Geometry

1. Geometry

2. Chapter 9 Material

3. Basic Terms • Point • Segment • Line • Ray • Angle

4. Angle Terms A • Vertex • Sides • Names • Measures: • Decimal Degrees • Degrees, Minutes, Seconds V B

5. Angle Measure • Convert 32.5˚ to D˚M’S”. • Convert 95.265˚ to D˚M’S”. • A full revolution is _____ degrees.

6. Types of Angles • Acute (< 90˚) • Right (= 90˚) • Obtuse (> 90˚) • Straight (= 180˚)

7. Line Relationships • Intersecting • Perpendicular • Parallel • Skew

8. Angle Relationships • Adjacent • Vertical • Complementary (sum is 90˚) • Supplementary (sum is 180˚) 137˚ 48˚

9. Transversal Angles • Interior/Exterior • Alternate interior/alternate exterior • Corresponding • Same side interior/same side exterior 75˚

10. Polygons • A simple closed figure made of line segments • A regular polygon has all sides equal in length and all angles equal in measure.

11. Types of Polygons 3 Triangle 9 Nonagon 4 Quadrilateral 10 Decagon 5 Pentagon 11 Undecagon 6 Hexagon 12 Dodecagon 7 Heptagon N N-gon 8 Octagon

12. Types of Triangles • Triangles are classified according to relationships between sides: • Scalene • Isosceles • Equilateral

13. Types of Triangles • Triangles are also classified according to angles: • Acute • Equiangular • Right • Obtuse

14. Triangle Fact • The measures of the interior angles of any triangle add to _____ degrees. 78˚ 40˚

15. Types of Quadrilaterals • Parallelogram • Rectangle • Rhombus • Square • Kite • Trapezoid • Isosceles Trapezoid

16. 68˚ 65˚ Quadrilateral Fact • The measures of the interior angles of any quadrilateral add to _____ degrees.

17. Other Polygons: Angle Sum • We can generalize on the sum of the measures of the interior angles of any polygon. • Find the sum of the interior angles in a: dodecagon

18. Another Interesting Fact: The sum of the measures of the exterior angles of any polygon is always _______ degrees. • Find the measure of each interior angle of a regular dodecagon

19. Diagonals • Line segments connecting non-consecutive vertices • General formula:

20. Three-Dimensional Shapes

21. Three-Dimensional Shapes • 5 Platonic solids: Made completely with congruent regular polygons • Tetrahedron • Hexahedron • Octahedron • Dodecahedron • Icosahedron

22. Three-Dimensional Shapes • Drawing in 3-D “1101” • Cube • Prism: Rectangular and Triangular • Pyramid: Square and Triangular • Circular Cylinder • Circular Cone • Sphere

23. Chapter 11 Material

24. Metric Measurement Prefix Chart: T G M k h dk ROOT d c m µ n p Roots: Length ― meter (m) Capacity ― liter (L) Mass ― gram (g)

25. Length facts: 12 in. = 1 ft 3 ft = 1 yd 36 in. = 1 yd 1760 yd = 1 mi 5280 ft = 1 mi Mass facts: 16 oz = 1 lb 2000 lb = 1 T Capacity facts: 4 qt = 1 gal Customary Measurement

26. Temperature Conversions

27. Perimeter • The perimeter of any triangle (or any other type of polygon) is the sum of the measures of the lengths of its sides. 5 ft 30 m 3 ft 4 ft Assume this is a regular hexagon.

28. 16.8 cm 9 ft 11.5 cm 12 ft Area of a Triangle • The common area formula is • Find the area of each triangle.

29. Quadrilaterals: Area & Perimeter • Square: A = P = 4b • Rhombus: A = bh P = 4b • Parallelogram: A = bh P = 2(a + b) • Rectangle: A = bh P = 2(b + h) • Trapezoid: A = P = a + b + c + d

30. 20 ft 8 ft 10 ft 90 ft Assume this is a square. 5.4 m 7.65 m Quadrilaterals: Area & Perimeter • Find the area and perimeter of each figure.

31. Quadrilaterals: Area & Perimeter • A rectangular field has dimensions 275 ft by 145 ft. If fence costs $1.79 per running foot, find the total cost of fencing the field. • If a bag of seed costs $10.95 and covers an average of 5,000 square feet, find the total cost of seeding this field.

32. s a Area of a Regular Polygon • General formula: a = length of apothem n = # of sides s = length of a side

33. Circle Circumference • The distance around the circle Formulas: C = 2 π r C = π d d r r = radius, d = diameter, π = pi (a number close to 3.14159 . . .)

34. Circle Circumference • The earth has a radius of approximately 3,960 miles. Find the distance around the earth along the equator. • A bicycle tire has a diameter of 26 inches. Find how far the bike travels in 1 full revolution of the tire.

35. d r Circle Area • A measure of the size of the region inside the circle Formulas: r = d ∕ 2

36. Circle Area • You measure a circle’s diameter to be 5 feet. Find the circle’s area. • If the area of a circle is 250 square meters, find the radius of the circle.

37. The Pythagorean Theorem • The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. c a b

38. The Pythagorean Theorem • Do the following represent lengths of sides of a right triangle? • 6 cm, 8 cm, 10 cm • 10 ft, 10 ft, 20 ft • 4 mi, 5 mi, 7 mi • 7 in., 24 in., 25 in.

39. 75 ft 93 yd 16.8 cm 67 yd 11.5 cm The Pythagorean Theorem • Find the missing lengths.

40. Three-Dimensional Shapes

41. Rectangular Prism • Volume: V = l w h • Surface Area: A = 2 l w + 2 w h + 2 l h l = length w = width h = height h w l

42. 8 ft 18 ft 24 ft Rectangular Prism • Find the volume and surface area of the following room.

43. r h Right Circular Cylinder • Volume: a measure of space inside a 3-dimensional shape

44. Right Circular Cylinder • Find the volume if r = 24 m and h = 40 m. • Find the diameter of a cylindrical tank 15 ft high with a capacity of 136,000 gallons. (1 cubic foot holds approximately 7.48 gallons)

45. Right Circular Cylinder • Surface area: • Find the lateral (L) and total (T)surface areas if r = 5 feet and h = 9 feet.

46. Challenge! • Orient an 8.5” by 11” piece of paper vertically and horizontally, folding to make a right circular cylinder. Compare volumes, lateral surface areas, and total surface areas. • Which is greater ― the circumference of a tennis can lid or the height of the tennis can?

47. Sphere • Volume: • Surface Area: • The earth has a radius of approximately 3,960 miles. Find the surface area and volume of the earth.

48. Geometry is all around us!