The term is commonly used in computer graphics to describe the three-dimensional region which is visible on the screen, the “viewing frustum”, which is formed by a clipped pyramid; in particular, frustum culling is a method of hidden surface determination.

In the aerospace industry, frustum is the common term for the fairing between two stages of a multistage rocket (such as the Saturn V), which is shaped like a truncated cone.

Each plane section is a floor or base of the frustum. Its axis if any, is that of the original cone or pyramid. A frustum is circular if it has circular bases; it is right if the axis is perpendicular to both bases, and oblique otherwise.

The height of a frustum is the perpendicular distance between the planes of the two bases.

Cones and pyramids can be viewed as degenerate cases of frusta, where one of the cutting planes passes through the apex (so that the corresponding base reduces to a point). The pyramidal frusta are a subclass of the prismatoids.

Two frusta joined at their bases make a bifrustum.

Let’s checkout the formula!

Volume

The volume of a conical or pyramidal frustum is the volume of the solid before slicing the apex off, minus the volume of the apex:

where B₁ is the area of one base, B₂ is the area of the other base, and h₁, h₂ are the perpendicular heights from the apex to the planes of the two bases.

Considering that

the volume can also be expressed as the product of the height h = h₂−h₁ of the frustum, and the Heronian mean of their areas:

Heron of Alexandria is noted for deriving this formula and with it encountering the imaginary number, the square root of negative one.

In particular, the volume of a circular cone frustum is

where π is 3.14159265…, and R₁, R₂ are the radii of the two bases.

The volume of a pyramidal frustum whose bases are n-sided regular polygons is

where a₁ and a₂ are the sides of the two bases.

Surface area

For a right circular conical frustum

and

where R₁ and R₂ are the base and top radii respectively, and s is the slant height of the frustum.

The surface area of a right frustum whose bases are similar regular n-sided polygons is

where a₁ and a₂ are the sides of the two bases.

Anyone else  excited as I am over how awesome sauce this Math Monday was? Comeback next week when we talk about Rhombicuboctahedron!

Al-Jayyani’s The book of unknown arcs of a sphere in the 11th century introduced the general law of sines.The plane law of sines was later described in the 13th century by Nasīr al-Dīn al-Tūsī. In his On the Sector Figure, he stated the law of sines for plane and spherical triangles, and provided proofs for this law.

According to Glen Van Brummelen, “The Law of Sines is really Regiomontanus’s foundation for his solutions of right-angled triangles in Book IV, and these solutions are in turn the bases for his solutions of general triangles.”

Regiomontanus was a 15th-century German mathematician.

The Law of Sines says that in any given triangle, the ratio of any side length to the sine of it’s opposite angle is the same for all three sides of the triangle. The Law of Sines is written formally as:

This is true for any triangle, not just right triangles!

a, b and c are sides.

A, B and C are angles.

(Side a faces angle A, side b faces angle B and side c faces angle C).

So if you divide side a by the sine of angle A it is equal to side b divided by the sine of angle B, and also equal to side c divided by the sine of angle C.

This is just beyond cool but what is it used for?

A triangle has three sides and three angles. The Law of Sines is one of the tools that allows us to solve the triangle. That is, given some of these six measures we can find the rest. Depending on what you are given to start, you may need to use this tool in combination with others to completely solve the triangle.

Now that we’ve got the basics lets check this bad boy out in action!

a/sin A = 8 / sin (62.2°) = 8 / 0.885… = 9.04…

b/sin B = 5 / sin (33.5°) = 5 / 0.552… = 9.06…

c/sin C = 9 / sin (84.3°) = 9 / 0.995… = 9.05…

The answers are almost the same!
(They would be exactly the same if I used perfect accuracy).

So to wrap this up, if we are given one side and its opposite angle we can find the “law of Sines” ratio for the triangle. Then, using that ratio and the other given elements, we can solve the triangle.

Triangles are AWESOME SAUCE!

A Greek scholar named Pythagoras, who lived around 500 BC, was also fascinated by triangles with these special side ratios. He studied them a bit closer and found that the two shorter sides of the triangles squared and then added together, equal exactly the square of the longest side. And he proved that this doesn’t only work for the special triangles, but for any right triangle. Today we would write it somehow like this: a2 + b2= c2. In the time of Pythagoras they didn’t use letters yet to replace variables. (They weren’t introduced until the 16th century by Vieta.) Instead they wrote down everything in words, like this: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).

We can’t be sure if Pythagoras really was the first person to have found this relationship between the sides of right triangles, since no texts written by him were found. In fact, we can’t even prove the guy lived. But the theorem a2 + b2= c2 got his name. Another Greek, Euclid, wrote about the theorem about 200 years later in his book called “Elements”. There we also find the first known proof for the theorem. Now there are about 600 different proofs.

Today the Pythagorean theorem plays an important part in many fields of mathematics. For example, it is the basis of Trigonometry, and in its arithmetic form it connects Geometry and Algebra.

Don’t forget to bookmark my blog I do Math every Monday!

“In any moment of decision, the best thing you can do is the right thing, the next best thing you can do is the wrong thing, and the worst thing you can do is nothing.”  – Theodore Roosevelt

One of the places where indecisiveness had become an issue in my business was my ever growing project list. A few years ago I started to feel like the weight of the world was on my shoulders because of all the “stuff” I needed to get done.  I wanted to make the right decision all the time to avoid the pain associated with a wrong decision but I wasn’t avoiding it I was creating it!

To get my momentum rolling, I took on the mantra —

Do it, Delegate it or Delete it

Do It

Nike said it best with Just Do It! Don’t wait for the perfect opportunity, it will never come. Don’t wait for the right answers to come, you may be waiting forever. The sooner you make the decision the sooner you put it into action, so what if it was the wrong one now you know and can quickly move in the right direction. Doing nothing will never move you in the right direction.

Delegate It

This was a tough one for me in the beginning because I want everything to be just right. I quickly learned being a control freak perfectionist will kill momentum along with wasting tons of valuable time. Start building a core group of individuals you can trust with your tasks. They don’t need to be business partners or employees, for years I bartered services and outsourced when possible. Don’t be afraid to leverage your position, there is always someone willing to do the mundane tasks in exchange for your experience and knowledge. This is something we do in our “Circle of Awesome Sauce” (core group) constantly, everyone wins in this situation.

Delete It

My wife loves to watch the show Hoarders. It drives me crazy and I actually get stressed out seeing all the chaos people have created for themselves by never getting rid of stuff. The same holds true for our desktops, whiteboards and endless collection of note paper. We can hoard ideas to the point where enough of them will create that same chaos as the TV show. The more we hoard the wider we spread our focus and the more time we’ll need to dig ourselves out. Two of biggest benefits I found from just deleting projects was it narrowed my focus and took the weight of the world off my shoulders. When in doubt throw it out!

Below you can see the simple outline I use to prevent myself from slowing the momentum and losing focus.

Do This Every 90 Days:

1. Grab your to-do or project list
2. Go over every item and be decisive – either Do it, Delegate it, or Delete it
3. Develop Smart Objectives to stay on track for those you keep.

The more decisive you are, the more momentum you gain and remember momentum builds success.

I hope you found some of the information insightful and are able to use it for yourself. I had a great time and look forward to sharing more information like this with you.

What are some of the things you do to encounter more positive interactions even with those you find most difficult to interact with?

Specific – What is the precise outcome? Who is responsible?
Measurable – How will progress be measured?
Attainable – Can the objective be achieved with a reasonable amount of effort?
Realistic – Do you have the ability and resources to get the job done?
Time dependent – What is the start or finish date?

I hope you find this video helpful and I wish you a very prosperous New Year!

You now have a map that goes with all that passion, go out and show this world what a ton of will and a little skill can accomplish.

Missed David’s webinar last night? That’s ok he was kind enough to record it for me in case anyone missed it.

Click here to watch the replay but make sure you have about 75 minutes of quiet time as this is top quality content. Trust me you won’t want to be disturbed.

Whether we know it or not a continued pattern of allowing negativity to control us leads to a pattern of failure. I’m not perfect and I’d be lying if I told you sometime it takes me awhile to realize I’m allowing the negativity to control me. It’s never too late to stop yourself and ask the question, while it’s better to always to remain in control our human sin nature prevents us from doing so all the time. This is a skill I’m still working on perfecting but I’m already seeing huge improvements with the way I handle situations.

People have different ways of getting themselves back on track. What do you do when these things happen in your life?

Maybe you never thought about taking control back before today, do you think the technique I am using can help you?

Maybe you can be in the next Nerd Dance video. All I need is your permission you can give it to me in the comments below. Ta Da!