Print this Page
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 crosssectional areas are the same.
 Cavalieri’s principle applies to all solids.
 Constant – A value in a formula that does not change (unlike a variable).
 CrossSectional Area – The area of a horizontal slice of a threedimensional solid.
 Triangular Prism – A threedimensional solid consisting of two parallel congruent triangles and all the points between them.
 Volume – A measure of the amount of space inside a threedimensional figure.
Review
Notes
Volume – What is It? 
 You know that the area of a plane figure measures the amount of twodimensional space inside it.
 The volume of a solid measures the amount of threedimensional 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


Volume of a Prism 
 Formula:
 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).
 Every prism is also defined by its third dimension, called height.
 The height of a prism is the length of its altitude, which is always perpendicular to the base(s).
 Right and oblique prisms have the same volume IF, and only if, their bases and heights are the same.



Volume of a Cylinder 
 Formula:
 Its base is a Circle, so…

 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.



Volume of a Pyramid 
 One third times the volume formula for a prism (Bh).
 This formula works for any pyramid.
 Written:

 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 .
 Step 4: Multiply the product from step 3 by the height of the pyramid.
 Finding the volume of a pyramid is just like finding of the volume of the prism that has the same base and height.



Volume of a Cone 
 One third times the area of its base (B) times the length of its altitude (h):

 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 .
 Step 4: Multiply the product from step 3 by the height of the cone.
 Finding the volume of a cone is just like finding of the volume of the cylinder that has the same base and height.



Cavalieri’s Principle 
 Cavalieri’s principle, if figures have equal crosssectional 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 crosssectional areas are the same.
 Cavalieri’s principle applies to all solids.
 Ask yourself…
 Do the solids have the same height?
 Do their crosssections have the same area?
 If the answer to BOTH of these questions is YES, then the two solids have equal volume!

Examples
Important!
Practice (Apex Study 11.6)
 Try practice problems on Pg 15, 25
 Mandatory: write and answer problems on Pg 14, 26
 2 Quizzes
Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=4849