building a unique sandbox for his children. He plans on building a dodecagonal sandbox with an apothem length of 2ft. What is the total area of the sandbox? ͧ ͕ͨ͢15° Ɣ 2 2 ͧ 360 2͕ͨ͢15° Ɣ Ɣ Ɣ 30° 2 15° 15° 12 4͕ͨ͢15° Ɣ ͧ 1.07 ͚ͨ Ɣ ͧ 2 1 ̻ Ɣ ͕͊ 2 1 ̻ Ɣ 2 ʚ12 · 1.07ʛ 2 75° 75° ̻ Ɩ 12.84 ͚ͨͦ Big Ideas Geometry https://518a613669c95d087a62- ee257dca653275bec786ff52fb0c62c0.ssl.cf1 .rackcdn.com/1208/table_7a_1200__square.j pg