How to find the Area of a Parallelogram: Geometry

Hey math peeps! In today’s post, we are going to go over how to find the area of a parallelogram. There is an easy formula to remember, A=bh, but we are going to look at why this formula works in the first place and then solve a few examples. Just a quick warning: The following examples do use special triangles and if you are need of a review, check out the posts here for 45 45 90 and 30 60 90 special triangles. Also, don’t forget to watch the video and try the practice problems below. Thanks so much for stopping by and happy calculating!

Area of a Parallelogram Formula:

How to find the Area of a Parallelogram

Why does the Formula for Area of a Parallelogram work?

Did you notice that the formula for area of a parallelogram above, base times height, is the same as the area formula for a rectangle?  Why?

If we cut off the triangle that naturally forms along the dotted line of our parallelogram, rotated it, and placed it on the other side of our parallelogram, it would naturally fit like a puzzle piece and create a rectangle! Check it out below:

How to find the Area of a Parallelogram

Now that we know where this formula comes from, let’s see it in action in the examples below:

Example #1:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Fill in the formula with values found on our parallelogram, b=12 inches h=4 inches, and multiply them together to get 48 inches squared.

How to find the Area of a Parallelogram

That was a simple example, but lets try a harder one that involves special triangles.

Example #2:

How to find the Area of a Parallelogram

Step 1: Write out the formula:

Step 2:  Label the values found on our parallelogram, b=10 ft and notice that we are going to need to find the value of the height.

Step 3: In order to find the value of the height, we need to remember our special triangles! We are not given the value of the height, but we are given some value of the triangle that is formed by the dotted line.  Let us take a closer look and expand this triangle:

Step 4: We can add in the missing 45º degree value so that our triangle now sums to 180º.

Step 5: Remember 45 45 90 special triangles(If you need a review click the link). Because that is exactly what we are going to need to find the value of the height! Below is our triangle on the left, and on the right is the 45 45 90 triangle ratios we need to know to find the value of the height.

Based on the above ratios, we can figure out that the height value is the same value as the base of the triangle, 2.

Step 6: If we place our triangle back into the original parallelogram, we can plug in our value for the height, h=2, into our formula to find the area:

How to find the Area of a Parallelogram

When you’re ready, check out the practice questions below!

Practice Questions:

Find the area of each parallelogram:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Constructing a Perpendicular Bisector

Hi everyone and welcome to MathSux! In this post we are going to be constructing a perpendicular bisector, a line that cuts a line segment in half and creates four 90º angles. It’s a super fast and super simple construction! If you’re looking for more constructions, don’t forget to check more out here. Thanks so much for stopping by and happy calculating! 🙂

What is the Perpendicular Bisector of a line ?

  • Cut’s our line AB in half at its midpoint, creating two equal halves.
  • This will also create four 90º angles about the line.
Constructing a Perpendicular Bisector

What is happening in this GIF?

Step 1: First, we are going to measure out a little more than halfway across the line AB by using a compass.

Step 2: Next we are going to place the compass on point A and swing above and below line AB to make a half circle.

Step 3: Keeping the same distance on our compass, we are then going to place the point of the compass onto Point B and repeat the same step we did on point A, drawing a semi circle.

Step 4: Notice the intersections above and below line AB!? Now, we want to connect these two points by drawing a line with a ruler or straight edge.

Step 5: Yay! We now have a perpendicular bisector! This cuts line AB right at its midpoint, dividing line AB into two equal halves.  It also creates four 90º angles.

Still got questions? No problem! Don’t hesitate to comment with any questions below or check out the video above. Thanks for stopping by and happy calculating! 🙂

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Want to see how to construct a square inscribed in a circle? Or maybe you want to construct an equilateral triangle? Click on each link to view each construction!

Inscribed Angles & Intercepted Arcs: Geometry

Ahoy math friends and welcome to MathSux! In this post, we are going to go over inscribed angles and intercepted arcs. We’ll break down the main basic rule for inscribed angles and the three theorems associated with this rule. If you are looking for more circle theorems, check out these posts on the Intersecting Secants Theorem and Central Angles Theorem. Also, don’t forget to check out the video and practice questions to truly master the topic below. Happy calculating! 🙂

Inscribed Angles:

When two chords come together to touch the outline of a circle, they create something called an inscribed angle. An inscribed angle is equal to half the value of the arc length.

Inscribed Angles & Intercepted Arcs

Inscribed Angle Theorems:

There are three inscribed angle theorems to know based on the rule stated above, check them out below!

Theorem #1: (Intercepted Arcs) In a circle when inscribed angles intercept the same arc, the angles are congruent.

Inscribed Angles & Intercepted Arcs

Theorem #2: In a circle when an angle is inscribed by a semicircle, it forms a  90º angle.

Theorem #3: When a quadrilateral is inscribed in a circle, opposite angles are supplementary (add to 180º). (The proof below shows angles A and C as supplementary, but this proof would also work for opposite angles B and D).

Inscribed Angles & Intercepted Arcs

Let’s look at how to apply these rules with an Example:

a) Step 1: To find the value of angle CDB we need to look at our given information. We know that angle CAB=85º, notice that this follows theorem number 3, “When a quadrilateral is inscribed in a circle, opposite angles are supplementary.” Therefore, we must subtract 110º from 180º to find the value of angle CDB.

b) Step 2: For finding angle ABD, we’re going to use the same theorem we used in part a, opposite supplementary angles of an inscribed quadrilateral are supplementary.

c) Step 3: Next, to find the value of arc ABD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to half the value of its arc. Then use some basic algebra to solve for arc ABD.

d) Step 4: To find arc ACD, we need to use the basic inscribed angle theorem that tells us an inscribed angle is equal to  the value of its arc, then use algebra to solve similar to part c.

If this looks confusing, check out the video above! And when you are ready master this topic with the practice questions below!

Practice Questions:

Solutions:

Still got questions?  No problem! Check out the video above or comment below for any questions and follow for the latest MathSux posts. Happy calculating! 🙂

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Square Inscribed in a Circle Construction

Greetings math friends and welcome to MathSux! In this week’s post, we are going to take a step by step look on how a square inscribed in a circle construction works! We got videos, we got GIF’s, and we got a step by step written explanation below, the choice of learning this construction is up to you! Happy Calculating! 🙂

Square Inscribed in a Circle Construction

How to Construct a Square Inscribed in a Circle:

Step 1: Draw a circle using a compass.

Step 2: Using a ruler, draw a diameter across the length of the circle, going through its midpoint.

Step 3: Open up the compass across the circle. Then take the point of the compass to one end of the diameter and swing the compass above the circle, making a mark.

Step 4: Keeping that same length of the compass, go to the other side of the diameter and swing above the circle again making another mark until the two arcs intersect.

Step 5: Repeat steps 3 and 4, this time creating marks below the circle.

Step 6: Connect the point of intersection above and below the circle using a ruler. This creates a perpendicular bisector, cutting the diameter in half and forming 90º angles.

Step 7: Lastly, use a ruler to connect each corner point to one another creating a square.

Still got questions? No problem! Don’t hesitate to comment with any questions. Happy calculating! 🙂

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Looking for something similar to square inscribed in a circle construction? Check out this post here on how to construct and equilateral triangle here!

30 60 90 Special Triangles: Geometry

Hi everyone and welcome to MathSux! In this post we are going to break down 30 60 90 degree special triangles. What is it? Where did it come from? What are the ratios of it’s side lengths and how to do we use them? You will find all of the answers to these questions below. Also, don’t forget to check out the video below and practice questions at the end of this post. Happy calculating! 🙂

Want to make math suck just a little bit less? Subscribe to my Youtube channel for free math videos every week! 🙂

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What is a 30 60 90 Triangle and why is it “Special”?

The 30 60 90 triangle is special because it forms an equilateral triangle when a mirror image of itself is drawn, meaning all sides are equal!  This allows us to find the ratio between each side of the triangle by using the Pythagorean theorem. Check it out below!

30 60 90 Special Triangles

Now let’s draw a mirror image of our triangle.  Next, we can label the length of the new side opposite 30º “a,” and add this new mirror image length with the original we had to get, a+a=2a.

30 60 90 Special Triangles
30 60 90 Special Triangles

If we look at our original 30 60 90 triangle, we now have the following values for each side based on our equilateral triangle:

30 60 90 Special Triangles
30 60 90 Special Triangles

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two legs. This creates a ratio that applies to all 30 60 90 triangles!

30 60 90 triangle side lengths

How do I use this ratio?

30 60 90 triangle side lengths

Knowing the above ratio, allows us to find any length of any and every 30 60 90 triangle, when given the value of one of its sides.

Let’s try an Example:

30 60 90 triangle side lengths

-> First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

->Notice we are given the value of a, which equals 4, knowing this we can now fill in each length of our triangle based on the ratio of a 30 60 90 triangle.

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now let’s look at an Example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

-> Notice we are given the value of the hypotenuse, 2a=20. Knowing this we can find the value of a by dividing 20 by 2 to get a=10. Once we have the value of a=10, we can easily find the length of the last leg based on the 30 60 90 ratio:

30 60 90 triangle side lengths
30 60 90 triangle side lengths

Now for our last Example, when we are given the side length across from 60º and need to find the other two missing sides.

30 60 90 triangle side lengths

->First let’s look at our ratio and compare it to our given triangle.

30 60 90 triangle side lengths

-> In this case, we need to use little algebra to find the value of a, using the ratio for 30 60 90 triangles.

30 60 90 triangle side lengths

Now that we have one piece of the puzzle, the value of a, let’s fill it in our triangle below:

Finally, let’s find the value of the length of the hypotenuse, which is equal to 2a.

Practice Questions:

Find the value of the missing sides of each 30 60 90 degree triangle.

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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Looking to review 45 45 90 degree special triangles? Check out this post here!

NYE Ball Fun Facts: Volume & Combinations

Greetings math friends, and Happy New Year! In today’s post we’re going do something a little different and take a look at the math behind the very famous and very shiny New Year’s Eve Ball that drops down every year at midnight.  We’ll break down the shape, the volume, and the number of those dazzling Waterford crystals (and no this post isn’t sponsored) and look at some NYE Ball Fun Facts.

NYE Ball Fun Facts

Shape: Geodesic Sphere

Yes, apparently the shape of the New Year’s ball is officially called a “Geodesic Sphere.”  It is 12 feet in diameter and weighs 11,875 pounds.

Volume (Estimate): 288π ft3

If we wanted to estimate the volume of the New Year’s Ball we would could use the formula for volume of a sphere:

NYE Ball Fun Facts

Number of Waterford Crystals: 2,688

Talk about the ultimate shiny bauble! The NYE ball lights up the night with all 2,688 crystals in the shape of different sized triangles, each with heights of 5.75 inches or 4.75 inches.

NYE Ball Fun Facts

Number of Lights: 48 light emitting diodes (LED’s)

On each triangle, there are 48 LEDs: 12 red, 12 blue, 12 green, and 12 white, for a total of 32,256 LEDs on the entire NYE ball itself.

NYE Ball Fun Facts

Permutations and Combinations:

Permutations: With this many lights and colors, there are over a billion potential permutations of colors on the entire NYE ball.

Combinations: Let’s break down one triangle with 48 LED lights each with 12 red, 12 blue, 12 green, and 12 white LEDs. How many possible combinations of lights are possible if we were to choose 7 blue, 5 red, 10 green, and 1 white turned on all at the exact same time?

We end up with the combination formula below:

NYE Ball Fun Facts

That means that there are 496,793,088 possible ways that 7 blue lights, 5 red lights, 10 green lights, and 1 white light can be lit up on a triangle that is part of the entire NYE ball!

Interested in more NYE fun facts?  Check out the sources of this article here.

NYE Fact Sheet from: timessquarenyc.org

NYE Ball picture: Timesquareball.net

If you like finding the volume of the NYE ball maybe, you’ll want to find the volume of the Hudson Yards Vessal in NYC here.  Happy calculating and Happy New Year from MathSux!

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Central Angles Theorems: Geometry

Hi everyone, and welcome to MathSux! In this post, we are going to go over the Central Angles Theorems of circles. We’ll go over the theorems associated with central angles and then solve a quick example. Make sure to test your understanding of central angles and arcs with the practice questions at the end of this post. And, if you want more, don’t forget to check out the video below, happy calculating!

Central Angles and Arcs:

Central angles and arcs form when two radii are drawn from the center point of a circle.  When these two radii come together they form a central angle. A central angle is equal to the length of the arc. When it comes to measuring the central angle, the central angle is always equal to arc length and vice versa:

Central Angles = Arc Length

central angles theorems

Central Angle Theorems:

There are a two central angle theorems to know, check them out below!

Central Angle Theorem #1:

central angles theorems

Central Angle Theorem #2:

central angles theorems

Let’s look at how to apply these rules with an Example:

central angles theorems

Let’s do this one step at a time.

central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems
central angles theorems

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

*Also, if you want to check out Intersecting Secants click this link here!

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TikTok Math Video Compilations

Happy December everyone! With crazy 2020 coming to an end, I thought I would share some TikTok math video compilations of Algebra, Geometry, Algebra 2/Trig, and Statistics for a quick review of all our videos posted throughout the year. Enjoy these TikTok math video compilations and happy calculating! 🙂

Want to make math suck just a little bit less? Subscribe and follow us for FREE fun colorful math videos and lessons every week! 🙂

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TikTok Math Video Compilations

Algebra:

Within algebra, you will find arithmetic sequences, combining like terms, box and whisker plots, geometric sequences, solving radical equations, completing the square, 4 ways to factor quadratic equations, piecewise functions and more!

Geometry:

Within Geometry, you will find, how to construct an equilateral triangle, a median of a trapezoid, area of a sector, how to find perpendicular and parallel lines through a given point, SOH CAH TOA right triangle trigonometry, reflections, and more!

Algebra 2/Trig.

Within Algebra 2/Trig., you will find, how to expand a cubed binomial, how to divide polynomials, how to solve log equations, imaginary numbers, synthetic division, unit circle basics, how to graph y=sin(x), and more!

Statistics:

Within statistics, you will find, box and whisker plots, how to find the variance, and, the probability of flipping a coin 2 times!

For full length video, don’t forget to check out our free math video index page! Thanks for stopping by! 🙂

Rotations about a Point: Geometry

rotations about a point

Happy Wednesday math friends! In this post we’re going to dive into rotations about a point! In this post we will be rotating points, segments, and shapes, learn the difference between clockwise and counterclockwise rotations, derive rotation rules, and even use a protractor and ruler to find rotated points. The fun doesn’t end there though, check out the video and practice questions below for even more! And as always happy calculating! 🙂

What are Rotations?

Rotations are a type of transformation in geometry where we take a point, line, or shape and rotate it clockwise or counterclockwise, usually by 90º,180º, 270º, -90º, -180º, or -270º.

A positive degree rotation runs counter clockwise and a negative degree rotation runs clockwise.  Let’s take a look at the difference in rotation types below and notice the different directions each rotation goes:

rotations 90 degrees

How do we rotate a shape?

There are a couple of ways to do this take a look at our choices below:

  1. We can visualize the rotation or use tracing paper to map it out and rotate by hand.
  2. Use a protractor and measure out the needed rotation.
  3. Know the rotation rules mapped out below.  Yes, it’s memorizing but if you need more options check out numbers 1 and 2 above!

Rotation Rules:

rotations 90 degrees

Where did these rules come from?

To derive our rotation rules, we can take a look at our first example, when we rotated triangle ABC 90º counterclockwise about the origin. If we compare our coordinate point for triangle ABC before and after the rotation we can see a pattern, check it out below:

rotations

The rotation rules above only apply to those being rotated about the origin (the point (0,0)) on the coordinate plane.  But points, lines, and shapes can be rotates by any point (not just the origin)!  When that happens, we need to use our protractor and/or knowledge of rotations to help us find the answer. Let’s take a look at the Examples below:

Example #1:

rotations

Step 1: First, let’s look at our point of rotation, notice it is not the origin we rotating about but point k!  To understand where our triangle is in relation to point k, let’s draw an x and y axes starting at this point:

rotations

Step 2: Now let’s look at the coordinate point of our triangle, using our new axes that start at point k.

Step 2: Next, let’s take a look at our rule for rotating a coordinate -90º and apply it to our newly rotated triangles coordinates:

rotations

Step 3: Now let’s graph our newly found coordinate points for our new triangle .

rotations about a point

Step 4: Finally let’s connect all our new coordinates to form our solution:

rotations about a point

Another type of question with rotations, may not involve the coordinate plane at all! Let’s look at the next example:

Example #2:

rotations about a point

Step 1: First, let’s identify the point we are rotating (Point M) and the point we are rotating about (Point K).

rotations about a point

Step 2: Next we need to identify the direction of rotation.  Since we are rotating Point M 90º, we know we are going to be rotating this point to the left in the clockwise direction.

Step 3: Now we can draw a line from the point of origin, Point K, to Point M.

rotations about a point

Step 4: Now, using a protractor and ruler, measure out 90º, draw a line, and notice that point L lands on our 90º line. This is our solution! (Note: For help on how to use a protractor, check out the video above).

rotations about a point

Ready for more? Check out the practice questions below to master your rotation skills!

Practice Questions:

  1. Point B is rotated -90º about the origin. Which point represents newly rotated point B?    

2. Triangle ABC is rotated -270º about point M.  Show newly rotated triangle ABC as A prime B prime C prime.

3. Point G is rotated about point B by 180º. Which point represents newly rotated point B?

rotations about a point

4.  Segment AB is rotated 270º about point K.  Show newly rotated segment AB.

Solutions:

Still got questions on how to rotations about a point? No problem! Don’t hesitate to comment with any questions or check out the video above for even more examples. Happy calculating! 🙂

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Looking to brush up on your rotations skills? Check out this post here!

Geometry: 45º 45º 90º Special Triangles

45 45 90 triangle

Greetings math folks! In this post we are going to go over 45º 45º 90º special triangles and how to find the missing sides when given only one of its lengths. For even more examples, check out the video below and happy calculating! 🙂

Why is it “special”?

 The 45º 45º 90º triangle is special because it is an isosceles triangle, meaning it has two equal sides (marked in blue below).  If we know that the triangle has two equal lengths, we can find the value of the hypotenuse by using the Pythagorean Theorem.  Check it out below!

45 45 90 triangle

Now we can re-label our triangle, knowing the length of the hypotenuse in relation to the two equal legs. This creates a ratio that applies to all 45º 45º 90º triangles!

45 45 90 triangle

How do I use this ratio?

45 45 90 triangle

Knowing the above ratio, allows us to find any length of a 45º 45º 90º triangle, when given the value of one of its sides.

Let’s try an example:

45 45 90 triangle
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides
45 45 90 triangle sides

Now let’s look at an example where we are given the length of the hypotenuse and need to find the values of the other two missing sides.

45 45 90 triangle sides
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula
45 45 90 triangle formula

Now try mastering the art of the 45º 45º 90º special triangle on your own!

Practice Questions: Find the value of the missing sides.

45 45 90 triangle formula
45 45 90 triangle formula

Solutions:

45 45 90 triangle formula

Still got questions? No problem! Don’t hesitate to comment with any questions or check out the video above. Happy calculating! 🙂

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