Area of an Object

Surface Area of 3d Objects

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Math III​

Recall, area is a two dimensional representation of how much we can fill the given section.

Example: how much carpet would a floor need or how much space is on a blank piece of paper.

Surface area of a 3d object is the total area around the outside of the object.

Think of it like perimeter or circumference but it is area instead.

Some 3d objects have faces, one side of the object, if you find the area of all the faces and add them together you would get the surface area.

some objects do not have faces, but curved sections. They have special formulas to find the surface area.

what is the area of the surface of our planet earth?

a 6 sided rectangular prism can be separated into its 6 sides or faces

To find the surface area of a rectangular prism, just add the areas of all six faces.

Notice that opposite sides of the prism are the same. we can find the area of just three faces and multiply by 2.

SA=2 * (L*H + W*H + L*W)

How would we find the surface area in this example? we see a width of 3 and a length of 2 and a height of 6.

SA= 2 * (W*H + L*H + L*W)

SA= 2*(3*6 + 2*6 + 3*2)

remember, use order of operations

2*(18+12+6) = 2*(36) = 72

Multiple Choice

What would be the surface area of each rectangular prism?

cube one is 1 by 1 by 1 and cube two is 2 by 2 by 2?

• cube one sa= 6 and cube two sa=24

• cube one sa= 3 and cube two sa=12

• cube one sa= 1 and cube two sa=2

• cube one sa= 1 and cube two sa=4

add up the area of the 5 faces(sides)

There are several formulas for finding surface area of a triangular prism.

This one has you find the area of one of the triangles(BH/2) then multiply by 2, since you have 2 triangles, then add in the areas of the other 3 sides.

A=B*H+L*B+2*L*S

B=base h=height L=length S=side

Another way to find the surface area of a triangular prism

Add up the 3 edges of the triangle, the perimeter, and multiply that by the length. this will get you the area of the side faces, then add in the area of the two basses.

SA=B*H + L*P

where p=(s1+s2+3),

perimeter of triangle

How would we find the surface area in this example? let us add up the area of the 2 bases and the perimeter of the triangle times the length.

SA=B*H + L*P

SA= 24 + 10(5+5+8) = 204

Multiple Choice

what is the surface area of this triangular prism with base 4 and height 3 and length 11 and perimeter 12. Hint: (B*H + L*P)

• 144

• 132

• 60

• 660

To find the surface area of a cylinder, we add up the area of the two circle bases and the area of the rectangle that makes up the center of the cylinder.

Notice if we unwrap the center it becomes a rectangle.

The center part of the cylinder forms a rectangle with the same height at the height of the cylinder.

the length of the rectangle is the distance around the cylinder, which is the circumfrence of the circle.

We see that the formula when we add the areas of the 2 bases and the area of the rectangle is SA=2πr² + 2πrh

recall that area of a circle is πr² and circumfrence of a circle is 2πr and area of a rectangle(L*W), in this case L=2πr and W=H

How would we find the surface area of this cylinder? We see the radius is 4 and the height is 9.

Apply the formula we learned SA=2πr² + 2πrh

2*π*4² + 2*π*4*9

32π+72π = 104π

Multiple Choice

What is the surface area of this cylinder with radius 3 and height 9? hint: sa = 2πr² + 2πrh

• 72π

• 4π

• 18π

• 54π

when finding surface area you can always just add up all the areas of the faces.

5 faces for rectangular pyramids and 4 faces for triangular pyramids.

but formulas make the math go faster.

The base of a pyramid could be any shape, but let us look at square bases.

The formula for a square base pyramid is, SA = a + (P*S)/2

where a is the area of the base,

p is the perimeter of the base,

s is the slant height

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slant height goes from the top vertex to midpoint of an edge.

What information do we need to solve the surface area of this square base pyramid?

We see the base is a square with edge length of 5 and we see the slant height is 9. thus the area is 5*5=25 and the perimeter is 4*5=20

sa= a + (p*s)/2

sa= 25 + (20*9)/2 = 25 + 90 = 115

Multiple Choice

what is the surface area of this square pyramid with

base edge 6 and slant height 12?

hint: A+(P*S)/2

• 180

• 144

• 36

• 72

Notice how the blue part of the cone would look unfolded. To find the surface area we again add the 2 sections together.

area of circle(π*r²) PLUS

area of sector(π*r*s)

s is the slant height

what information do we need to find the surface area of this cone?

radius = 4 and slant height = 5

SA=(π*r² + π*r*s)

SA= π*16 + π*20) = 36π

Multiple Choice

What is the surface area of a cone with radius 2 and slant height 6? Hint: π*r² + π*r*s

• 16π

• 12π

• 24π

• 8π

Poll

How confident are you with this lesson?

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