Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Warm-up: Dimensions
Dimension
A dimension is roughly defined as the ____________________________
needed to describe any point on an object.
One
Dimensional
A one dimensional object requires only ________ coordinate to
describe any point within it. They have lengths, but not _______
___________________. A line is a one-dimensional object. If we
have a line of length 5, we can specify any point on the line by
stating how far from the beginning or end of the line it is.
length
[units]
Two
Dimensional
surface area
[units2]
Three
Dimensional
volume
[units3]
Two dimensional objects require ______ coordinates to describe
a point. A _______________ is a two-dimensional object. A ______
________________ are needed to describe a point on the object.
Two dimensional objects have lengths and areas, but not
_______________.
Three coordinates are needed to describe a point in a __________
_____________ object. A cube is an example of a three dimensional
object. A _______________________________ are needed to describe a
point on the object. Three-dimensional objects have lengths,
areas and volumes.
Summary
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Think
Can you explain why a dot has zero dimensions?
Think
How many dimensions make up our physical world?
Background Information: 2D Shapes and Area
Area
Review
Area is the size of a ____________. In other words, it is the amount
of space in a 2-dimensional object such as a rectangle or circle.
Name: _________________________
Area Formula: _______________
Name: _________________________
Area Formula: _______________
Name: _________________________
Area Formula: _______________
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Surface Area of 3D Shapes
Math 10 Plus Notes
Practice
Date:
What is the area of the triangle below?
4cm
10 cm
Diameter
A diameter is the ______________ straight line between two points
on a circle (it passes through the ______________ of the circle).
Radius
The radius is _____________ of the diameter. It is the length from
the center of a circle to ____________________________________.
Practice
What is the area of a circle that has a radius of 6 cm?
Surface
Area
Surface area is the ____________________ of the surfaces of a threedimensional shape.
Think
1) How many squares make up the surface of a cube?
2) If the area of a square is A = l x l or A = l2, what is the total
area of a cube?
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Surface Area of 3D Shapes
Date:
Math 10 Plus Notes
Background Information: Prism and Cylinder Surface Area
Review surface area of prisms, cylinders and cubes.
(Put after next page)
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
3D Shape Names
Cone
Cylinder
Cube
Sphere
Prism
[Rectangular]
Pyramid [Rectangular Base]
Prism
[Triangular]
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Surface Area of Pyramids
Activity
Using the 3D figures provided, complete the table below.
Shape
Square
Pyramid
Rectangular
Pyramid
Triangular
Pyramid
Sketch
Net
Individual
Sides
Surface
Area
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Pyramid
A pyramid is a 3D object that has triangular sides or faces and
a polygon base (ie. square, rectangle, triangle, etc.).
Apex
The apex is the point at which ____________________________________.
Height
The height of a pyramid is the distance from _______________ to
the _________________.
Slant
Height
Slant height of a pyramid refers to the height of ____________
_____________________________________.
Example
Find the surface area of the square-based pyramid below.
1) Draw each side in the 3D shape:
4x
+ 1x
2) Calculate the area of each side:
3) Add the areas of each side to find the total surface area:
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Surface Area of 3D Shapes
Math 10 Plus Notes
Practice
Date:
Find the surface area of the triangular pyramid below.
1) Draw the sides that made up the 3D shape:
2) Calculate the surface area of each side:
3) Find the total surface area:
Practice
A square based pyramid has base length 8 cm and a slant
height of 6 cm.
1) Sketch the pyramid.
2) Find the surface area of the pyramid.
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Lateral
Area
Lateral area is the surface area of the triangular faces of a
pyramid. In other words, it is the total surface area of _________
_________________________________________. Lateral comes from the
Latin word “latere “ which means side.
Example
Find the lateral area of the rectangular pyramid below:
Practice
A rectangular pyramid has a base length of 5 in and a
corresponding slant height of 9 in., a base width of 7 in. and a
corresponding slant height of 8 in. Sketch the 3D figure and
find its lateral area.
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Finding Slant Height Given Height
In order to calculate the surface area of a pyramid, we need to
know the ____________ height of each triangular face. Sometimes,
we are only given the height of the pyramid (the distance from
the base to the apex) and we need to use this to find the slant
height before we can calculate surface area.
Recall
The Pythagorean Theorem
a
c
b
Try This
a 2  b2  c 2
a 2  b2  c
Using the diagram below and the Pythagorean theorem,
discuss in your groups how you might calculate the slant
height given the pyramid height.
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Surface Area of 3D Shapes
Math 10 Plus Notes
Example
Date:
Find the slant height(s) of the objects below
a)
b)
Practice
Find the surface area of a triangular based pyramid with base
length 8 yards (all three sides) and a slant height of 10 yards.
Practice
Find the surface area of a square based pyramid with base
length 3 inches and pyramid height of 5 inches.
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Practice
Find the surface area of the rectangular pyramid below.
Practice
Find the surface area of the rectangular pyramid below.
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Surface Area of Cones
Circumference
The distance around the ________________ of a circle.
Cone
A cone is a 3-dimensional object that has a _____________ base
and a curved _______________ surface. The height (h) of a cone is
the perpendicular distance from the base to the apex (the top
point) of the cone. The slant height of a cone is the _____________
distance on the curved surface from a point on the
circumference of the base to the apex
Derivation
If you were to take the top (sides) of a cone and lay it flat, you
would end up with the shaded area below. [Cut out a blank
piece of paper to prove this to yourself].
Notice that the surface of the side of the cone forms part of a
circle whose radius is the slant height of the cone.
We know that the area of a circle is r 2 . The radius of the
circle above is the slant height, represented by s, of the cone.
We can write the equation for the area of the circle above as
s 2 . If we can find how much of the circle is made up by the
shaded region, we will know the lateral area of the cone.
Recall
The circumference of a circle is equal to ____________ where r is
the radius of the circle.
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Surface Area of 3D Shapes
Date:
Math 10 Plus Notes
Derivation
The section of the circle above wraps around the circle that
Continued
forms the base of the cone. This means that the length from A
to B of the larger circle is equal to 2r .
The ratio of the area of the shaded section to the area of the
larger circle is the same as the ratio of the circumference of
the shaded section to the circumference of the larger circle.
[Note: We will learn how to show this in Math 12, but for now
we will just use this information to help us derive the area of
the shaded section.]
Let x represent the unknown area of the shaded section of the
larger circle:
x
AB

area of the large circle circumference of large circle
We know the area of the large circle is 2s 2 and the
circumference of the large circle is 2s . The length of AB is
equal to the circumference of the smaller circle which
forms the base of the cone, so 2r :
x
2r

2s 2
2s
Re-arranging to solve for x:
x( 2 s )  r ( 2 s 2 )
x  rs
Therefore, the lateral surface area of a cone is rs where ‘r’ is
the radius of the circular base and ‘s’ is the slant height of the
cone.
Think
What is the formula for the total surface area of a cone?
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Example
What is the surface area of a cone that has a diameter of 4 feet
and a height of 7 feet?
Note
The slant height of a cone is calculated similarly to slant height
of a pyramid. However, it is slightly easier because the base of
the right-angled triangle is always the radius of the cone.
Practice
A cone has a radius of 10 millimeters and a height of 8
millimeters. Sketch the cone and find its surface area.
Practice
A cone has a slant height of 5 meters and a radius which is half
of the height. Sketch the cone and find the surface area.
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Surface Area of 3D Shapes
Date:
Math 10 Plus Notes
Finding Unknown Dimensions Given Surface Area
Recall
Example
Practice
We can solve for an unknown value by isolating it (ie. ________
______________________). In order to “un-do” an operation that is in
the way of isolating the unknown, we simply perform the
opposite operation:
y+4 = 6 : There is a 4 added to y, so we must subtract four to
isolate it. If we subtract four from the left, we must
also subtract four from the right.
y + 4 – 4 = 6 – 4 ; y =2
Solve 3x - 2 = 10
The same logic applies to SA formulas for 3D shapes:
Example
A square based pyramid has a surface area of 256.7 m2 and a
base length of 7.2 m. What is the slant height of the pyramid?
Practice
Find the length of a cube with SA = 150 units2.
Practice
What is the slant height of a cone that has a radius of 5 inches
and a surface area of 200 inches2?
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Practice
What is the height of a cylinder that has surface area of 314 m2
and radius of 10 m?
Practice
A pyramid has an isosceles (all sides equal) triangle for its
base. The length of the base is 7 cm and the total surface area
is 156.9 cm2. What is the slant height of the pyramid?
Hint: You first need to find the height of the base pyramid.
Surface Area Practice Problems
You may want to discuss these problems within your group.
Practice
Salima made ten conical (cone-shaped) party hats out of
cardboard. How much cardboard was used if each hat has a
radius of 14 cm and a slant height of 25 cm?
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Surface Area of 3D Shapes
Math 10 Plus Notes
Date:
Practice
A tent has a rectangular base of 3 yards by 5 yards and a
height of 2.5 yards. The tent needs a canvas rain cover for the
sides. How much canvas will be required to make the rain
cover? Draw a diagram.
Practice
A square-based pyramid has a surface area of 154 cm2. A cone
has a base radius of 3 cm. The cone and pyramid have equal
surface areas. What is the slant height of the cone?
What is the height of the cone?
Think
If you double the height of a right prism, do you double its
surface area?
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