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SURFACE AREA
In this unit, you will examine the surface area of several solids. First, you will examine the nets of solids (two-dimensional representations of solids) and calculate the surface area by examining the areas of each face of the solid. You will then develop and apply formulas for the surface area of cubes, prisms, cylinders, pyramids, cones, and spheres. Finally, you will examine how to draw and interpret three-dimensional figures sketched on a two-dimensional surface.
Surface Area and Nets
Surface Area of Prisms and Cylinders
Surface Area of Pyramids and Cones Surface Area of Spheres
Drawing Three-Dimensional Figures Isometric Dot Paper
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
10 in
Surface Area of Prisms and Cylinders
Surface Area of a Rectangular Prism
The surface area of a solid is the sum of the areas of all surfaces of a figure.
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 1 : Find the surface area of a 16 inch × 10 inch × 14 inch(rectangular prism).
Method 1 : Use the formula A = l x w to find the areas of the surfaces.
2 faces 2 2 faces 2 2 faces
Front and Back ( ): 16 14 448 Top and Bottom ( ): 16 10 320 Sides ( ): 10 14 280
Add to find the total surface area: 448 320 280 1048 The surface area f
o
× × =
× × =
× × =
2
a 16 inch 10 inch 14 inch rectangular prism is 1048 in
× ×
16 in
10 in
14 in
Method 2 : Draw a net for the prism and label the dimensions for each face. Find the area of each face, and then add to find the total surface area.
2
Back: 16 14 224 Bottom: 16 10 160 Front: 16 14 224 Top: 16 10 160 Side: 10 14 140 Side: 10 14 140
Total: 160 140 160 140 224 224 1048
The surface area of the box is 1048 in.
× =
× =
× =
× =
× =
× =
bottom
16 in
10 in
14 in
side 14 in front
back (^) 14 in
16 in 14 in
top side
Step 2 : Find the area of the triangular bases.
Use 1 to find the area of the bases 2 (Base 1) 1 8 6 24 2 (Base 2) 1 8 6 24 2
A bh
A
A
⋅ ⋅
⋅ ⋅
Step 3 : Find the area of each of the rectangular lateral faces. Use to find the area of the sides. (Side 1) 6 15 90 (Side 2) 8 15 120 (Side 3) 10 15 150
A bh A A A
= × =
= × =
= × =
Step 4 : Find the sum of all the areas.
Add: 24 + 24 + 90 + 120 + 150 = 408.
The surface area of the triangular prism is 408 in 2.
Surface Area of a Cube
The surface area of a cube is the total area of all of the square faces measured in square units.
A cube is a special rectangular prism because the lengths of all of its edges are the same and all of its faces have the same area.
Area is measured in square units.
Let’s develop the formula to compute the surface area of a cube.
Step 1 : One face of a cube is a square. Its area is found by multiplying the length ( e ) and the width ( e ). (Note: e represents the length of one edge of the cube.)
A (one face) = e × e = e^2
Step 2 : A cube has six (6) faces and they all have the same area. 2
2
(one face)
Surface Area ( ) = (six faces) 6
A e
SA A e
= ×
SA = 6 e^2
To find the surface area of a cube, multiply the area of one face ( e^2 ) times 6.
Square Unit
e
Surface Area of a Cylinder
The surface area of a cylinder is determined by adding the lateral area to the area of the two circular bases. The lateral area, the body of the cylinder, is rectangular when laid flat.
Let’s examine how the formula is derived for finding the surface area of a cylinder.
*The base of the rectangular area is equal to the circumference of either of the circular bases.
To calculate the surface area of a cylinder, calculate the area of the three parts of the cylinder: the top, the bottom, and the body.
Top: Circle Bottom: Circle Body
A = π× r^2 A = π× r^2
A bh A C h A r h A rh
= ×
= ×
h (height)
A = π r^2
A = π r^2
C = 2 π r (base)*
Therefore, SA = π r^2 + π r^2 + 2 π rh.
This formula simplifies to SA = 2 π r^2 + 2 π rh
Thus, the surface area of a cylinder can be found by using the formula:
SA = 2 π r^2 + 2 π rh
The lateral area of a cylinder (the area of the body) is found using LA = 2 π rh.
Example 4 : Find the surface area of a cylinder with a radius that measures 2 inches and a height that measures 3 inches.
2 2
SA r rh SA SA SA SA
The surface area of the cylinder is 62.8 square inches.
Example 5 : Find the lateral area of the cylinder above. 2 2(3.14)(2)(3)
LA rh LA LA
The lateral area (area of the body) of the cylinder is 37.68 square inches
Step 2 : Find the area of the square base. Use formula A = l × w to find the area of the base.
2
(Square Base) 12 12 (Square Base) 144 in
A l w A A
= ×
= ×
Step 3 : Find the area of the triangular surfaces.
1 2 (Side 1) 1 12 15 90 2 (Side 2) 1 12 15 90 2 (Side 3) 1 12 15 90 2 (Side 4) 1 12 15 90 2
A bh
A
A
A
A
⋅ ⋅
⋅ ⋅
⋅ ⋅
⋅ ⋅
Step 4 :
Add to find the total surface area: 144 + 90 + 90 + 90 + 90 = 504.
The surface area of the square pyramid is 504 in 2.
Surface Area of a Cone
cone - A cone is a three-dimensional figure that has a circular base and one vertex. The lateral face is a circle sector.
base – The base of a cone is a circle.
height – The height of a cone is a segment that has an endpoint at the vertex and is perpendicular to the base.
slant height – The slant height of a right cone is the length of any segment that joins the vertex to the edge of the base.
lateral surface area – The lateral surface area of a cone is the area of the curved surface.
To find the lateral surface of a cone, use the following formula:
LA = π rl
*Note: The development of this formula is left to study in a more advanced mathematics course.
Example 2 : Find the lateral surface area of a party hat that has a radius of three inches and a slant height of six inches. Formula for Lateral Area of a Cone (3)(6) Substitution ( 3, 6) 18 Simplify 56.52 Simplify
LA rl LA r l LA LA
The lateral area of the party hat (cone) is 56.52 square inches.
height
Cone
slant height
r = 3 in
l = 6 in
Surface Area of Spheres
A sphere is a 3-dimensional figure with all points equidistant from a fixed point called its center.
The center of a sphere is the fixed point from which all points on a sphere are a given distance.
A radius of a sphere is a segment that has one endpoint on the sphere and the other at the center of the sphere.
A diameter of a sphere is a chord that passes through the center of the sphere.
A great circle is the circle formed when a circle is sliced such that the slice contains the center of the sphere. The equator is the Earth’s great circle.
A hemisphere is half a sphere. A great circle divides a sphere into two congruent hemispheres.
great circle
r d
When examining the surface area of a sphere, it takes four areas of its great circle to cover the sphere.
The surface area of a sphere is four times the area of its great circle.
SA = 4 π r^2
Example 1 : Find the surface area of a sphere with a radius of 2 inches. Round the answer to the nearest whole square inch.
The surface area of the sphere is approximately 50 square inches.
2 2
4 (2 ) -Substitution 16 -Simplify 50.24 -Simplify
SA r SA SA SA
Surface Area = 4 × Base Area
= 4 × π r^2
r = A =^ πr
2
B
r
4 ×
great circle
Drawing Three-Dimensional Figures
Three-dimensional figures have faces, edges, and vertices. A face is a flat surface. An edge is where two faces meet. A vertex is where three edges meet.
Using Isometric Dot Paper to Sketch Solids
Isometric dot paper will be used to draw various three-dimensional figures.
Example 1 : Use isometric dot paper to sketch a rectangular prism that is four units long, three units wide, and three units tall.
Step 1 : Draw the edges of the bottom face. (4 units by 3 units, parallelogram)
Step 2 : Draw the vertical line segments from the vertices of the base_. (3 units high)_
Step 3: Draw the top face by connecting the vertical lines. (4 units by 3 units, parallelogram)
ff faacacceee
vveveerrrttteeexxx
eededdgggeee
Example 2 : Draw a unit cube using three axes that form120 degrees on isometric dot paper.
Step 1 : Pick a point on the isometric dot paper. Draw a set of three axes that form 120 degrees.
Step 2 : Draw a unit cube where the three axes intersect.
Example 3 : Draw unit cubes from different points of view.
Sample 1 : Unit cubes from one point of view.
