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11.6 – Volume

Key Terms

  • Cavalieri’s principle – Prisms with different shapes as bases can also have the same volume, as long as their heights and cross-sectional areas are the same.
    • Cavalieri’s principle applies to all solids.
  • Constant – A value in a formula that does not change (unlike a variable).
  • Cross-Sectional Area – The area of a horizontal slice of a three-dimensional solid.
  • Triangular Prism – A three-dimensional solid consisting of two parallel congruent triangles and all the points between them.
  • Volume – A measure of the amount of space inside a three-dimensional figure.

Review

Remember
  • Inverses
    • Multiplying by \frac{1}{3} is the same as dividing by 3.
  • Dimensions
    • Perimeter is a one-dimensional measurement, usually l (length).
      • length is sometimes written as x instead of l.
    • Area and surface area are two-dimensional measurements, usually l and w (length and width).
      • width is sometimes written as y instead of w; so, l • w becomes x • y.
    • Volume is a three-dimensional measurement, usually l, w, and h (length, width, and height).
      • height is sometimes written as z instead of h; so, l • w • h becomes x • y • z.

GeoB 11.6 Review

GeoB 11.6 Volume Units

  • Constants
    • The constant in the area of a rectangle (or square) is 1
      • Both x and y (or l and w) can change.  They are variables.
    • The constant in the area of a circle is \pi
      • The radius can change.  It is a variable.
    • The constant in the area of a triangle is \frac{1}{2}
      • Both x and y (or l and w) can change.  They are variables.
      • Why is the base of a triangle half of a rectangle?

GeoB 11.6 Why Triangle Half a Rectangle

GeoB 11.6 Constants of Area

Notes

Volume – What is It?
  • You know that the area of a plane figure measures the amount of two-dimensional space inside it.
  • The volume of a solid measures the amount of three-dimensional space inside it.
  • Measured
    • Volume = length • width • height
  • Formula
    • The volume of a solid is the area of one base (B) times its height.
    • Volume = Bh

GeoB 11.6 Volume Formula

GeoB 11.6 Volume Ex

 

Volume of a Prism
  • Formula: V=Bh
    1. By definition, the base must be one of two parallel faces of the prism. It does not matter if a prism is standing on its base or lying on its side. The base size and shape remain the same (congruent and parallel).
    2. Every prism is also defined by its third dimension, called height.
      1. The height of a prism is the length of its altitude, which is always perpendicular to the base(s).
      2. Right and oblique prisms have the same volume IF, and only if, their bases and heights are the same.
GeoB 11.6 Vol Prism GeoB 11.6 Prism Volume Ex

 

Volume of a Cylinder
  • Formula: V=Bh
    • Its base is a Circle, so…
    • V=(\pi r^{2})h
    • V=\pi r^{2}h
  • How to Find the Volume of Any Cylinder
    • Step 1: Write the volume formula.
    • Step 2: Find the area of one base.
    • Step 3: Find the height.
    • Step 4: Multiply the results from steps 2 and 3
  • Right and Oblique cylinders have the same volume IF, and only if, their bases and heights are the same.
GeoB 11.6 Vol Cylinder GeoB 11.6 Vol Cylinder Ex

 

Cubes
GeoB 11.6 CubicsGeoB 11.6 Cubic Volume ExGeoB 11.6 Cubic Ex Volume
  • Splitting a Cube in half
    • Two triangular prisms are formed.
    • They are congruent.
GeoB 11.6 Cube into 2 Prisms
  • Splitting a Cube into thirds
    • Three pyramids are formed.
    • They are congruent and oblique.
    • They are called Yangma.
      • Yangma is an ancient Chinese name for a pyramid with a rectangular base and a vertex perpendicular to one of the vertices of the base.
GeoB 11.6 Cube into 3 Pyramids


GeoB 11.6 Cube into 3 Pyramids2

 

Volume of a Pyramid
  • One third times the volume formula for a prism (Bh).
  • This formula works for any pyramid.
  • Written: v=\frac{1}{3}Bh
  • How to Find Volume of Any Pyramid
    • Step 1: Write the volume formula.
    • Step 2: Find the area of its base.
    • Step 3: Multiply that base area by \frac{1}{3}.
    • Step 4: Multiply the product from step 3 by the height of the pyramid.
  • Finding the volume of a pyramid is just like finding \frac{1}{3} of the volume of the prism that has the same base and height.
GeoB 11.6 Vol Pyramid GeoB 11.6 Pyramid Volume Ex

 

Volume of a Cone
  • One third times the area of its base (B) times the length of its altitude (h): v=\frac{1}{3}Bh
  • How to Find Volume of Any Cone
    • Step 1: Write the volume formula.
    • Step 2: Find the area of its base.
    • Step 3: Multiply that base area by \frac{1}{3}.
    • Step 4: Multiply the product from step 3 by the height of the cone.
  • Finding the volume of a cone is just like finding \frac{1}{3} of the volume of the cylinder that has the same base and height.
GeoB 11.6 Vol Cone GeoB 11.6 Cone Volume Ex

 

Cavalieri’s Principle
  • Cavalieri’s principle, if figures have equal cross-sectional areas and heights, then they also have equal volumes.
    • Prisms with differently shaped bases will have the same volume as long as their heights and cross-sectional areas are the same.
    • Cavalieri’s principle applies to all solids.

 

  • Ask yourself…
    • Do the solids have the same height?
    • Do their cross-sections have the same area?
    • If the answer to BOTH of these questions is YES, then the two solids have equal volume!

 

Summary
  • Cylinder & Cones have round (disc) bases
    • B=\pi r^{2}
    • Prisms & Pyramids have polygon bases
    • Rectangle: B=l\bullet w
    • Triangle: B=\frac{1}{2} l\bullet w

GeoB 11.6 Solids Vol Chart GeoB 11.6 Cone Cylinder Vol

Examples

  • Ex 1. What is the volume of the prism below?

GeoB 11.6 Q1-01

Answer: 168\; units^{3}

  • Ex 2. What is the volume of the prism below?

GeoB 11.6 Q1-02

Answer: 324\; units^{3}

  • Ex 3. What is the volume of the cylinder below?

GeoB 11.6 Q1-03

Answer: 2160\; \pi units^{3}

  • Ex 4. What is the volume of the cylinder below?

GeoB 11.6 Q1-05

Answer: 288\; \pi units^{3}

  • Ex 5. What is the volume of the cube below?

GeoB 11.6 Q1-07

Answer: 4x^{2}

  • Ex 6. The volume of a cylinder with a base of radius r is the area of the base times the length of its height (h). Which of the following is the formula for the volume of a cylinder?
    • Answer: v=\pi r^{2}h
  • Ex 7. The volume of a cylinder is the base (B) times the length of its height (h). Which of the following is the formula for the volume of a cylinder?
    • Answer: V = Bh
  • Ex 8. It is true that Cavalieri’s principle states that two solids with equal heights and cross-sectional areas at every level have equal volumes.
  • Ex 9. What is the volume of the cone below?

GeoB 11.6 Q2-01Answer: 144\; \pi units^{3}

  • Ex 10. What is the volume of the regular pyramid below?

GeoB 11.6 Q2-03Answer: 3408\; units^{3}

  • Ex 11. Which solid has a greater volume?

GeoB 11.6 Q2-05Answer: They are equal

  • Ex 12. It is true that a cone has one-third times the volume of a cylinder with the same base and altitude.
  • Ex 13. It is true that a prism has three times the volume of a pyramid with the same base and altitude.
  • Ex 14. The volume formula for a right pyramid is V=\frac{1}{3}Bh. What does B represent?
    • Answer: B is the “Area of base”
  • Ex 15. Find the volume of the pyramid below.

GeoB 11.6 Q2-08Answer: 400\; units^{3}

  • Ex 16. Find the volume of the cone shown below.

GeoB 11.6 Q2-09Answer: 768\; \pi units^{3}

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