1. Unit 9 Understanding 3D Figures
Geometry

2. Unit Outline 10.1 Areas of Parallelograms and Triangles (0.5 day)
10.2 Areas of Trapezoids, Rhombuses, and Kites (1.5 day)
10.3 Areas of Regular Polygons (2 days)
11.1 Space Figures and Cross Sections (2 days)
11.4 Volumes of Prisms and Cylinders (1 day)
11.5 Volumes of Pyramids and Cones (2 days)
11.6 Surface Area and Volumes of Spheres (1 day)
11.7 Areas and Volumes of Similar Solids (2 days)

3. 10.1 Areas of Parallelograms and Triangles
Unit 9 – Understanding 3D Figures

4. Vocabulary Base of a Parallelogram
Any one of the various sides of the figure
Altitude
Corresponding segment perpendicular to the line containing the base, drawn from the side opposite the base
Height
Length of an altitude
Area of Parallelogram
𝐴𝑟𝑒𝑎=(𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)

5. Finding the Area of a Parallelogram
What is the area of ABCD?
Solution: 𝐴𝐵 is the correct base to use for the given altitude
A = bh Substitute and simplify
A = (8)(16)
A = 128 in2

6. Find the Missing Measurement
What is the value of x?
Solution:
Step 1. Find the area of the parallelogram using the altitude perpendicular to 𝐿𝑀
A = bh Substitute and simplify
A = (9)(3) = 27 m2
Step 2. Use the area of the parallelogram to find the value of x.
A = bh Substitute
27 = 4.5x Simplify
x = 6 m

7. Your Turn! Find the value of h for each parallelogram.
Find the area of each parallelogram.
Find the value of h for each parallelogram.

8. Vocabulary Base of a Triangle Any of the sides of a triangle Height
Length of a corresponding altitude to the line containing the base
Area of Triangle
𝐴𝑟𝑒𝑎= 1 2 (𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)

9. Finding the Area of a Triangle
A triangle has an area of 18 in2. The length of its base is 6 in. What is the corresponding height?
Solution: Draw a sketch of the triangle to visualize the problem.
A = ½ bh Substitute
18 = ½ (6)h Simplify
18 = 3h
h = 6 in
The height of the triangle is 6 in.

10. Your Turn!
A triangle has height 11 in. and base length 10 in. Find its area.
A triangle has area 24 m2 and base length 8 m. Find its height.
The figure at the right consists of a parallelogram and a triangle. What is the area of the figure?

11. Questions?
Instructor

12. 10.2 Areas of Trapezoids, Rhombuses, and Kites
Unit 9 – Understanding 3D Figures

13. Area of a Trapezoid 𝐴𝑟𝑒𝑎= 1 2 (ℎ)( 𝑏 1 + 𝑏 2 )
Vocabulary
Area of a Trapezoid 𝐴𝑟𝑒𝑎= 1 2 (ℎ)( 𝑏 1 + 𝑏 2 )

14. Finding the Area of a Trapezoid
What is the area of trapezoid WXYZ?
Solution: Draw an altitude to divide the trapezoid into a rectangle and a 30-60-90 triangle. In a 30-60-90 triangle, the length of the longer leg is times the length of the hypotenuse.
ℎ= =4 3 𝑖𝑛
Use the formula for the area of a trapezoid.
𝐴𝑟𝑒𝑎= 1 2 (ℎ)( 𝑏 1 + 𝑏 2 )
𝐴𝑟𝑒𝑎= 1 2 (4 3 )(12+16)
𝐴𝑟𝑒𝑎= 𝑖𝑛 2

15. Your Turn!
Find the area of each trapezoid. If necessary, leave your answer in simplest radical form.

16. Area of a Rhombus or a Kite 𝐴𝑟𝑒𝑎= 1 2 𝑑 1 𝑑 2
Vocabulary
Area of a Rhombus or a Kite 𝐴𝑟𝑒𝑎= 1 2 𝑑 1 𝑑 2

17. Finding the Area of a Kite
Find the area of the kite.
Solution:
Step 1. Find each diagonal length.
𝑑 1 =9+9=18 𝑓𝑡
𝑑 2 =12+4=16 𝑓𝑡
Step 2. Use the formula for the area of a kite.
𝐴𝑟𝑒𝑎= 1 2 𝑑 1 𝑑 2
𝐴𝑟𝑒𝑎= 1 2 (18) (16)
𝐴𝑟𝑒𝑎=144 𝑓𝑡 2

18. Finding the Area of a Rhombus
What is the area of rhombus ABCD?
Solution: First find the length of each diagonal.
d1 = = 14 Diagonals of a rhombus bisect each other.
82 = 72 + x2 Pythagorean Theorem
64 = 49 + x2 Simplify.
𝑥= 15
𝑑 2 =2𝑥= 𝑐𝑚
Use the formula for the area of a rhombus.
𝐴𝑟𝑒𝑎= 1 2 𝑑 1 𝑑 2
𝐴𝑟𝑒𝑎=
𝐴𝑟𝑒𝑎= 𝑐𝑚 2

19. Your Turn!
Find the area of each figure. Leave your answer in simplest radical form.

20. Questions?
Instructor

21. 10.3 Areas of Regular Polygons
Unit 9 – Understanding 3D Figures

22. Vocabulary Radius of a Regular Polygon
Distance from the center to a vertex
Apothem
Perpendicular distance from the center to a side

23. Finding Angle Measures
What are m1, m2, and m3?
Solution: To find m1 , divide the number of degrees in a circle by the number of sides.
𝑚∠1= =45
The apothem bisects the vertex angle, so you can find m2 using m1.
𝑚∠2= 1 2 𝑚∠1= =22.5
Find 3 using the fact that 3 and 2 are complementary angles.
𝑚∠3=90−22.5=67.5

24. Your Turn!
Each regular polygon has radii and apothem as shown. Find the measure of each numbered angle.

25. Vocabulary
Theorem 10-1
If two figures are congruent, then their areas are equal.
Area of a Regular Polygon
𝐴𝑟𝑒𝑎= 1 2 𝑎𝑝

26. Finding the Area of a Regular Polygon
What is the area of a regular quadrilateral (square) inscribed in a circle with radius 4 cm?
Solution: Draw one apothem to the base to form a 45°-45°-90° triangle. Using the 45°-45°-90° Triangle Theorem, find the length of the apothem.
4=𝑎 The hypotenuse = 2 ∙𝑙𝑒𝑔 in a triangle
𝑎= Simplify
𝑎= ∙ =2 2 𝑐𝑚 Rationalize the denominator
The apothem has the same length as the other leg, which is half as long as a side. To find the square’s area, use the formula for the area of a regular polygon.
𝐴𝑟𝑒𝑎= 1 2 𝑎𝑝= 1 2 (2 2 )(16 2 )
𝐴𝑟𝑒𝑎=32 𝑐𝑚 2

27. Your Turn!
Find the area of each regular polygon with the given apothem, a, and side length, s.
pentagon, a = 4.1 m, s = 6 m
octagon, a = 11.1 ft, s = 9.2 ft

28. Your Turn!
Find the area of each regular polygon. Round your answer to the nearest tenth.

29. Questions?
Instructor

30. 11.1 Space Figures and Cross Sections
Unit 9 – Understanding 3D Figures

31. Vocabulary Polyhedron
A 3-dimensional figure whose surfaces are polygons
Face
Each polygon of the polyhedron
Edge
Segment formed by the intersection of two faces
Vertex
Point where three or more edges intersect

32. Vocabulary Euler’s Formula
The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E).
F + V = E + 2
In two dimensions, Euler’s Formula reduces to F + V = E + 1.

33. Using Euler’s Formula
What does a net for the doorstop at the right look like? Label the net with its appropriate dimensions.
Solution: Draw the net and then verify Euler’s Formula.
Faces (F) = 5
Vertices (V) = 10
Edges (E) = 14
F + V = E + 1 =
15 = 15

34. Your Turn! Draw a net the 3-dimensional figure.

35. Vocabulary
Cross-section
Intersection of a solid and a plane

36. Drawing a Cross-section
Draw the horizontal cross-section for a triangular prism.
Solution: To draw a cross section, visualize a plane intersecting one face at a time in parallel segments. Draw the parallel segments, then join their endpoints and shade the cross section.

37. Your Turn!
Draw and describe the cross section formed by intersecting the rectangular prism with the plane described.
A) a plane that contains the vertical line of symmetry
Solution: See board for cross-section; the cross-section is a rectangle
B) a plane that contains the horizontal line of symmetry

38. Questions?
Instructor

39. 11.4 Volumes of Prisms and Cylinders
Unit 9 – Understanding 3D Figures

40. Vocabulary
Volume
The space that a figure occupies; It is measured in cubic units
Cavalieri’s Principle
If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume.

41. Vocabulary Volume of a Prism
The volume of a prism is the product of the area of the base (B) and the height of the prism (h).
𝑉=𝐵ℎ

42. Finding the Volume of Rectangular Prisms
Determine the volume of the rectangular prism shown.
Solution:
𝑉=𝐵ℎ
𝑉= 𝑠 2 ℎ
𝑉= 6 2 ∙12
𝑉=422 𝑖𝑛 2

43. Find the Volume of Triangular Prisms
What is the volume of the triangular prism?
(Hint: Sometimes the height of a triangular base in a triangular prism is not given. Use what you know about right triangles to find the missing value. Then calculate the volume as usual.)
Solution:
hypotenuse = 18 cm Given
short leg = 9 cm 30°-60°-90° triangle theorem
long leg = cm 30°-60°-90° triangle theorem
𝑉= (9)(9 3 )(12)
𝑉= 𝑐𝑚 3

44. Your Turn!
Find the volume of each object. Round to the nearest tenth.

45. Vocabulary Volume of a Cylinder
The volume of a cylinder is the product of the area of the base and the height of the cylinder.
𝑉=𝐵ℎ, 𝑜𝑟 𝑉=𝜋 𝑟 2 ℎ

46. Finding the Volume of a Cylinder
Determine the volume of the cylinder shown.
Solution:
𝑉=𝜋 𝑟 2 ℎ
𝑉=𝜋 (3) 2 (12)
𝑉= 𝑖𝑛 2

47. Your Turn! Find the volume of each figure.
the cylindrical part of the measuring cup

48. Vocabulary Composite Space Figure
A 3-dimensional figure that is the combination of two or more simpler figures.

49. Finding the Volume of a Composite Figure
What is the approximate volume of the bullnose aquarium to the nearest cubic inch?
Solution: View the figure as a rectangular and half-cylindrical prism.
Step 1. Find the volume of the rectangular prism.
𝑉 1 =𝐵ℎ
𝑉 1 =(24)(36)(24)
𝑉 1 =20,736
Step 2. Find the volume of the half-cylinder.
𝑉 2 = 1 2 𝜋 𝑟 2 ℎ
𝑉 2 = 1 2 𝜋 (12) 2 (24)
𝑉 2 =5,429
Step 3. Combine the two volumes for total volume.
𝑉=20,736+5,429=26,165 𝑖𝑛 3

50. Your Turn!
Find the volume of each composite figure to the nearest tenth.

51. Questions?
Instructor

52. 11.5 Volumes of Pyramids and Cones
Unit 9 – Understanding 3D Figures

53. Vocabulary Volume of a Pyramid
The volume of a pyramid is one third the product of the area of the base and the height of the pyramid.
𝑉= 1 3 𝐵ℎ

54. Finding Volume of a Pyramid
What is the volume of the square pyramid?
Solution: Sometimes the height of a triangular face in a square pyramid is not given. Here the slant height and the lengths of the sides of the base are given. Use what you know about right triangles to find the missing value. Then calculate the volume as usual.
7 2 + 𝑥 2 = 25 2
49+ 𝑥 2 =625
𝑥 2 =576
𝑥=24 𝑐𝑚
Volume of the pyramid
𝑉= 1 3 𝐵ℎ
𝑉= (14)(14)(24)
𝑉=1568 𝑐𝑚 3

55. Your Turn!
Find the volume of each pyramid. Round to the nearest tenth.

56. Vocabulary Volume of a Cone
The volume of a cone is one third the product of the base (B) and the height of the cone (h).
𝑉= 1 3 𝐵ℎ, 𝑜𝑟 𝑉= 1 3 𝜋 𝑟 2 ℎ

57. Find the Volume of a Cone
What is the volume of the cone?
Solution:
Step 1. Find the height of the cone
13 2 = ℎ
169= ℎ 2 +25
ℎ 2 =144
ℎ=12 𝑐𝑚
Step 2. Volume of the cone
𝑉= 1 3 𝜋 𝑟 2 ℎ
𝑉= 1 3 𝜋 (12)
𝑉=100𝜋 𝑐𝑚 2 = 𝑐𝑚 2

58. Your Turn!
Find the volume of each figure. Round answers to the nearest tenth.

59. Questions?
Instructor

60. 11.6 Surface Area and Volumes of Spheres
Unit 9 – Understanding 3D Figures

61. Vocabulary
Sphere
Set of all points in space equidistant from a given center
Volume of a Sphere
The volume of a sphere is four thirds the product of pi and the cube of the radius of the sphere.
𝑉 = 4 3 𝜋 𝑟 3

62. Finding the Volume of a Sphere
What is the volume of the sphere?
Solution: Substitute r = 5 into the volume formula.
𝑉 = 4 3 𝜋 𝑟 3
𝑉 = 4 3 𝜋 (5) 3
𝑉 = 𝜋
𝑉 = 𝑖𝑛 3

63. Your Turn!
Find the volume and surface area of a sphere with the given radius or diameter. Round your answers to the nearest tenth.

64. Your Turn!
A sphere has the given volume. Find its radius to the nearest tenth.
A) mi3
B) 808 cm3
C) 72 m3

65. Your Turn!
The sphere at the right fits snugly inside a cube with 18 cm edges. What is the volume of the sphere?
Leave your answers in terms of π.

66. Questions?
Instructor

67. 11.7 Areas and Volumes of Similar Solids
Unit 9 – Understanding 3D Figures

68. Vocabulary Similar Solids
Have the same shape, and all corresponding dimensions are proportional.

69. Identifying Similar Solids
Are the two rectangular prisms similar?
Solution: Use a ratio to compare each side of the prisms.
ℎ𝑒𝑖𝑔ℎ𝑡: 8𝑚 12𝑚 = 2 3
𝑙𝑒𝑛𝑔𝑡ℎ: 2𝑚 3𝑚 = 2 3
𝑤𝑖𝑑𝑡ℎ: 4𝑚 6𝑚 = 2 3
The ratio of each corresponding pair of sides is equal, therefore the two prisms are similar solids.

70. Your Turn!
Are the given pairs of figures similar?

71. Vocabulary Volumes of Similar Solids
If the scale factor of two similar solids is a:b, then the ratio of their volumes is a3:b3.

72. Finding the Scale Factor
The pyramids shown are similar, and they have volumes of 216 cu. in. and 125 cu. in. What is the scale factor relating the two triangular prisms?
Solution:
𝑎 3 𝑏 3 =
𝑎 𝑏 = 6 5

73. Your Turn!
Each pair of figures is similar. Use the given information to find the scale factor of the smaller figure to the larger figure.
Two cubes have sides of length 4 cm and 5 cm. Find the ratio of volumes.

74. Questions?
Instructor