Unit 9
Understanding 3D Figures
Geometry 2014-15
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)
10.1 Areas of Parallelograms
and Triangles
Unit 9 – Understanding 3D Figures
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)
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
A = (8)(16)
A = 128 in2
Substitute and simplify
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=6m
Your Turn!
 Find the area of each parallelogram.
 Find the value of h for each
parallelogram.
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
1
 𝐴𝑟𝑒𝑎 = 2 (𝑏𝑎𝑠𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡)
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.
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?
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
10.2 Areas of Trapezoids,
Rhombuses, and Kites
Unit 9 – Understanding 3D Figures
Vocabulary
Area of a Trapezoid
1
𝐴𝑟𝑒𝑎 = (ℎ)(𝑏1 + 𝑏2 )
2
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
3
2
times the length of the hypotenuse.
3
ℎ=
8 = 4 3 𝑖𝑛
2
Use the formula for the area of a trapezoid.
1
𝐴𝑟𝑒𝑎 = (ℎ)(𝑏1 + 𝑏2 )
2
1
𝐴𝑟𝑒𝑎 = (4 3)(12 + 16)
2
𝐴𝑟𝑒𝑎 = 56 3 𝑖𝑛2
Your Turn!
 Find the area of each trapezoid. If necessary, leave your answer in simplest radical form.
Vocabulary
Area of a Rhombus or a Kite
1
𝐴𝑟𝑒𝑎 = 𝑑1 𝑑2
2
Finding the Area of a Kite
 Find the area of the kite.
Solution:
Step 1. Find each diagonal length.
𝑑1 = 9 + 9 = 18 𝑓𝑡
𝑑2 = 12 + 4 = 16 𝑓𝑡
Step 2. Use the formula for the area of a kite.
1
𝐴𝑟𝑒𝑎 = 𝑑1 𝑑2
2
1
𝐴𝑟𝑒𝑎 = (18) (16)
2
𝐴𝑟𝑒𝑎 = 144 𝑓𝑡 2
Finding the Area of a Rhombus
 What is the area of rhombus ABCD?
Solution: First find the length of each diagonal.
d1 = 7 + 7 = 14
Diagonals of a rhombus bisect each other.
82 = 72 + x2
Pythagorean Theorem
64 = 49 + x2
Simplify.
𝑥=
15
𝑑2 = 2𝑥 = 2 15 𝑐𝑚
Use the formula for the area of a rhombus.
1
𝐴𝑟𝑒𝑎 = 2 𝑑1 𝑑2
1
𝐴𝑟𝑒𝑎 = 2 14 2 15
𝐴𝑟𝑒𝑎 = 14 15 𝑐𝑚2
Your Turn!
 Find the area of each figure. Leave your answer in simplest radical form.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
10.3 Areas of Regular Polygons
Unit 9 – Understanding 3D Figures
Vocabulary
Radius of a Regular Polygon
 Distance from the center to a vertex
Apothem
 Perpendicular distance from the center to a side
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.
360
𝑚∠1 =
= 45
8
The apothem bisects the vertex angle, so you can find m2
using m1.
1
1
𝑚∠2 = 𝑚∠1 = 45 = 22.5
2
2
Find 3 using the fact that 3 and 2 are complementary
angles.
𝑚∠3 = 90 − 22.5 = 67.5
Your Turn!
 Each regular polygon has radii and apothem as shown. Find the measure of each
numbered angle.
Vocabulary
Theorem 10-1
 If two figures are congruent, then their areas are equal.
Area of a Regular Polygon
1
 𝐴𝑟𝑒𝑎 = 2 𝑎𝑝
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.
The hypotenuse = 2 ∙ 𝑙𝑒𝑔 in a 45-45-90 triangle
4=𝑎 2
𝑎=
4
2
𝑎=
4
2
Simplify
∙
2
2
= 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.
1
1
𝐴𝑟𝑒𝑎 = 𝑎𝑝 = (2 2)(16 2)
2
2
𝐴𝑟𝑒𝑎 = 32 𝑐𝑚2
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
Your Turn!
 Find the area of each regular polygon. Round your answer to the nearest tenth.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
11.1 Space Figures and Cross
Sections
Unit 9 – Understanding 3D Figures
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
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.
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
5 + 10 = 14 + 1
15 = 15
Your Turn!
 Draw a net the 3-dimensional figure.
 Draw a net the 3-dimensional figure.
Vocabulary
Cross-section
 Intersection of a solid and a plane
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.
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
Solution: See board for cross-section; the cross-section is a
rectangle
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
11.4 Volumes of Prisms and
Cylinders
Unit 9 – Understanding 3D Figures
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.
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
Finding the Volume of Rectangular Prisms
 Determine the volume of the rectangular prism shown.
Solution:
𝑉 = 𝐵ℎ
𝑉 = 𝑠2ℎ
𝑉 = 62 ∙ 12
𝑉 = 422 𝑖𝑛2
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 = 9 3 cm
30°-60°-90° triangle theorem
1
𝑉=
(9)(9 3)(12)
2
𝑉 = 841.8 𝑐𝑚3
Your Turn!
 Find the volume of each object. Round to the nearest tenth.
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 = 𝜋𝑟 2 ℎ
Finding the Volume of a Cylinder
 Determine the volume of the cylinder shown.
Solution:
𝑉 = 𝜋𝑟 2 ℎ
𝑉 = 𝜋(3)2 (12)
𝑉 = 339.3 𝑖𝑛2
Your Turn!
 Find the volume of each figure.
the cylindrical part of the measuring cup
Vocabulary
Composite Space Figure
 A 3-dimensional figure that is the combination of two or more simpler figures.
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.
𝑉1 = 𝐵ℎ
𝑉1 = (24)(36)(24)
𝑉1 = 20,736
Step 2. Find the volume of the half-cylinder.
1
𝑉2 = 𝜋𝑟 2 ℎ
2
1
𝑉2 = 𝜋(12)2 (24)
2
𝑉2 = 5,429
Step 3. Combine the two volumes for total volume.
𝑉 = 20,736 + 5,429 = 26,165 𝑖𝑛3
Your Turn!
 Find the volume of each composite figure to the nearest tenth.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
11.5 Volumes of Pyramids and
Cones
Unit 9 – Understanding 3D Figures
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.
1
𝑉 = 𝐵ℎ
3
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.
72 + 𝑥 2 = 252
49 + 𝑥 2 = 625
𝑥 2 = 576
𝑥 = 24 𝑐𝑚
Volume of the pyramid
1
𝑉 = 𝐵ℎ
3
1
𝑉=
(14)(14)(24)
3
𝑉 = 1568 𝑐𝑚3
Your Turn!
 Find the volume of each pyramid. Round to the nearest tenth.
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).
1
1 2
𝑉 = 𝐵ℎ, 𝑜𝑟 𝑉 = 𝜋𝑟 ℎ
3
3
Find the Volume of a Cone
 What is the volume of the cone?
Solution:
Step 1. Find the height of the cone
132 = ℎ2 + 52
169 = ℎ2 + 25
ℎ2 = 144
ℎ = 12 𝑐𝑚
Step 2. Volume of the cone
𝑉=
1 2
𝜋𝑟 ℎ
3
1
𝑉 = 𝜋 5 2 (12)
3
𝑉 = 100𝜋 𝑐𝑚2 = 314.2 𝑐𝑚2
Your Turn!
 Find the volume of each figure. Round answers to the nearest tenth.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
11.6 Surface Area and
Volumes of Spheres
Unit 9 – Understanding 3D Figures
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.
4 3
𝑉 = 𝜋𝑟
3
Finding the Volume of a Sphere
 What is the volume of the sphere?
Solution: Substitute r = 5 into the volume formula.
4 3
𝑉 = 𝜋𝑟
3
4
𝑉 = 𝜋(5)3
3
500
𝑉 =
𝜋
3
𝑉 = 523.6 𝑖𝑛3
Your Turn!
 Find the volume and surface area of a sphere with the given radius or diameter. Round
your answers to the nearest tenth.
Your Turn!
 A sphere has the given volume. Find its radius to the nearest tenth.
 A) 1436.8 mi3
 B) 808 cm3
 C) 72 m3
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.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us
11.7 Areas and Volumes of
Similar Solids
Unit 9 – Understanding 3D Figures
Vocabulary
Similar Solids
 Have the same shape, and all corresponding dimensions are proportional.
Identifying Similar Solids
 Are the two rectangular prisms similar?
Solution: Use a ratio to compare each side of the prisms.
8𝑚
2
ℎ𝑒𝑖𝑔ℎ𝑡:
=
12𝑚 3
2𝑚 2
𝑙𝑒𝑛𝑔𝑡ℎ:
=
3𝑚 3
4𝑚 2
𝑤𝑖𝑑𝑡ℎ:
=
6𝑚 3
The ratio of each corresponding pair of sides is equal, therefore
the two prisms are similar solids.
Your Turn!
 Are the given pairs of figures similar?
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.
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:
𝑎3 216
=
𝑏3 125
𝑎 6
=
𝑏 5
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.
Questions?
Instructor Email: MartinJ@lake.k12.fl.us