8.1: Apply the Pythagorean Theorem
Homework: pg.423 #1-10
The Pythagorean Theorem is named after the Greek philosopher and mathematician Pythagoras. The Pythagorean
Theorem applies to all right triangles.
Right triangle – a triangle with one 90° angle.
Hypotenuse – the longest side of a right triangle. This side is opposite the 90° angle.
Pythagorean Relationship:
The following diagram illustrates the Pythagorean Relationship quite clearly:
The area of the large square – square C is equal to the sum of the areas of squares A
and squares B.
Area C  Area A  Area B
Algebraically we can write:
Suppose the length of square A is a units, the length of square B is b units and the
length of square C is c units. Without much effort, we should be able to conclude that
the Area C  c 2 , Area B  b 2 , Area A  a 2 .
Rewriting the above equation for the Area C we get: c 2  a 2  b 2
Be careful not to get too caught up with the formula c 2  a 2  b 2 - because it will change on you! For example,
Pythagorean relationship between the sides a, b, and c will be different in the following triangles!
a
b
b
a
c
a2  b2  c2
c
b2  a2  c2
c
a
b
c2  a2  b2
For this reason, we state the Pythagorean Theorem as follows:
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of
the lengths of the two shorter sides.
Ex. 1:
Calculate the length of the missing side in each triangle. Round your answers to the nearest tenth of a unit.
a)
b)
Ex. 2:
Determine the area of the following right triangle. Round your answers to the nearest tenth of a square unit.
Ex. 3:
Calculate the length of the line segment joining the points M(-2,3) and E(5,6). Use the following grid to help.
8.2: Perimeter & Area of Composite Figures
Homework: pg.432 #1,2,4,13,18
A composite figure is made up of one or more simple shapes. To determine the total area of a composite figure, add
and/or subtract areas. To determine the perimeter of a composite figure, add the distances around the outside of
the figure.
Before doing so, we should review the some basic formulas for common shapes:
Shape
Area Formula
Shape
1
A  bh
2
A  lw
(rectangle)
Area Formula
(triangle)
A  bh
A  r 2
(parallelogram)
(circle)
A
1
a  bh
2
(trapezoid)
The perimeter of a shape is the distance around the outside of the figure. A circle’s perimeter is called the
circumference.
Shape
Perimeter
Shape
P  2l  w
(rectangle)
C  2r
(circle)
Ex. 1: Solve for any unknown variables, and then calculate the perimeter and area of each figure:
a)
b)
Ex. 2: Calculate the perimeter and area of each of the following figures:
a)
b)
Perimeter
8.3: Surface Area and Volume of Prisms and Pyramids
Homework: pg.441 #1-4, 6, 9, 11, 16
To package the products economically, manufactures need to know the amount of material required for the package.
To do this, they need to know the surface area of the package. The amount of space a package occupies is the volume.
Definitions:
Pyramid – a polyhedron whose base is a polygon and whose faces are triangles that meet at a
common vertex.
Lateral faces – the faces of a prism or pyramid that are not bases.
Investigation #1: Volume of a Pyramid
1. Fill a pyramid with liquid, and then dump it into a rectangular prism of equal height and base area.
2. Repeat this process until the rectangular prism is full.
3. What conclusion can you draw about the relationship between the volume of a pyramid and the volume of a prism
with the same base and height?
Investigation #2: Modeling Surface Area of a Pyramid
1. A square-based pyramid and its net are shown:
a) What is the shape of the base? Write a formula for its area.
b) What is the shape of each lateral face? Write a formula for the area of one lateral face.
c) Write an expression for the surface area of the pyramid. Simplify the expression to give a formula for the surface
area of a pyramid.
2. How would the results in #1 change if you were developing a formula for the surface area of a hexagon-based
pyramid? An octagon-based pyramid? Describe how to find the surface area of any pyramid.
Key Concepts:
Surface area is a measure of how much material is required to cover or construct a three-dimensional
object. Surface area is expressed in square units.
The surface area of a prism or pyramid is the sum of the areas of the faces.
Volume is a measure of how much space a three-dimensional object occupies. Capacity is the maximum
volume a container can hold. Volume and capacity are measured in cubic units.
The litre (L) is a measure of capacity or volume often used for liquids. 1 L  1000 cm3 or 1mL  1cm3
For a prism, Volume  area of base  height . For a pyramid, Volume 
Ex. 1: Draw a net for each object, and then determine the surface area of each object:
a)
b)
1
area of base  height
3
Ex. 2: Determine the volume of each object.
a)
b)
Ex. 3:
A rectangular prism has length 9 cm, width 10 cm, and a capacity of 600 mL. What is the height of the prism?
Ex. 4:
A pyramid with height 1.5m and base dimensions of 3m by 3m is placed on top of a rectangular prism with base
dimensions 3m by 3m and height 2m. Find the volume and surface area of the resulting three dimensional object.
8.4/8.5: Surface Area & Volume of A Cone
Homework: pg.447 #1-3,5,10 & pg.454 #1-4,7,10
A cone is a three dimensional object with a circular base and a curved lateral surface that
extends from the base of a point called the vertex.
The height of a cone is the perpendicular distance from the vertex to the base. The slant height
of a cone is the distance from the vertex to a point on the edge of the base.
The lateral area is formed by folding a sector of a circle. The radius of the circle used becomes
the slant height, s, of the cone formed.
The area of this curved surface is  rs , where r is the radius of the base of the cone.
The area of the circular base is  r 2
The surface area of a cone can be found by adding the lateral area of the cone and the base area. Given a cone with
radius r, and slant height s, the surface area of a cone can be found using the formula: SAcone   rs   r 2 .
Ex. 1: Calculate the surface area of a cone with slant height 7 cm, and radius 5 cm.
Ex. 2:
One cone has base radius 4cm and height 6cm. Another cone has a base radius 6cm and height 4cm.
a) Do the cones have the same slant height?
b) Do the cones have the same surface area? If not, predict which cone has the greater surface area. Explain your
reasoning.
c) Determine the surface area of each cone to check you prediction. Were you correct?
Investigation #1: Volume of a Cone
Using a cone with same base radius and height as a cylinder, compare the relative volumes of the two containers by
filling the cone repeatedly and dumping it into the cylinder.
Conclusion:
Ex. 3:
Find the volume of each of the following cones:
a)
b)
Ex. 4:
A cone has volume 263.9 cm3 and base radius 6 cm. What is the height of the cone?
Ex. 5:
A cone has volume 424.1 cm3 and height 5 cm. What is the base radius of the cone?
8.6: Surface Area of a Sphere & 8.7: Volume of a Sphere
Homework: pg.459 #1,4,6 & pg.465 #1,3,4,7,8,9,15
The surface area of a sphere with radius r is given by the formula SAsphere  4 r 2 .
Ex. 1: Find the surface area of the sphere with diameter 32cm.
Ex. 2: Determine the radius of a sphere with surface area 707cm2
Investigation:
Given a cylinder of radius r and height h, we can find its volume using the
formula V   r 2 h (the area of the base multiplied by the height).
Using a sphere with the same radius as a cylinder, and whose diameter is
equal to the height of the cylinder, can we find a relationship between these
two volumes?
Fill the sphere with sand or water, and dump it into the cylinder. What fraction of the cylinder is filled with sand/water?
Conclusion:
Ex. 3: A ball has an outside circumference of 17.97 cm. Calculate the volume, to the nearest cubic centimetre.
Ex. 4: A sphere just fits inside a cube with edges 15 cm long.
a) Calculate the volume of the sphere.
b) Calculate the volume of the cube.
c) How much empty space is there in the cube?
Ex. 5: Calculate the radius of a sphere whose volume is equal to 80cm3.
Ex. 6: A cylindrical corn silo with spherical top measures 10m high with radius 2.5 m.
a) Calculate the surface area of the corn silo
b) Calculate the volume of the corn silo.
c) If 1L of paint covers 10m2, calculate the amount of paint needed.