### Elegant Algebraic Expressions

Sunday, October 18, 2009
Heey everyone! This was an assignment from last week and I just found out about it this weekend. Sorry if it's late Mr. Backe. I haven't been on blogger for awhile so I'm not caught up with all the blogging assignments. Anyways, Mr. Backe asked us the find elegant algebraic expressions for the surface area of the following: cube, rectangular prism, cylinder, triangular prism and a composite of any of these shapes. He also wants us to answer these questions: How does symmetry help us solve some of these surface area problems? What happens if a part of any of these shapes is missing? How do I find surface area then? So here it goes....

Surface Area:

Cube
Finding the surface area of a cube is as easy as finding the area of a square and you just multiply it by 6 because there are six faces in a cube (six squares).
Area formula for a square is: s x s or s squared.
Surface Area formula for a cube: 6s squared

Example: The side = 5.

SA= 6(5 squared)
SA= 6(25)
SA= 150 squared units.

Rectangular Prism
You only have to calculate 3 sides of a rectangular prism, lh+2lw+2wh, and multiply it by 2 because of symmetry.
Surface Area formula for a rectangular prism: 2(lh+2lw+2wh)

Example: The length = 5, width = 9 and the height = 4.

SA= 2(lh+2lw+2wh)
SA= 2(5)(4)+2(5)(9)+2(9)(4)
SA= 2(20)+2(45)+2(36)

SA= 40+90+72
SA= 202 squared units

Cylinder

To find the total surface area of a cylinder you first have to find the area of the two circles and then the area of the rectangle by multiplying the area of the circles to the height of the rectangle.
Surface Area formula for a cylinder: 2
πr squared + 2πrh

Example: radius = 3 and height is = 9.

SA= 2πr squared + 2πrh
SA= 2
π(3 squared) + 2π(3)(9)
SA= 2
π(9) + 2π(27)
SA= 2(28.26) + 2(84.78)
SA= 56.52 + 169.56
SA= 226.08 squared units

Triangular Prism
Finding the surface area of a triangular prism seems hard but it's actually really easy. First you have to do the Pythagorean Theorem,
a2 + b2 = c2, then calculate the 2 triangles, 2(bh/2), then the sides or the "roofs", 2(h)(hyp), and then last but not least the bottom/ base, bh, and add it all up.
Surface Area formula for Triangular Prism: SA of base/bottom + SA of 2 triangles + SA of 2 roofs/sides

Example: height = 15, height of triangle = 6, and base is = 9.

2 Triangles

SA= 2(bh/2)

SA= (9)(6)

SA= 54 squared units

Base/Bottom
SA= bh
SA= (9)(15)
SA= 135 squared units

2 Roofs/Sides
=(h)c2= a2 + b2
hypotenuse
c2= a2 + b2
c2= 4.5 squared + 6 squared
* How did I get 4.5? Divide the base, 9,by 2.

c2= 20.25 + 36
c2= 56.25 (square root)
c2= 7.5

whole roof
= 2(hyp)(h)
= 2(7.5)(15)
= 2(112.5)
= 225 squared units

SA= 1 base/bottom +2 triangles + 2 roofs/sides
SA= 135 + 54 +225
SA= 414 squared units

Composite
A composite figure is a figure (or shape) that can be divided into more than one of the basic figures. In other words, it`s a shape or a figure with more than one shape and it can also be divided. It can be a rectangular prism with another rectangular prism, smaller or bigger, a rectangular prism with a triangular prism so it looks like a house, a cube with another cube or it can be a rectangular prism missing a corner or a middle. To find the surface area of a composite figure you have to look for different surface areas of different objects and then add it all up or subtract it if there's a hole. You also have to subtract the area of the faces touching because it doesn't count because it's not showing.
Surface Area formula for this composite object: (SA of big rectangular prism + SA of small rectangular prism) - 2 (SA of one face of the small rectangular prism)

Example: (big rectangular prism) height = 4, length = 19 and width = 12. (small rectangular prism) height = 2, length = 10 and width = 5.

SA of big rectangular prism
= 2(lh+2lw+2wh)

= 2(19)(4)+2(19)(12)+2(12)(4) = 2(76)+2(228)+2(48) = 152+456+96
= 704 squared units

SA of small rectangular prism
= 2(lh+2lw+2wh)

= 2(10)(2)+2(10)(5)+2(5)(2)

= 2(20)+2(50)+2(10)

= 40+100+20
= 160 squared units

Area that is touching
= (lw)
= (10)(5)
= 50 squared units

Calculate...
SA formula for this composite object = (SA of big rectangular prism + SA of small rectangular prism) - 2 (SA of one face of the small rectangular prism)
= (704 + 160)- 2(50)
= (864)- 2(50)
= 864 - 100
= 764 squared units

How does symmetry help us solve some of these surface area problems? I think that symmetry helps us solve surface area problems easier. if you know where and what kind of symmetry an object, shape or figure has then it will be easier for us to solve and find the surface area. instead of calculating everything we can just multiply it by 2s, 4s, etc. depending on the symmetry it has. I don't know if that made sense and I'm sorry if I didn't.

What happens if a part of any of these shapes is missing? How do I find surface area then?
I don't really know how to explain this but I'm going to try my best. If a part of any of these shapes are missing you have to....
Look for the surface area of the whole thing and ignore the missing part for awhile then when you're done, find the surface area of the missing face and subtract that from the total surface area. That's what I would do. If that didn't make sense here's an example with a rectangular prism with no top:
Surface Area formula for a rectangular prism with no top part = SA of rectangular prism - SA of missing face/top part or 2(lh+2lw+2wh) - lw

Example: length = 10, width = 5 and height = 3.
SA = 2(lh+2lw+2wh) - lw
SA = 2(10)(3)+2(10)(5)+2(5)(3) - (10)(5)
SA = 2(30)+2(50)+2(15) - 50
SA = 60+100+30 - 50
SA = 140 squared units

Don't forget to comment! (; Tell me what you think about my post and point out my mistakes and everything. Sorry again if it's late ):