Elegant Algabraic Expressions

Hi 9-05! Mr. B has assigned us all some homework for the weekend. We're supposed to answer a few questions about surface area, so I'm going to answer mine in the most blunt way possible.

WHAT IS THE MOST ELEGANT ALGEBRAIC EXPRESSION FOR ...

... surface area of a cube?

Well, you see, a cube has 6 faces and each of those 6 faces are equal. Since each of those faces are equal, whatever length one side is would be squared.

i.e. 3 by 3 cube

The most elegant algebraic expression for a cube would be 6s²

Example:
SA= 6s²

SA= 6(3²)
SA= 6(9)
SA= 54u²

... surface area of a rectangular prism?
Being a prism, a rectangular prism has 6 faces. Although, unlike a cube, a rectangular prism does not have all equal sides. A particularly good way to see how to calculate surface area of a rectangular prism wold be to see how its net looks like.

Seeing the measurements is a good way of seeing what parts of the rectangular prism have plane symmetry. The net above, however, is not the only way to portray the net of a rectangular prism. It's actually kind of confusing, but the picture itself is quite clear.

The most elegant algebraic expression for a rectangular prism is SA= 2(lw) + 2(hw) + 2(hl)

Example:

h=5

l=4

w=2

SA= 2(lw) + 2(hw) + 2(hl)

SA= 2(4)(2) + 2(5)(2) + 2(5)(4)

SA= 2(8) + 2(10) 2(20)

SA= 16 + 20 + 40

SA= 76u²

*remember to always add in the u² after your answer because we're dealing with 3D shapes

... surface area of a cylinder?

I'm sure all of you are familiar with how we learned this formula last year, where we just found the area of the two circles and the rectangle of the cylinder and added them all together. As you all should know, we learned a different way to calculate surface area of a cylinder.

During one of our math classes we learned that the circumference of the circle of the cylinder is equal to the width of its rectangle. To test this out just roll up a rectangular peice of paper and look at the circle and see that its width goes all around the circle, or rather the circumference.

Since we're calculating surface area of a cylinder, we know we need to have the formula for area of two circles. The formula for a circle is πr², so the formula for two circles would simply be 2(πr²). Now we need to find the area of the rectangle of the cylinder. We know that the width is equal to the circumference of the circle, and area of a rectangle is (lw) so we multiply 2πr by height, which is like the length of a rectangle.

So the most elegant algebraic expression for a cylinder is SA= 2(πr²) + 2πrh

COMPOSITE SHAPES
I actually didn't know what a composite shape was at first, but I found that it's a figure that can be divided into two or more shapes. This was what Mr. B has been telling us about for the past couple of days, but I didn't know what the technical term was then.
I think the best way to calculate the surface area for composite shapes is to look at the figure from all angles and add it all up because this can apply to all composite shapes. Some composite shapes have holes and spaces in them and this can make calculating surface area for composite shapes confusing.

HOW DOES SYMMETRY HELP US SOLVE SOME OF THESE SURFACE AREA PROBLEMS?
A figure usually has 6 faces; 3 sets of 2 symmetrical faces. Plane of symmetry are imaginary lines drawn on polyhedrons to show that both sides are symmetrical. Imagine these lines as if they cut right through the polyhedron. Symmetry helps us solve for surface area because if we see one face we know that on the opposite side there should be another face just like it. This rule usually applies for regular shapes, however. Understanding symmetry allows us to see that if one side has a measurement of so-and-so, then the other side should have that same length as well.

WHAT HAPPENS IF A PART OF ANY OF THESE SHAPES IS MISSING? HOW DO I FIND SURFACE AREA THEN?

Most polyhedrons usually have plane of symmetry. We can use plane of symmetry to find the missing part of a shape by finding its symmetrical other.

Example of Plane of Symmetry:

I hope in one way or another this has helped! Please feel free to comment! :)