Unit 4: Circles and VolumeVocabulary Builder

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Volume of Cylinders

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

Polyhedra Euler’s Formula net

edge

Cross section

Vertex

Face Horizontal plane Vertical plane

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Formula for volume

Substitutionh = 9

Base is a right triangle, legs 3 and 4.2

Multiply

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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1

2

3

4

5

1

2

3

4

5

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Volume of Cylinders

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G.GMD.1 Give informal arguments for

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Volume of Cylinders

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G.GMD.1 Give informal arguments for

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G.GMD.1 Give informal arguments for

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Ratio

Similar

Proportion

Scale factor

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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Lateral area

Volume

Linear dimensions

Surface area

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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Unit 4: Circles and Volume4.7 Volume of Prisms

(Rectangular, Triangular, and Cylinders)

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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Volume

cubic

congruent

Right rectangular prism bases

length widthheight

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 = 𝑙𝑤ℎ

𝑉 = 10.5 × 8 × 12

𝑉 = 1008𝑐𝑚3

𝑉 = 𝑙𝑤ℎ

𝑉 = 15 × 5 × 3

𝑉 = 225𝑐𝑚3

𝑉 = 142.5𝑐𝑚3 𝑉 = 456𝑐𝑚3

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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height bases

multiply

base

height

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 =7 × 18 × 35

2

𝑉 = 2205𝑚𝑚3

𝑉 =22 × 15.3 × 19

2

𝑉 = 3195.4𝑐𝑚3

Must find the height!

𝑥2 + 9.52 = 182

𝑥2 = 233.75

𝑥 = 15.3

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𝑉 = 1925𝑚3 𝑉 = 120𝑚3

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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capacity

area

height

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 = 𝜋𝑟2ℎ

𝑉 = 𝜋(8.5)2(7)

𝑉 = 1588.9𝑚3

𝑉 = 𝜋𝑟2ℎ

𝑉 = 𝜋 4.5 2(8.5)

𝑉 = 540.7𝑐𝑚3

𝑉 = 125.7𝑚𝑚3

𝑉 = 18472.6𝑐𝑚3

𝑉 = 505.8𝜋 𝑉 = 172.1𝜋

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G.GMD.1 Give informal arguments for

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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Unit 4: Circles and Volume4.8 Volume of Pyramid and Spheres

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Volume of Sphere

Volume of Cones

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

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One-third

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 =8 × 8 × 10

3

𝑉 = 213.3𝑖𝑛3

𝑉 =400 × 36

3

𝑉 = 4800𝑦𝑑3

𝑉 = 15𝑐𝑚3 𝑉 = 48𝑐𝑚3

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

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height

slant

Pythagorean Theorem

𝑥2 + 52 = 132

𝑥 = 12

𝑉 =10 × 10 × 12

3𝑉 = 400𝑖𝑛3

𝑥2 + 6.52 = 12.82

𝑥 = 11

𝑉 =13 × 13 × 11

3𝑉 = 619.7𝑚𝑚3

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G.GMD.1 Give informal arguments for

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G.GMD.3 Use volume formulas to solve

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𝑥2 + 0.62 = 2.82

𝑥 = 2.73

𝑉 =1.2 × 1.2 × 2.73

3𝑉 = 1.3𝑐𝑚3

𝑥2 + 2.52 = 14.22

𝑥 = 13.9

𝑉 =5 × 5 × 13.9

3𝑉 = 116.5𝑚3

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Volume of Sphere

Volume of Cones

Volume of Rectangular and Triangular Pyramid

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

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Cones

One-third

circle

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Volume of Sphere

Volume of Cones

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 =𝜋(7)2(24)

3

𝑉 = 1232𝑓𝑡3

𝑉 = 392𝜋

𝑉 =𝜋(4)2(12)

3

𝑉 = 201𝑚3

𝑉 = 64𝜋

𝑉 = 98𝑖𝑛3 𝑉 = 5131.3𝑚3

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Volume of Sphere

Volume of Cones

Volume of Rectangular and Triangular Pyramid

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

Center

radius

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Volume of Sphere

Volume of Cones

Volume of Rectangular and Triangular Pyramid

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

𝑉 =4𝜋(5)3

3

𝑉 = 523.6𝑖𝑛3

𝑉 = 166.7𝜋

𝑉 =4𝜋(3)3

3

𝑉 = 113.1𝑐𝑚3

𝑉 = 36𝜋

𝑉 = 11494𝑖𝑛3 𝑉 = 7238.2𝑚3

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Volume of Sphere

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Volume of Rectangular and Triangular Pyramid

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

problems

Composite Solid

Composite Solid = Volume 1 + Volume 2

𝑉 =1

2∙4𝜋(6)3

3

𝑉 = 1922.7𝑖𝑛3

𝑉 = 144𝜋 + 468𝜋

+ 𝜋 6 2(13)

𝑉 = 612𝜋

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Volume of Sphere

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G.GMD.1 Give informal arguments for

geometric formulas

G.GMD.3 Use volume formulas to solve

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Composite Solid

Composite Solid = Volume 1 + Volume 2

𝑉 =(4)(4)(6)

3

𝑉 = 112𝑖𝑛3

𝑉 = 32 + 80

+ (4)(4)(5)

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Volume of Sphere

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G.GMD.1 Give informal arguments for

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Volume of Sphere

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Cross Sections3-Dimensional

Figures

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

polyhedron

edgeface vertex diagonal

Cross section

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Cross Sections3-Dimensional

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

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Cross Sections3-Dimensional

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

polygon

circles

congruent

similar

circle triangle square

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

rectangle triangle triangle

axis

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

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G.GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects and rotations

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Cavalieri’s Principle Density

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

𝑽 = 𝝅𝒓𝟐𝒉

𝑉 = 𝜋(5)2(12)

𝑉 = 942.5𝑚3

𝑉 = 300𝜋

Cylinder A:

𝑉 = 𝜋(5)2(12)

𝑉 = 942.5𝑚3

𝑉 = 300𝜋

Cylinder B:

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Cavalieri’s Principle Density

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

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Cavalieri’s Principle Density

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

mass

sizedensity

mass volume

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Cavalieri’s Principle Density

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

𝑫 =𝑴

𝑽

𝑫 =𝑴

𝑽

Rock # 1 because the density is greater than rock #2

=45𝑔

15𝑐𝑚3= 3𝑔/𝑐𝑚3

=60𝑔

30𝑐𝑚3= 2𝑔/𝑐𝑚3

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

𝑫 = 𝑴× 𝑽 =3.2𝑔

𝑚𝐿× 2𝑚𝐿 = 80𝑔

𝑫 =𝑴

𝑽=

500𝑔

240𝑐𝑚3= 21𝑔/𝑐𝑚3

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids

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Cavalieri’s Principle Density

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G.GMD.2 Give an informal argument using Cavalieri’s Principle for the formulas for the

volume of a sphere and other solids