Scale Factor

Here, I will be using similar polygons (circles) and showing their scale factors for units and units squared. Here are two ordinary circles, one with radius of 6, and one with radius of 10. Their scale factor is 6 divided by 10, and that simplified is 3:5.

Now when we find the circumferences of the two circles, their ratio should be the same as the ratio of the radii.

The circumferences of the two circles are 37.68 and 62.86. When we divide 37.68 by 62.86, we get .60, which is equal to 3 divided by 5. 

The areas of the circles should be in the scale factor of k squared, in other words, 3 squared / 5 squared.  The scale factor for the areas of the circles should be 9:25.

The area for the first circle is 112.99 square cm, and the area for the second circle is 314.45 square cm. If we divide 112.99 by 314.45, we get .36, which is the same as 9/25. This demonstrates that if we have any two similar polygons, any length measured in units is in the scale factor k, and anything measured in units squared is in the scale factor k squared. (~DM5)

Proof of Parallels

My brother recently had surgery and will be restricted to a wheelchair for the next six weeks. Because of this, we need wheelchair ramps on our steps so he has access to the house. The wheels on a wheelchair are parallel, so the ramps must be parallel or the wheelchair cannot successfully scale the ramps. I can prove that the wheels are parallel, and that the ramps are close enough to parallel to allow the wheelchair to travel up and down them.

Because of the angle at which the pictures were taken, the ramps appear slightly crooked, but I can assure you that they are not. If you zoom out, you can see this.

(~CB1)

Measuring Lateral Area and Volume of a Cone

After finding a three dimensional cone in the art room, I decided to put it to use to find its circumference and area, as well as use Theorem 12-7 and 12-8 to find its lateral area and volume.

First, I began by measuring the cone's height (h) and slant height (l).

Next, I measured the radius, and used that to calculate both the circumference of the base and the area.

After collecting all the needed information, I put it to use to calculate lateral area using Theorem 12-7, and volume using Theorem 12-8.

Theorem 12-7: The lateral area of a cone equals half the circumference of the base times the slant height.

Theorem 12-8: The volume of a cone equals one third the area of the base times the height of the cone.

(~RO5)

Tessellations in Boston

While we were in Boston for our D.C. trip, I noticed a sidewalk that contained tessellations. The shapes fit in perfectly with no gaps or spaces. The pattern would be able to continue indefinitely. The basic definition of a tessellation is a random shape repeated over and over covering a plane with out any gaps or overlaps. Another word is tiling.

   

















Some further research shows that there are two main types of tessellations: regular and semi-regular.

Regular tessellations are patterns made by repeating a regular polygon. In that case there are only three types: squares, triangles and hexagons. For regular tessellations, the pattern is always the same at the vertex.

A semi-regular tessellation consisted of two or more regular polygons. As with the regular tessellations, the patterns must be the same at each vertex. (~IR1)

Surface Area of the Pentagon

When I was in Washington D.C. last week I noticed that the pentagon was just a regular pentagonal prism. With this I searched online to figure out the height and side length of the Pentagon. From these dimensions I was able to figure out the Lateral Area and the Surface Area of the Pentagon. (~AL1)

Circumference, Area, and Pie

After taking a picture of this fruit pie (because everyone takes a picture of their food before they eat it these days) I noticed that this pie was a perfect example of a circle, and using some simple formulas, I could find its circumference and area.

After measuring a radii of the circle, I was able to calculate the area and circumference using the formulas shown above. 

But, why stop there? It also became apparent that I could inscribe a polygon inside the circle, and calculate the area of that as well. In this case, I used a trapezoid.

As shown above, I was able to calculate the area of this trapezoid using Theorem 11-5.

It's seems that everything, even a fruit pie, has some relation to math, and more specifically: geometry. 

-RO5

Hexagonal Prism

This is a hexagonal prism that I bought at a souvenir shops in Washington D.C. It is really cool and shows a bunch of famous monuments and buildings in D.C. which are on each side of the prism. The Washington Monument is in the middle. I saw this and remembered calculating lateral area, surface area, and volume, with prisms just like this one.

(~SV1)