A. Measurement Systems
Math 10: Foundations and Pre-Calculus
 FP10.3
 Demonstrate understanding of SI and imperial units of
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measurement including:
linear measurement
surface area of spheres, and right cones, cylinders, prisms,
and pyramids
volume of spheres, and right cones, cylinders, prisms, and
pyramids
relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of
measurement including:
 linear measurement
 relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of
measurement including:
 linear measurement
 relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of
measurement including:
 surface area of spheres, and right cones, cylinders, prisms,
and pyramids
 relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of
measurement including:
 volume of spheres, and right cones, cylinders, prisms, and
pyramids
 relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of
measurement including:
 surface area of spheres, and right cones, cylinders, prisms,
and pyramids
 volume of spheres, and right cones, cylinders, prisms, and
pyramids
 relationships between and within measurement systems.
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
 FP10.3
 Demonstrate understanding of SI and imperial units of




measurement including:
linear measurement
surface area of spheres, and right cones, cylinders, prisms,
and pyramids
volume of spheres, and right cones, cylinders, prisms, and
pyramids
relationships between and within measurement systems.
 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