How to Make a Paper Cube Using Origami

Greeting math peeps and welcome to MathSux! In today’s post, we are going to have a bit of fun when finding the volume of a cube. We’ll start by going over how to make a paper cube using origami, then we will measure the dimensions of our real-world cube and find its volume. Hope you’re all feeling crafty and ready to take on this project with the Japanese art style of origami! If you don’t have any origami paper, please fill free to print out the origami guideline sheets included at the end of this post to follow along! Also, if the below written tutorial isn’t your style, watch the video up on YouTube and shown below. Hope you’re all having a great week and can find some fun in this post! Happy calculating!

How to Make a Paper Cube Using Origami:

How to Make a Paper Cube Using Origami

Step 1: Begin with six sheets of square origami paper (or cut out a square from any type of regular computer paper). If you want to really follow along, this paper is 4 inches by 4 inches and I have included a print out below.

How to Make a Paper Cube Using Origami

Step 2: Fold the top of the origami paper in half, then unfold.

How to Make a Paper Cube Using Origami

Step 3: Fold one side of the paper halfway to the center.

How to Make a Paper Cube Using Origami

Step 4: Fold the left side to the center of the paper and rotate the paper horizontally.

How to Make a Paper Cube Using Origami

Step 5: Take the bottom right corner and fold it to the top center point of the rectangle.

Math and art

Step 6: Now fold the top left corner and bring it to the bottom edge, so both folds align in the middle.

Math and art

Step 7: Next, Open both folds towards the center, undoing our last two steps.

geometry and origami

Step 8: Open the first flap on top, and fold in the top right triangle.

Math and art

Step 9: Unfold the bottom flap, and fold in the bottom left triangle.

Geometry and paper folding

Step 10: Now, we are going to fold down the top left corner to the center of the paper (just above the bottom flap).

Origami

Step 11: Fold up the bottom flap.

Math and Origami

Step 12: Now take the bottom right corner, and tuck it in under the top flap, towards the center.

geometry and origami

Step 13: Flip your piece of paper around.

How to Make a Paper Cube Using Origami

Step 14: Fold the corner, towards the center.

How to Make a Paper Cube Using Origami

Step 15: Now, fold in the bottom left edge, towards the center, to the top edge.

How to Make a Paper Cube Using Origami

Step 16: Unfold, the left and right edges we made to get a shape like we have above.

How to Make a Paper Cube Using Origami

Step 17: Repeat steps 1 through 16 5 more times! To have a total of six of the above shapes. We will need all of them to piece together our origami cube.

How to Make a Paper Cube Using Origami

Step 18: Connect each of our shapes, by placing the edge of each in top left the corner “pocket” of each shape.

How to Make a Paper Cube Using Origami

Step 19: Once they are all connected, we finally have created our origami cube!

Want to follow along with everything!? Check out the printable origami paper below to create your very own box!

Volume of a Cube:

Now that we have made our cube using paper folding, we can measure one side of our cube and find its volume! Please note that these measurements are based on the 4″X 4″ origami paper I used. If you would like to follow along with your own project, just download and print the activity sheet above!

Volume of a Cube

Did you create your origami cube with different-sized origami paper? Let us know what measurement you got for volume in the comments below!

Looking for more Origami + Math? Check out this post here to fold and find the volume of a real-world pyramid and rectangular prism. And for another math + crafty post, learn how to make a Mobius Band here!

What is your favorite way to combine math and the real world? Or math and art in general? Let me know in the comments and happy calculating!

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Looking for another math challenge? Check out this post on derivatives here!

How to Construct the Altitudes of a Triangle ⊿

Hi everyone and welcome back to another week of MathSux! For this week’s math lesson, I bring to you four constructions in one in How to Construct the Altitudes of a Triangle. In the video below, we will:

  • Define what an altitude is
  • Find the altitude of an acute, obtuse, and right triangles
  • Discover and define the orthocenter

I hope you find the below video helpful and interesting! Let me know in the comments below and happy calculating!

How to Construct the Altitudes of a Triangle :

An Altitude is a perpendicular line drawn from the vertex of a triangle to the opposite side, creating a 90º angle.

How to Construct the Altitudes of a Triangle

In the video above, we will look at how to find the altitude of an acute obtuse, and right triangle.

How to Find the Orthocenter of Triangle with a Compass:

The Orthocenter is a point where all three altitudes meet within a triangle.

How to Construct the Altitudes of a Triangle

In order to find the orthocenter using a compass, all we need to do is find the altitude of each vertex. The point at which they meet is the orthocenter. Check out the video above to see how this works step by step.

More Constructions?

If you’re looking for more constructions out there, be sure to check out the MathSux Constructions Playlist on YouTube for even more tutorials! I hope you find this video helpful and hope it motivates you to learn even more about constructions (or you know get through your homework/pass that test). Whatever reason you have to learn about constructions I hope this helps! Stay positive and happy calculating!

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

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

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And if you’re looking for even more geometry constructions, check out the link here!

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Similar Triangles

Legs of a Right Triangle (when an altitude is drawn)

Legs of a Right Triangle (when an altitude is drawn)

Greetings math peeps and welcome to another week of MathSux! In todays post we are going to explore how to find the legs of a right triangle when an altitude is drawn from the vertex to its hypotenuse. We are going to take this step by step on how to solve a problem like this. Although, I go over the long way to solve this problem, there is also short cut many people use called the “geometric means” which is also briefly mentioned in this post (under the Tip! section). Whichever method you choose, do what makes most sense to you! Happy calculating!

Similar Right Triangles (with Altitude drawn):

When two triangles have equal angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but if they have the same angles and the sides are in proportion, they are similar! Triangles can be proven similar by AA, SAS, or SSS. For more on similar triangles, check out this post here.

There is a special type of scenario that happens with similar right triangles. When an altitude is drawn from the vertex of a right triangle, it forms two smaller triangles, which creates three right triangles that are similar to the original triangle, based on Angle Angle (AA). Check out the example to see how it works!

In triangle ABC, an altitude is drawn from angle A to its hypotenuse BC. Notice that this creates three 90º right triangles in total (ABC, ABD, and ADC). 

Drawing an altitude created three total right triangles, broken out below:

If you take a closer look, at the triangles above you’ll notice that these two new triangles (ABD and ADC) share an angle with the original bigger triangle ABC and a 90º angle. This makes each of the new triangles similar to the original triangle by AA. We can therefore say that:

How are the New Right Triangles Similar?

1) Triangle ABC ~ ADC

2) Triangle ABC ~ ADB

Ready for an Example?  Lets check out the one below!

Legs of a Right Triangle

Step 1: To find the length of the missing legs of a right triangle, first, let’s separate each right triangle to see what we’re working with, along with the values of each length that was provided.

Legs of a Right Triangle

Step 2: We want to find the length of side AC, so let’s use the triangles whose sides include AC.  This leads us to use triangles ABC and ADC.

Legs of a Right Triangle

Step 3:  Now, let’s set up our proportion to find our missing side.  To make it easier for ourselves let’s first flip and rotate triangle ADC to line up with triangle ABC.

Legs of a Right Triangle
Legs of a Right Triangle

Think you’re ready to try some practice questions on your own? Check out the ones below!

Practice Questions:

Solutions:

Still got questions? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating!

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And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Similar Triangles

How to Construct the Altitude of a Triangle with Compass

Mobius Bands

Greeting math friends and welcome to another fabulous week of MathSux! Today we are going to look at how to make a Mobius Strip otherwise known as “mobius bands.” Whatever you call them, they are a one of a kind shape that leads to all sorts of questions and math exploration. So let’s get to it! Let’s start off by explaining what a mobius strip is:

What are Mobius Bands?

A mobius strip (or band) is a two-dimensional shape that only has one surface.  Invented by German mathematician August Ferdinand Mobius and also independently by Johann Benedict Listing in the 1800s, it is considered a mathematical phenomenon! We are going to create our own mobius strip today to understand why this shape is so fascinating!

*Note: You may also see different spelling of the mobius band: including Möbius or Moebius

How to Make a Mobius Strip:

Mobius Bands

Step 1: First, let’s gather out materials. For this activity we will need a piece of paper, scissors, tape, and a pencil.

Mobius Bands

Step 2: Let’s cut out a rectangle. The size of the one shown above is 11 X 1 inches. Once you have your rectangle, take one end of the rectangle and give it a half twist.

Mobius Bands

Step 3: Now take the half twisted piece and attach it to the other end of the rectangle with a piece of tape. Behold the beauty of our mobius strip!

How to make a mobius strip

Step 4: Now we are going to take a pen or pencil and draw a line going all around the mobius strip without lifting the pen. See if you can move he mobius strip along, while the pencil remains where it is.  Notice anything special happen? We created a line around the entire shape without lifting our pencil!

Real-World Mobius:

Have you been pondering where we can find Mobius bands in the real world? I thought so! Take a look at the list below:

  • Printer ink cartridges
  • Serpentine Belt in a car
  • VHS tapes (if anyone remembers or knows what those are)
  • Can you think of anymore? Let me know in the comments below!

Worksheet:

I also made this FREE worksheet to go along with the above video and lesson for anyone interested. Let me know if this is helpful! 🙂

If you’re looking for more fun math projects, check out my “Just for Fun” page here. And if you want to get to see the latest MathSux content, don’t forget to follow along with us and subscribe via the links below. Thanks so much for stopping by and happy calculating! 🙂

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Math Jewelry Review: Hanusa Design

Math Jewelry Review Interview

Hi Everyone and welcome to MathSux! I was sent three pieces of math jewelry from the jewelry brand, Hanusa Design. Math jewelry?! What is that?! Each piece you see here was inspired by mathematical art and created using 3D printing. I’m wearing the mobius necklace above and below you’ll also see mini pi and golden ratio earrings. For the full un-boxing and math jewelry review check out the video below and if you’re interested and want to learn more about Hanusa designs and the 3-d printing process, keep reading for the full interview I had with founder, Chirs Hanusa himself in this blog post.

Math Jewelry Review Interview

Math Jewelry Review:

Hanusa Design Interview with Founder, Chris Hanusa:

1.What made you start Hanusa Design? What led you to making jewelry as someone interested in math?

My adventure into 3D printing started in 2015 when I was updating a course in Mathematica I was teaching at Queens College. I was intrigued by 3D printing and I noticed that it was possible to use Mathematica as 3D design software, so I included a 3D design project as part of the class. As my students and I explored 3D printing, I recognized the universal appeal of the beauty and precision of mathematical concepts, and turned these ideas into jewelry. In turn, I founded Hanusa Design in 2017.

2. The jewelry is made through a 3D printing process. Can you explain the process from start to end? Is there a difference between the use of metal vs. nylon?

The design process starts with a mathematical concept that I’ve seen in my research, in mathematical texts, or as “found math” in the real world. I use Mathematica to do the 3D design, using three-dimensional coordinates, parametric functions, and aesthetic choices that turn the idea into a 3D model. The model is then exported directly from Mathematica to an STL file, which is basically a way to represent the boundary of the 3D object as a collection of triangles. The STL files are then sent to a 3D printer.

Once there, the same STL file can be used to create a nylon or metal piece of jewelry. The colorful nylon pieces are created using a SLS (selective laser sintering) process, where a thin layer of nylon powder is spread out and precisely fused to the previous layer using a laser. The excess powder is removed and then I hand dye the models using fabric dye. In contrast, the metal pieces are created using a lost-wax casting process. First, the models are 3D printed in high-resolution wax, then a plaster mold is created around the wax, and then the wax is replaced by molten metal.

3.I saw on your website that you are a mathematician and mathematical artist. Do you teach mathematical art at a university? If so, what types of topics do you cover? What is your favorite form of mathematical art?

I do teach two different courses that involve mathematical art. I teach a class called Mathematical Design that explores art that is created with functions, parametric functions, and polar functions using Desmos. This year I hope to give my Mathematical Design students the opportunity to use the Queens College Makerspace to take their digital art and bring it into reality using a laser cutter, a sewing machine, or a pen plotter. My other class is called Mathematical Computing. In this class I teach my students the computational software Mathematica, including how to use the software to do 3D modeling.  By the end of the semester, the students have designed and 3D printed a mathematical sculpture.

I suppose my favorite type of mathematical art is the visualization of complex mathematical concepts. It’s hard to understand certain concepts, like constructions in the fourth (or higher) dimension. Any picture or sculpture that helps clarify these difficult ideas is important, and it’s even better when it’s created with an eye to the aesthetic. I highly recommend any work by Henry Segerman.

4.I saw on your website that Hanusa Design has been featured in both New York Fashion Week (NYFW) and MoMath.  In what capacity?  Are you being featured in this week’s NYFW Fall 2021?

A wide variety of Hanusa Design jewelry has been available in the gift shop at the National Museum of Mathematics since 2018, including my dangling cubes earrings and interlocking octahedron necklace. I was asked to participate in a New York Fashion Week-adjacent show in Spring 2019 and enjoyed the experience. I am looking forward to eventually participate in New York City Jewelry Week.

5.Where can we find Hanusa Design, in stores or online? 

A variety of museums, galleries, and stores stock Hanusa Design Jewelry. As I mentioned before, it is available at the National Museum of Mathematics in New York, NY. It can also be found at the Exploratorium in San Francisco, CA, the Queens Museum in Queens, NY, Gallery North in Setauket, NY, Because Science in Vienna, VA, and in the Wolfram Store in Champaign, IL.

Hanusa Design Discount:

Hanusa has been kind enough to give MathSux readers an exclusive 10% off discount with the code ‘MATHROCKS’ now through December 31st 2021. I know its a bit early but this would be the perfect gift for the holiday season which is coming around the corner! Check out the full collection on their website here for even more designs and colors!

MathSux Giveaway:

math jewelry review

I’m going to be giving away a pair of pi earrings to one lucky MathSux reader! All you have to do is watch the YouTube video above, subscribe to MathSux, and comment below.

What do you guys think of Hanusa Design? Would you wear mathematical jewelry? What about the 3D printing process? Don’t forget to check out the video above for the full math jewelry review. Let me know what you guys think and happy calculating! 🙂

If you’re looking for more mathematical reviews, check out my review on the NumWorks calculator here.

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Angle Bisector Definition & Example

Hi everyone and welcome to another fabulous week of MathSux! I bring to you the first construction of the back to school season! In this post, we are going to go over the angle bisector definition and example. First, we will define what an angle bisector is, then we’ll take our handy dandy compass and straight edge to construct an angle bisector that will bisect an angle for any size! Check out the video and GIF below for more and happy calculating! 🙂

What is an Angle Bisector?

A line that evenly cuts an angle into two equal halves, creating two equal angles.

Angle Bisector Example:

Angle Bisector Definition & Example

Step 1: Place the point of your compass on the point of the angle.

Step 2: Draw an arc that intersects both lines that stem form the angle you want to bisect.

Step 3: Take the point of your compass to where the lines and arc intersect, then draw an arc towards the center of the angle.

Step 4: Now keeping the same distance on your compass, take the point of your compass and place it on the other point where both the line and arc intersect, and draw another arc towards the center of the angle.

Step 5: Notice we made an intersection!? Where these two arcs intersect, mark a point and using a straight edge, connect it to the center of the original angle. 

Step 6: We have officially bisected our angle into two equal 35º halves.

*Please note that the above example bisects a 70º angle, but this construction method will work for an angle of any size!🙂

What do you think of the above angle bisector definition & example? Do you use a different method for construction? Let me know in the comments below! 🙂

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

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Looking for more constructions? Check out how to construct a square inscribed in a circle and an equilateral triangle by clicking on their respective links! And if you’re looking for even more geometry constructions, check out the link here!

AnAngle Bisector Definition & Example

How to Construct a Perpendicular Line through a Point on the Line

Greetings math peeps and welcome to another week of MathSux! In this post, we will learn how to construct a perpendicular line through a point on the line step by step. In the past, we learned how to construct a perpendicular bisector right down the middle of the line, but in this case we will learn how to create a perpendicular line through a given point on the line (which is not always in the middle). Following along with the GIF or check out the vide below. Thanks for stopping by and happy calculating! 🙂

What are Perpendicular Lines ?

  • Lines that intersect to create four 90º angles about the two lines.
How to Construct a Perpendicular Line through a Point on the Line

What is happening in this GIF?

Step 1: First, we are going to gather materials, for this construction we will need a compass, straight edge, and markers.

Step 2: Notice that we need to make a perpendicular line going through point B that is given on our line.

Step 3: Open up our compass to any distance (something preferably short though to fit around our point and on the line).

Step 4: Place the compass end-point on Point B, and draw a semi-circle around our point, making sure to intersect the given line.

Step 5: Open up the compass (any size) and take the point of the compass to the intersection of our semi-circle and given line.  Then swing our compass above the line.

Step 6: Keeping that same length of the compass, go to the other side of our point, where the given line and semi-circle connect.  Swing the compass above the line so it intersects with the arc we made in the previous step.

Step 7: Mark the point of intersection created by these two intersecting arcs we just made and draw a perpendicular line going through Point B!

Best Geometry Tools!

Looking to get the best construction tools? Any compass and straight-edge will do the trick, but personally, I prefer to use my favorite mini math toolbox from Staedler. Stadler has a geometry math set that comes with a mini ruler, compass, protractor, and eraser in a nice travel-sized pack that is perfect for students on the go and for keeping everything organized….did I mention it’s only $7.99 on Amazon?! This is the same set I use for every construction video in this post. Check out the link below and let me know what you think!

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! And if you’re looking for even more geometry constructions, check out the link here!

The Original Spirograph: Math + Art

Happy Summer everyone! Now that school is out, I thought we could have a bit of fun with Math and Art! In this post, we will go over how to make a the original spirograph (by hand) step by step using a compass and straight edge. Follow along with the video below or check out the tutorial in pictures in this post. Hope everyone is off to a great summer. Happy calculating! 🙂

What is a Spirograph?

The childhood toy we all know and love was invented by Denys Fisher, a British Engineer in the 1960’s.But the method of creating Spirograph patterns was invented way earlier by engineers and mathematicians in the 1800’s.

The Original Spirograph (by hand):

The Original Spirograph

Step 1: Gather materials, for this drawing, we will need a compass and straight edge.

The Original Spirograph

Step 2: Using our compass, we are going to open it to 7 cm and draw a circle.

The Original Spirograph

Step 3: Next, we are going to open the compass to 1cm, making marks all around the circle, keeping that same distance on the compass.

The Original Spirograph

Step 4: Draw a line connecting two points together (any two points some distance apart will do).

The Original Spirograph

Step 5: Now, we are going to move the straight edge forward by one point each and connect the two points with another line.

The Original Spirograph

Step 6: Continue this pattern of moving the ruler forward by one point and connecting them together all the way around.

Step 7: We have completed our Spirograph drawing! Try different sized circles, points around the circle, colors, and points of connections to create different types of patterns and have fun! 🙂

Spirograph Deluxe Art Set:

Want to try the one and only toy spirograph on your own!? Check out this Deluxe Spirograph set that brings mathematics and art together! Let your artistic creativity run free by experimenting with different-sized spirograph tools and colorful pens! Great for kids or math nerd adults, and easily available at Amazon for $23.99. Let me know what you think if you end up getting a spirograph set or if you already have one!

Still got questions or want to learn more about Math+ Art? No problem! Don’t hesitate to comment with any questions below. Thanks for stopping by and happy calculating! 🙂

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For more Math + Art, check out this post on Perspective Drawing here. And for another math + art project, check out this post on Mobius Bands!

Geometry Cheat Sheet & Review

Greeting math peeps! As promised here is the Geometry Cheat Sheet and Review made just for you to prepare for finals. On this page, you’ll also find links to the Geometry lesson playlist, the NYS Geometry Common Core Regent’s Playlist, and the library of Geometry blog posts. Hope you find these resources helpful as the end of the school year approaches. Good luck on finals and happy calculating! 🙂

Geometry Cheat Sheet:

Download and print the below .pdf for a quick and easy guide of everything you need to know for finals; From formulas to shapes, it’s on here.

Geometry Review Playlist:

Looking for a more detailed review? Check out the Youtube playlist for Geometry below. It includes every MathSux video related to Geometry and will be sure to help you ace the test!

Geometry Common Core Regents Review:

This playlist is made especially for New York State dwellers as it goes over each and every question of the NYS Common Core Regents. Perfect if you are stuck on that one question! You will surely find the answer here.

Geometry Math Lessons for Review:

For anyone in search of blog posts and practice questions, check out MathSux’s entire Geometry library organized by topic here.

Geometry Cheat Sheet & Review

Still got questions? No problem! Don’t hesitate to comment with any questions below. Also, if you find you need some motivation, check out my 6 tips and tricks for studying math here! Thanks for stopping by and happy calculating! 🙂

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Get everything you need to know with this Geometry Cheat Sheet and Review! Download and print the pdf for reviewing Geometry or check out the video playlists for a more in-depth review of each topic. If you are living in NYS, you also might want to check out the NYS Regents Common Core Video as needed!

Similar Triangles: AA, SSS, & SAS

Happy Wednesday math peeps! In today’s post, we are going to go over Proving Similar Triangles, by going over:

1) What it means when two triangles are similar?

2) How to prove two triangles similar?

3) How to find missing side lengths given triangles are similar?

For even more practice, don’t forget to check out the video and practice problems below. Happy calculating! 🙂

What are Similar Triangles?

When two triangles have congruent angles and proportionate sides, they are similar.  This means they can be different in size (smaller or larger) but as long as they have the same angles and the sides are in proportion, they are similar! We use the “~” to denote similarity.

In the Example below, triangle ABC is similar to triangle DEF:

Similar Triangles

How can we Prove Triangles Similar?

There are three ways to prove similarity between two triangles, let’s take a look at each method below:

Angle-Angle (AA): When two different sized triangles have two angles that are congruent, the triangles are similar.  Notice in the example below, if we have the value of two angles in a triangle, we can always find the third missing value which will also be equal.

Similar Triangles

Side-Side-Side (SSS): When two different sized triangles have three corresponding sides in proportion to each other, the triangles are similar. 

Similar Triangles

Side-Angle- Aside (SAS): When two different sized triangles have two corresponding sides in proportion to each other and a pair of congruent angles between each proportional side, the triangles are similar. 

Similar Triangles

Let’s look at how to apply the above rules with the following Example:

Step 1: Since, we know the triangles ABC and DEF are similar, we know that their corresponding sides must be in proportion! Therefore, we can set up a proportion and find the missing value of length EF by cross multiplying and solving for x.

Similar Triangles

Practice Questions:

1) Are the following triangles similar?  If so, how? Explain.

2) Are the following triangles similar?  If so, how? Explain.

3) Given triangle ABC is similar to triangle DEF, find the side of missing length AB.

4) Given triangle ABC is similar to triangle PQR, find the side of missing length AC.

Solutions:

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

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Check out more posts on Geometry here!

And if you’re looking for more lessons on triangles, check out these related posts below:

Congruent Triangles

Similar Triangles

45 45 90 special triangles

30 60 90 special triangles

Construct the Altitude of a Triangle

Legs of a Right Triangle (when an altitude is drawn)