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SURFACE AREA
In this unit you will create and visualize three-dimensional shapes and figures. You will be given various views of a figure and determine its 3-D model. You will examine the properties of various solids including common 3-D shapes and the Platonic solids. You will calculate various problems that are related to determining the surface area of a variety of 3-D shapes.
Visualize Models Isometric Dot Paper
Three-Dimensional Figures Platonic Solids
Surface Area and Nets
Surface Area of a Prism Surface Area of a Cylinder
Surface Area of a Pyramid Surface Area of a Cone
Surface Area of a Sphere
Surface Area Formulas
Visualizing Models
Models may be constructed when given a diagram of different views of a three-dimensional figure.
We will examine a three dimensional model of cubes drawn from two-dimensional views.
Given below are five two-dimensional views of the same three-dimensional model built from cubes. The heavy segments between the squares represent columns of different heights.
Top View
View of Left Side
As the model is viewed from the top, it shows the tops of four columns. The heavy line segments represent that the columns are different heights.
- tops of four columns
- columns are different heights
The left view reveals that the figure is two blocks wide with one column being 3 blocks high and the other column being 1 block high. All surfaces are flush to the side (no heavy lines).
- side view from left
- two columns
- columns are different heights
Back View
Putting all the views together, the 3-D figure shown below is the figure described in the various views above.
The back view reveals that the figure is three blocks high and two blocks wide with all surfaces flush to the side (no heavy lines).
- back view
- two columns, three rows
- all the same height and width
corner view (perspective view) – A corner view is the view from a corner. Isometric dot paper may be used to draw the figure from a corner view.
The figure shown above is drawn on isometric dot paper from a different perspective.
Three-Dimensional Figures
Polyhedra
polyhedron – A polyhedron is a solid with all flat faces that enclose a single region of space.
Polyhedron
polyhedra – Polyhedra is the plural of polyhedron.
Prisms
prism – A prism is a polyhedron with two congruent faces that are polygons contained in parallel planes.
Prism
bases of a prism – The bases of a prism are its congruent faces that are polygons contained in parallel planes.
lateral faces of a prism – The lateral faces of a prism are the faces shaped like parallelograms that connect the edges of the bases of the prism.
lateral edges – Lateral edges are the line segments formed at the intersection of lateral faces. Lateral edges are parallel line segments.
regular prism – A regular prism is a prism whose bases are regular polygons.
Regular Prism
Lateral Face
Bases
Lateral Edge
cone – A cone is a solid that has one circular base, a vertex point that does not lie in the same plane as the base, and a curved lateral surface area that is made up of all points that lie on segments connecting the vertex and the edge of the base.
sphere – A sphere is a set of points in space that are equidistant from a given point.
slice – A slice is the shape formed when a plane intersects a solid.
Example 1 : What is the shape of the slice that is formed when a plane intersects the body of a cylinder laterally as shown below?
The shape of the slice is a rectangle. (The slide is outlined in red.)
cross section – A cross section of a solid is a slice made by a plane that is parallel to the base or bases of the solid.
Example 2 : What is the shape of the cross-section that is formed by the intersection of a cylinder and a plane that is parallel to the cylinder’s bases?
The shape of the cross-section is a circle.
Surface Area and Nets
surface area – Surface area is the sum of all the areas of a solid’s outer surfaces.
net - A net is a two-dimensional representation of a solid. The surface area of a solid is equal to the area of its net.
Example: Find the surface area of rectangular prism that measures 16 inches by 10 inches by 14 inches.
Method 1 :
Use the formula A = lw to find the areas of the surfaces.
2
Front and Back: (16 14) 2 448 Top and Bottom: (16 10) 2 320 Two Sides: (10 14) 2 280
Add to find the total surface area: 448 320 280 1048
SA 1048 in
× × =
× × =
× × =
The surface area of a 16 by 10 by 14 inch rectangular prism is 1048 square inches.
16 in
14 in
Method 2 :
Draw a net for the rectangular prism and label the dimensions of each face. Find the area of each face, and then add to find the total surface area.
The surface area of the rectangular prism is 1048 in.^2
bottom
16 in
10 in
14 in
side side 14 in front
back (^) 14 in
top 16 in 14 in
Side: 14 10 140 Bottom: 16 10 160 Side: 14 10 140 Top: 16 10 160 Front: 16 14 224 Back: 16 14 224
Total: 140 160 140 160 224 224 1048
× =
× =
× =
× =
× =
× =
A right rectangular prism and its net are drawn below to explore the formulas for the lateral and the surface area of prisms. We will use a rectangular prism to develop the formula.
The length of each edge of a base is represented by a , b , c , and d , respectively. The height of the prism is represented by h and the base area is represented by B. Perimeter is represented by P.
Lateral Area
lateral area – The lateral area of a prism is the sum of the areas of the lateral faces.
h
B
B
h
P
a b c d
a (^) b c d
h
P
a b c d
h
a (^) b
Sum of the Areas of Each Lateral Face ( ) Distributive Property Definition of Perimeter ( ) Substitution Commutative Property
L ah bh ch dh L h a b c d P a b c d L h P L Ph
*It can be shown that this formula applies to all right prisms such as the triangular prism, pentagonal prism, and so on.
Surface Area
surface area – The surface area of a prism is the sum of the areas of its bases and the lateral area.
The bases of a right prism are congruent; thus, the total surface area is found by adding the lateral area and the area of the two bases.
h
B
B
h
P
a b c d
a (^) b c
Lateral Area of a Right Prism The lateral area L of a right prism is the product of the perimeter P of its base and the height h of the prism.
L = Ph
Surface area = 2 × Base Area + Lateral Area
T = 2 B (^) + Ph
Find the perimeter of the base (right triangle).
6 8 10 Definition of Perimeter 24 Simplify
P
P
The perimeter of the base is 24 inches.
Step 2 : Find the lateral area (area of all the lateral faces).
Formula for determining the lateral area of right prisms. 24(15) Substitution ( 24, 15) 360 Simplify
L Ph L P h L
The lateral area of the right triangular prism is 360 square inches.
Step 3 : Find the area of the triangular bases.
(^1) Area formula for a Triangle 2 (^1) (8)(6) Substitution ( 8, 6) 2 24 Simplify
A bh
A b h A
The area of one triangle base is 24 square inches.
Step 4 : Find the surface area (total area of all faces)
2 Formula for Total Surface Area of Right Prisms 2(24) 360 Substitution 408
T B Ph T T
The surface area is 408 square inches.
An alternate way of find the surface area is as follows:
T = 2B + L 15(6) 15(8) 15(10) Sum of the Areas of Each Lateral Face 15(6 8 10) Distributive Property 15(24) Simplify 360 Simplify
2 Formula for Surface Area of Right Prisms 2(24) 360 Substitution ( 24, 360
L
L
L
L
T B L
T B L
T = 408 Simplify
