A. Measurement Systems
Math 10: Foundations and Pre-Calculus
Key Terms:
 Find and the definitions of
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each of the following
terms:
Imperial units
SI unit of measure
Apex
Unit Analysis
Right Pyramid
Right Cone
Slant Height
 Lateral Area
 Sphere
1. Remembering Imperial Measurement
 In 1976, Canada adopted the SI units to measure length.
 However, many trades and industries continue to use
imperial units
 The units for linear measurment in the imperial system are
the:
 inch, foot, yard and mile.
 Look at the chart on p. 6 of your text. It should look
familiar.
 Measure your pencil in imperial units and record its length.
 Next switch pencils with someone and see if they get the
same measurements? Were you right?
 Construct Understanding
p. 5
 Work with a partner or a group of three.
 How do we convert from one imperial measurement to
another imperial measurement?
 We use a Conversion Factor!
Example
1)
Examples
Example
Practice
 Ex. 1.1 (p.11) #1-18
#1-4, 7-20
2. Relating SI and Imperial Units
 Construct Understanding
p. 16
 Work with a partner or on your own.
 We will use the same conversion chart that we used in Math
A and W (it is on you formula sheet)
 How do we convert from imperial to SI measurements?
 We use a conversion factor!
Examples
Examples
Practice
 Ex. 1.3 (p.22) #4-16
#4, 7-18
3. Surface Area of Right Pyramids
and Right Cones
 A right pyramid is a 3D object that has triangular faces and a
base that is a polygon.
 The shape of the base determines the name of the pyramid.
 The triangular faces meet at a point called the apex
 The height of the pyramid is the perpendicular distance from
the apex to the center of the base.
 When the base of a right pyramid is a regular polygon, the
triangular faces are congruent.
 Then the slant height of the pyramid is the height of a
triangular face.
 The surface area of a right pyramid is the sum of the area of
the triangular faces and the base.
 This right square pyramid has a slant height of 12 cm and a
base length of 5cm. Find its area.
Examples
 We can determine a formula for any right pyramid with a
regular polygon as its base.
 A right cone is a 3D object with a circular base and curvered
surface.
 The height of a cone is perpendicular from apex to the base.
 The slant height of the cone is the shortest distance on the
curved side between the apex and a point on the
circumference of the base.
 Remember the formula for the surface area of a cone?
Examples
Practice
 Ex. 1.4 (p.34) #1-3, 5-19
#1-3, 8-21
4. Volume of Right Pyramids and
Right Cones
 Right Pyramids and Right Cones are related to right prisms
and right cylinders.
 Do you remember how to find the volume of a right prism
and right cylinder?
V = (area of base) x height
 Construct Understanding
 We will do this one as a class.
p.36
 Formula for the Volume of a Right Pyramid:
 How do you find the volume of a right Prism?
 So using the info we just found in the CU what is the formula
for the volume of a right pyramid (same base and height)?
 For example, what would be the formula to find the volume
of a right rectangular pyramid?
Examples
 Formula for the Volume of a Right Cone:
 How do you find the volume of a right Cylinder?
 So using the info we just found in the CU what is the formula
for the volume of a right cone (same base and height)?
Examples
Practice
 Ex. 1.5 (p.41) #1-20
#1-3, 8-22
5. Surface Area and Volume of a
Sphere
 The surface area of a sphere is related to the curved surface
area of a cylinder that encloses it
 The cylinder has the same diameter as the sphere, and a
height equal to its diameter
 How did we find the surface area of cylinder, SAc, with a
base radius r and height h. (rectangular cylinder)
 What if the cylinder has a height of 2r (diameter)?
 Therefore this is also the formula for the SA of a sphere with
a radius r.
 Formula for SA of a Sphere:
Examples
 We can use the formula for the SA of a sphere to develop the
formula for the Volume of a sphere
 Look at the picture below and visualize a sphere covered in
small congruent squares, and each square is joined to the
center of the sphere by lines to form a pyramid.
 Therefore if we add up the volume of all the square pyramids
we will have the volume of a sphere
 Formula for the Volume of Sphere:
Example
 When a sphere is cut in half, two hemispheres are formed.
Example
Practice
 Ex. 1.6 (p.50) #1-20
#1-2, 6-23
6. Solving Problems
 Find the area of the bin.
 A composite object comprises two or more distinct object.
 To determine the volume of a composite object, identify the
distinct objects, calculate the volume of each object, then add
the volumes together.
 To calculate the SA of a composite object, the first step is to
determine the faces that comprise the SA. Then calculate the
sum of the area of these faces.
Examples
Examples
Practice
 Ex. 1.7 (p. 59) #1-10
#3-13