Geometry: Area of a Sector

Youtube Area of a sector copy

Hi math friends, has anyone been cooking more during quarantine?  We all know there is more time for cookin’ and eatin’ cakes but have you ever been curious about the exact amount of cake you are actually eating?! Well, you’re in luck because today we are going to go over how to find the area of a piece of cake, otherwise known as the Area of a Sector!

Now, we’ll all be able to calculate just how much we are overdoing it on that pie! Hopefully, everyone is eating better than I am (I should really calm down on the cupcakes).  Ok, now to our question:

*Also, If you haven’t done so, check out the video that goes over this exact problem, and don’t forget to subscribe!

Screen Shot 2020-05-19 at 4.18.42 PMExplanation:

How do I answer this question? 

We must apply/adjust the formula for the area of a circle to find the area of the blue shaded region otherwise known as the sector of this circle.                                                    

How do we do this?    

Before we begin let’s review the formula for the area of a circle. Just a quick reminder of what each piece of the formula represents:

Screen Shot 2020-05-19 at 4.24.28 PMStep 1: Now let’s fill in our formula, we know the radius is 5, so let’s fill that in below:

Screen Shot 2020-05-19 at 4.26.08 PMStep 2: Ok, great! But wait, this is for a sector; We need only a piece of the circle, not the whole thing.  In other words, we need a fraction of the circle. How can we represent the area of the shaded region as a fraction?

Well, we can use the given central angle value, Screen Shot 2020-05-19 at 4.27.17 PM, and place it over the whole value of the circle,Screen Shot 2020-05-21 at 4.01.12 PM . Then multiply that by the area of the entire circle. This will give us the value we are looking for!

Screen Shot 2020-05-19 at 4.27.45 PMStep 3: Multiply and solve!Screen Shot 2020-05-19 at 4.28.38 PM

Ready for more? Try solving these next few examples on your own to truly master area of a sector!

Find the area of each shaded region given the central angle and radius for each circle:Screen Shot 2020-05-19 at 4.29.40 PM

Check the solutions below, when you’re ready:Screen Shot 2020-05-19 at 4.30.36 PMWhat do you think of finding the area of sector? Are you going to measure the area of your next slice of pizza?  Do you have any recipes to recommend?  Let me know in the comments and happy calculating! 🙂

 

Geometry: The Voluminous “Vessel” at Hudson Yards

Calling all NYC dwellers! Have you seen the new structure at Hudson Yards?  A staircase to nowhere, this bee-hive like structure is for the true adventurists at heart; Clearly, I had to check it out!

Where does math come in here you say?  Well, during my exploration, I had to wonder (as am sure most people do) what is the volume of this structure?  What do you think the volume of the Vessel is? (Hint: feel free to approximate!)

Screen Shot 2019-04-11 at 5.38.46 PM.png

Solution: I estimated the volume by using the formula of a three-dimensional cone. (Not an exact measurement of the Vessel, but close enough!) .

Screen Shot 2019-04-11 at 5.08.42 PM Screen Shot 2019-04-12 at 1.20.08 PM.pngWe can find the radius and height based on the given information above.  Everything we need for our formula is right here!

Screen Shot 2019-04-14 at 4.49.16 PM

Now that we have our information, let’s fill in our formula and calculate! Screen Shot 2019-04-11 at 5.14.58 PM.pngScreen Shot 2019-04-11 at 5.17.30 PM.png

Screen Shot 2019-04-11 at 5.18.27 PM.png

Screen Shot 2019-04-14 at 4.51.42 PM.png

Extra Tip! Notice that we labeled the solution with feet cubed Screen Shot 2019-04-14 at 4.53.49 PM.png, which is the short-handed way to write “feet cubed.”  Why feet cubed instead of feet squared? Or just plain old feet? When we use our formula we are multiplying three numbers all measured in feet:

radius X radius X (Height/3)

All three values are measured in feet! –> Feet cubed (Screen Shot 2019-04-14 at 4.53.49 PM.png)

  ____________________________________________________________________________________

Did you get the same answer? Did you use a different method or have any questions?  Let me know in the comments and happy mathing! 🙂

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Geometry: Intersecting Secants

Screen Shot 2018-12-25 at 12.32.32 PMScreen Shot 2018-12-25 at 12.32.38 PM

Screen Shot 2018-12-25 at 12.34.09 PMScreen Shot 2018-12-25 at 12.34.18 PMScreen Shot 2018-12-25 at 12.35.22 PM.png

Screen Shot 2018-12-25 at 12.35.52 PM.png

Screen Shot 2018-12-25 at 12.54.24 PM.png

*Extra Tip! Why does this formula work in the first place!??  If we draw lines creating and proving triangle RTQ and triangle RPS are similar by AA, this leads us to know that the two triangles have proportionate sides and can follow our formula!         ___________________________________________________________________________________

Still got questions?  Let me know in the comments and remeber having questions is a  good thing!

Also, happy holidays from Mathsux! May your December break be filled with family, food, happiness, and maybe some math problems! 🙂

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The Magic of the “Golden Ratio”

Walking around NYC, I was on a mission to connect mathematics to the real world.  This, of course, led me to go on a mathematical scavenger hunt in search of  the “Golden Ratio.” Hidden in plain sight, this often times naturally occurring ratio is seen everywhere from historic and modern architecture to nature itself. 

What is this all-encompassing “Golden Ratio” you may ask?
It’s a proportion, related to a never-ending sequence of numbers called the Fibonacci sequence, and is considered to be the most pleasing ratio to the human eye.  The ratio itself is an irrational number equal to 1.618……..(etc.).

Why should you care?
When the same ratio is seen in the Parthenon, the Taj Mahal, the Mona Lisa and on the shores of a beach in a seashell, you know it must be something special!
Screen Shot 2018-11-19 at 10.48.04 PM.png Screen Shot 2018-11-19 at 10.48.34 PM.png Screen Shot 2018-11-19 at 10.49.10 PM Screen Shot 2018-11-19 at 11.07.21 PM

Random as it may seem, this proportion stems from the following sequence of numbers, known as the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, …….

Do you notice what pattern these numbers form?
Capture(Answer: Each previous two numbers are added together to find the next number.)

The Golden RectangleThe most common example of the “Golden Ratio” can be seen in the Golden rectangle.  The lengths of this rectangle are in the proportion from 1: 1.618 following the golden ratio. Behold the beauty of the Golden Rectangle:

Screen Shot 2018-11-18 at 5.58.24 PM

How is the Fibonacci Sequence related to the Golden Ratio?                                               What if we drew a golden rectangle within our rectangle?

Screen Shot 2018-11-18 at 5.58.13 PM.png

Then drew another golden rectangle within that golden rectangle?

Screen Shot 2018-11-18 at 6.04.17 PM.png

And we kept doing this until we could no longer see what we were doing…….

Screen Shot 2018-11-19 at 8.38.34 AM

The proportion between the width and height of these rectangles is 1.618 and can also be shown as the proportion between any two numbers in the Fibonacci sequence as the sequence approaches infinity. Notice that the area of each rectangle in the Fibonacci sequence is represented below in increasing size:Screen Shot 2018-11-19 at 10.31.51 PM

Where exactly can you find this Golden Ratio in real life? Found in NYC! The Golden ratio was seen here at the United Nations Secretariat building in the form of a golden rectangle(s).  Check it out!

Golden Ratio Pic

Where have you seen this proportion of magical magnitude?  Look for it in your own city or town and let me know what you find! Happy Golden Ratio hunting! 🙂

If you’re interested in learning more about the golden ratio and are also a big Disney fan, I highly recommend you check out Donald Duck’s Math Magic!

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