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## Lesson 12.1 Exploring Solids

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**Lesson 12.1**Exploring Solids Today, we will learn to… > use vocabulary associated with solids > use Euler’s Formula**Polyhedron**~ a solid formed by polygons**Prisms**name the base are polyhedra named by their bases**Cylinders, Cones, & Spheres**with curved surfaces are NOT polyhedra**Parts of Solids**Faces Edges Vertex Bases vs Lateral Faces Height**+ =**6 8 14 F + V = E = 12**+ =**5 5 10 F + V = E = 8**Euler’s Theorem**# Faces + # Vertices = # Edges + 2 F + V = E + 2**1. In a polyhedron, F = 8 V = 10 E = ? find the**number of edges. F + V = E + 2 8 + 10 = E + 2 18 = E + 2 ___ edges 16**2. In a polyhedron, F = 5 V = ? E = 8 find the**number of vertices. F + V = E + 2 5 + V = 8 + 2 5 + V = 10 ___ vertices 5**3. In a polyhedron, F = ? V = 6 E = 9 find the**number of faces. F + V = E + 2 F + 6 = 9 + 2 F + 6 = 11 5 ___ faces**A soccer ball is a polyhedron with 32 faces (20 hexagons &**12 pentagons). How many vertices does this polyhedron have? hexagons pentagons 20(6) 12(5) = 60 = 120 # Edges = ½ (120 + 60) = 90 32 + V = 90 + 2 V = 60 vertices**A polyhedron can be**convex or concave.**4. Describe the cross**section shown. square**5. Describe the cross**section shown. pentagon**6. Describe the cross**section shown. triangle**A polyhedron is regular if all of its faces are congruent**regular polygons. Platonic Solids**Lesson 11.1**Angle Measures in Polygons Today, we are going to… > find angle measures in polygons**1**2 3 4 6 n-2 540˚ 720˚ (n-2)180˚ 1080˚ 180˚ 360˚ 60˚ 90˚ (n-2)180 n 108˚ 120˚ 135˚**Theorem 11.1**Polygon Interior Angles Theorem The sum of the measures of the interior angles is ____________ (n - 2)180°**(n - 2) 180°**n Corollary to Theorem 11.1 The measure of one interior angle in a regular polygon is…**1. Find the sum of the measures of the interior angles of**a 30-gon. (30 - 2) 180 (28) 180 5040˚**2.How many sides does a polygon have if the sum of the**interior angles is 3240˚? 180(n-2) = 3240 n – 2 = 18 n = 20 sides**1260˚**9 (9 - 2)180 9 = (7)180 9 3. What is the measure of each interior angle of a regular nonagon? 140**4. Find x.**How many angles? 7 Sum? 180(7 - 2) 180(5) 900˚ 900° – given angles = x 900 – 783 = 117**5. Find x.**How many angles? 5 Sum? 180(5 - 2) 180(3) 540˚ sum of all angles = 540 43x – 19 = 540 x = 13**Exploring Exterior Angles**GSP 1 2 5 3 4**Theorem 11.2**Polygon Exterior Angles Theorem The sum of the measures of one set of exterior angles in any polygon is _________ 360°**360°**n The measure of one exterior angle in any regular polygon is**360°**ext. The number of sides in any regular polygon is… # sides =**One exterior angle and its interior angle are always**________________. supplementary**360˚**10 6. What is the measure of each exterior angle in a regular decagon? 36˚**360˚**40° 7. How many sides does a regular polygon have if each exterior angle measures 40˚? = 9 sides**8. How many sides does a regular polygon have if each**interior angle measures 165.6˚? 165.6 1 180 (n-2) n = 360˚ 14.4 Don’t write this down, yet. First, find the measure of an exterior angle. 180 – 165.6 = 14.4˚ 180(n – 2) = 165.6 n = 25 sides 180n – 360 = 165.6 n – 360 =– 14.4 n n =25 sides**9. Find x.**? 70 90 ? 3x + 90 + 70 + 80 + 60 = 360° x = 20**180(3)**5 180(4) 6 A soccer ball is made up of 20 hexagons and 12 pentagons. What’s the measure of each interior angle of a regular pentagon? = 108° What’s the measure of each interior angle of a regular hexagon? = 120°**Lesson 11.2**Today, we are going to… > find the perimeter and area of regular polygons Perimeter & Areas of Regular Polygons**(10)**(5 3 ) 2 5 3 25 3 5 1. Find the area of this equilateral triangle 10 10 ≈ 43.3 60˚ 10 10**(24)**(12 3 ) 2 12 3 144 3 2. Find the area of this equilateral triangle 24 24 ≈ 249.4 24**2**s 2 3 Equilateral Triangles Area = s s s**A regular polygon’s area can be covered with isosceles**triangles.**apothem**side side Area = ½ side · apothem · # sides side · # sides Perimeter =**number of**isosceles triangles area of each isosceles triangle Area of Regular Polygons A =½(side)(apothem)(# sides) A = ½san**Perimeter of**Regular Polygons P =(side)(# sides) P = sn**Find the area of the polygon.**3. a pentagon with an apothem of 0.8 cm and side length of 1.2 cm A = ½ (s) (a) (n) A = ½ (1.2) (0.8) (5) A = 2.4 cm2**(s)(n)**Find the area of the polygon. 4. a polygon with perimeter 120 m and apothem 1.7 m A = ½ (s) (a) (n) A = ½ (1.7) (120) A = 102 m2**5. Find the central angle of the polygon.**360° 5 360° 6 360° 8 360° 2(5) 360° 2(6) 360° 2(8) 6. Find the measure of this angle 36° 30° 22.5° 72° 60° 45°