None of their interior angles is greater than 180. In that case, you get two trapezoids, and you can calculate the area of the hexagon as the sum of them. Since the sum of internal angles in one triangle is 180, it is concluded that 6 triangles, side by side, should measure up to 6x180=1080. 3! Become a Study.com member to unlock this answer! By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. How many triangles can be formed with the given information? You could also combine two adjacent triangles to construct a total of 3 different rhombuses and calculate the area of each separately. Solve My Task. The perimeter of an octagon is expressed in linear units like inches, cm, and so on. Alternatively, one can also think about the apothem as the distance between the center, and any side of the hexagon since the Euclidean distance is defined using a perpendicular line. Since the interior angles of each triangle totals 180, the hexagon's interior angles will total 4(180), or 720. In the adjoining figure of a pentagon ABCDE, on joining AC and AD, the given pentagon is divided into three triangles i.e. What is the sum of the interior angles of a hexagon? Three sprinters A, B, and C begin running from points A 1 , B 1 and C 1 respectively. The following properties of an octagon help us to identify it easily. $$N_0=\color{red}{\frac{n(n-4)(n-5)}{6}}$$ In photography, the opening of the sensor almost always has a polygonal shape. Then, after calculating the area of all the triangles, we add their areas to get the area of the octagon. Let's say the apothem is 73 cm. For example, suppose you divide the hexagon in half (from vertex to vertex). Triangle = 3 sides, 0 diagonal, 1 triangle, 2.) How many triangles can be formed by the vertices of a regular polygon of $n$ sides? The way that 120 angles distribute forces (and, in turn, stress) amongst 2 of the hexagon sides makes it a very stable and mechanically efficient geometry. How are probability distributions determined? In this case, there are 8 sides in an octagon. In a regular octagon, each interior angle is 135. These tricks involve using other polygons such as squares, triangles and even parallelograms. Why are physically impossible and logically impossible concepts considered separate in terms of probability? Each sprinter traverses her respective triangular path clockwise and returns to her starting point. This same approach can be taken in an irregular hexagon. 3. Most people on Quora agreed that the answer is 24, with each row containing six triangles. let me set of this numbers, where in every number corresponds with a number of sides of every polygon.. ( 3,4,5,6,7,8,9,10 ),,let me answer how many diagonal can be drawn from the fixed vertex?? In a regular hexagon, four triangles can be created using diagonals of the hexagon from a common vertex. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Functional cookies help to perform certain functionalities like sharing the content of the website on social media platforms, collect feedbacks, and other third-party features. We have discussed all the parameters of the calculator, but for the sake of clarity and completeness, we will now go over them briefly: Everyone loves a good real-world application, and hexagons are definitely one of the most used polygons in the world. The number of polygons with k sides that can be formed by joining them is C n k. Since no 3 vertices in given heptagon are collinear, then the number of triangles possible is C 7 3 = 35. Learn more about Stack Overflow the company, and our products. How many different triangles can be formed with the vertices of an octagon? Total of 35 triangles. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. If you don't remember the formula, you can always think about the 6-sided polygon as a collection of 6 triangles. Apothem is the line segment that is drawn from the center and is perpendicular to the side of the hexagon. How many different types of triangles can be formed with the vertices of a balanced hexagon? There are 20 diagonals in an octagon. 3 How many triangles can be formed by joining the vertices of Heptagonal? Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. Find the total number of diagonals contained in an 11-sided regular polygon. The best way to counteract this is to build telescopes as enormous as possible. Step-by-step explanation: 6 triangles are formed by the three diagonals through the center. In an 11-sided polygon, total vertices are 11. You also have the option to opt-out of these cookies. The sum of exterior angles of an octagon is 360. There are a total of 8 sides in an octagon, and those eight sides are parallel to their respective opposite side in the case of a regular octagon. Now, the 11 vertices can be joined with each other by 11C2 ways i.e. Regular or not? Solve Now. How many triangles are in a heptagon? On the circumference there were 6 and then 12 on the second one. None B. You can see a similar process in the animation above. Our hexagon calculator can also spare you some tedious calculations on the lengths of the hexagon's diagonals. Convex octagons bulge outwards, whereas concave octagons have indentations (a deep recess). Can a hexagon be divided into 4 triangles? The number of triangles is n-2 (above). The sum of the interior angles of an octagon can be calculated using the formula, Sum of interior angles of a polygon = (n - 2) 180, where 'n' represents the number of sides in the polygon. The formula to calculate the area of a regular hexagon with side length s: (3 3 s^2)/2. The cookie is set by GDPR cookie consent to record the user consent for the cookies in the category "Functional". Check out our online resources for a great way to brush up on your skills. Therefor the interior angles of the polygon must be the sum of all the triangles' interior angles, or 180 (n-2). The answer is 3, that is, approximately 1.73. The area of a triangle is \displaystyle 0.5\cdot b\cdot h. Since, How to determine greatest common monomial factor, How to find the height of a trapezium calculator, How to find the mean of a frequency distribution chart, Post office term deposit interest calculator, Va disabilty rate calculator with bilateral factor. What is the difference between Mera and Mujhe? By drawing a line to every other vertex, you create half as many equal areas (3 equal areas). 5 triangles made of 5 shapes. Now by subtracting n with nC2 ways, the formula obtained is n(n-3)/2. Pentagon = 5 sides, 5 diagonal formed, 40 triangles formed, 4.) Focus on your job You can provide multiple ways to do something by listing them out, providing a step-by-step guide, or giving a few options . We divide the octagon into smaller figures like triangles. vegan) just to try it, does this inconvenience the caterers and staff? A regular hexagon has a perimeter of 30 m. What is the area of the hexagon? Each is an integer and a^2 + b^2 = c^2 . In a regular octagon, by joining one vertex to the remaining non-adjacent vertices, 6 triangles can be formed. A regular octagon is an example of a convex octagon. Regular hexagon is when all angles are equal and all sides are equal. This fact makes it much easier to calculate their area than if they were isosceles triangles or even 45 45 90 triangles as in the case of an octagon. Assume you pick a side $AB$. When we plug in side = 2, we obtain apothem = 3, as claimed. This same approach can be taken in an irregular hexagon. The hexagon is an excellent shape because it perfectly fits with one another to cover any desired area. Hence no of triangles= n How do I connect these two faces together? If three diagonals are drawn inside a hexagon with each one passing through the center point of the hexagon, how many triangles are formed? How many triangles make a hexagon? This is because of the relationship apothem = 3 side. THE PENTAGON HAS 3 TRIANGLES. It only takes a minute to sign up. There will be a whole section dedicated to the important properties of the hexagon shape, but first, we need to know the technical answer to: "What is a hexagon?" Fill order form. What is the point of Thrower's Bandolier? The perimeter of a polygon is the total length of its boundary. Therefore, number of triangles $N_1$ having only one side common with that of the polygon $$N_1=\text{(No. No triangle. The interior angles of a triangle always sum to 180. For a random (irregular) hexagon, the answer is simple: draw any 6-sided shape so that it is a closed polygon, and you're done. The formula to calculate the area of a regular octagon is, Area of a Regular Octagon = 2a2(1 + 2); where 'a' is any one side length of the octagon. Where A means the area of each of the equilateral triangles in which we have divided the hexagon. In a convex 22-gon, how many. Can't believe its free would even be willing to pay for a pro version of this app. 1) no of triangles with only one side common with polygon, if we take any one side of a n-sided polygon and join its vertices to the remaining vertices, except the vertices adjacent to vertices of the line taken above, we get triangles with only one side as common i.e. The inradius is the radius of the biggest circle contained entirely within the hexagon. Every polygon is either convex or concave. =7*5=35.. $\forall \ \ \color{blue}{n\geq 3}$, Consider a side $\mathrm{A_1A_2}$ of regular n-polygon. Feel free to play around with different shapes and calculators to see what other tricks you can come up with. a) 5 b) 6 c) 7 d) 8. If we draw the other four missing chords and the one missing radius, we obtain too many triangles to count (I stopped at thirty). How many triangles exist in the diagonals intersections of an heptagon? How many triangles can we form if we draw all the diagonals . 3! Let's draw the angle bisectors of two adjacent interior angles, and call their point of intersection O: It is easy to see that OAB is equilateral - mBAF = mABC = 120, as interior angles of a regular hexagon. Math can be daunting for some, but with a little practice it can be easy! Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. We have to select 3 vertices out of n vertices (n=6 for hexagon) So, no of possible triangles : 6 C 3 = 6! six (33 s2)/2 where 's' is the side length. Q: In a convex 22-gon, how many diagonals can be drawn from one vertex? You will notice that with one or two chopsticks, for example, it is impossible to form a triangle, and that with three chopsticks only one triangle can be formed: While with 11 chopsticks four different triangles can be formed. If all of the diagonals are drawn from a vertex of a quadrilateral, how many triangles are formed? The number of triangles with no side common with regular polygon having $n$ number of sides $$=^nC_3-n-n(n-4)$$. If the shape is closed, made up of straight lines, and has eight sides, we call it an octagon. The easiest way is to use our hexagon calculator, which includes a built-in area conversion tool. Is it possible to rotate a window 90 degrees if it has the same length and width? How many axes of symmetry does an equilateral triangle have? Connect and share knowledge within a single location that is structured and easy to search. From bee 'hives' to rock cracks through organic chemistry (even in the build blocks of life: proteins), regular hexagons are the most common polygonal shape that exists in nature. 3 This rule works because two triangles can be drawn inside the shapes. How many diagonals are in a pentagon, an octagon, and a decagon? Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. How many equal sides does an equilateral triangle have? Another important property of regular hexagons is that they can fill a surface with no gaps between them (along with regular triangles and squares). What is the area of a regular hexagon inscribed in a circle of So, the area of hexagon will be 6 times this area because the hexagon is divided into 6 equilateral triangles. This cookie is set by GDPR Cookie Consent plugin. 2 What is the number of triangles that can be formed whose vertices are the vertices of an octagon? Avg. =20 I have no idea where I should start to think. Number of triangles contained in a hexagon = 6 - 2 = 4. How many triangles can be formed from the vertices of a polygon of $n$ sides if the triangle and the polygon may not share sides? How many obtuse angles does a rhombus have. The octagon in which one of the angles points inwards is a concave octagon. Then, you have two less points to choose from for the third vertex. 0 0 Similar questions But the DIAGONAL too is made from 3 points : 2vertices and 1 centre.. And here we make a line and not a triangle.. In case of a regular octagon, the perimeter can be divided by 8 to get the value of one side of the octagon. Exploring the 6-sided shape, Hexagon area formula: how to find the area of a hexagon. Log in, WhatsApp Guess the Toothpaste brand names puzzle, Guess Marwadi Names from whatsapp emoticons. Multiply the choices, and you are done. The sides of a regular octagon are of equal length. It is an octagon with unequal sides and angles. rev2023.3.3.43278. there are 7 points and we have to choose three to form a triangle, Learn Sentence Correction Strategies with 780 Scorer. of triangles corresponding to one side)}\text{(No. This result is because the volume of a sphere is the largest of any other object for a given surface area. In fact, it is so popular that one could say it is the default shape when conflicting forces are at play and spheres are not possible due to the nature of the problem. satisfaction rating 4.7/5. How many edges does a 20 sided polygon have? Answer with solution Again it is good to use symmetry here, we can brake this image into six small triangles each formed by one of the side of the hexagon and each of the triangle is divided in half by a line. Octagons that have equal sides are known as regular octagons, while irregular octagons have different side lengths. It only takes a minute to sign up. See what does a hexagon look like as a six sided shape and hexagon examples. A pentacle is a figure made up of five straight lines forming a star. Therefore, 6 triangles can be formed in an octagon. One of the biggest problems we experience when observing distant stars is how faint they are in the night sky. How to show that an expression of a finite type must be one of the finitely many possible values? Their length is equal to d = 3 a. It's frustrating. 3! If all of the diagonals are drawn from a vertex of a hexagon, how many triangles are formed? A place where magic is studied and practiced? using the hexagon definition. How many different triangles, if any, can be drawn with one 90 degrees angle and side lengths of 5 cm and 12 cm? In other words, an n-sided polygon has n-vertices which can be joined with each other in nC2 ways. How many acute angles are in a right triangle? For the regular hexagon, these triangles are equilateral triangles. Why the $\binom{6}{3}$ doesn't work to get 18 is obvious: you create triangles using intersection points. . The interior angles are greater than 180, that is, at least one angle is a reflex angle. Here is how you calculate the two types of diagonals: Long diagonals They always cross the central point of the hexagon. Great learning in high school using simple cues. However, you may visit "Cookie Settings" to provide a controlled consent. But, each diagonal is counted twice, once from each of its ends. Requested URL: byjus.com/question-answer/how-many-triangles-can-be-formed-by-joining-the-vertices-of-a-hexagon/, User-Agent: Mozilla/5.0 (Windows NT 10.0; Win64; x64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/103.0.0.0 Safari/537.36. OA is Official Answer and Stats are available only to registered users. What is the point of Thrower's Bandolier. What do a triangle and a hexagon have in common? This also explains why squares and hexagons tessellate, but other polygons like pentagons won't. A square will form corners where 4 squares meet, since 4 90 = 360. The sum of the interior angles of an octagon is 1080, and the sum of its exterior angles is 360. Equivalent Fractions in Hexagon Drawing a line to each vertex creates six equilateral triangles, which is six equal areas. $A_4, \ A_5,\ A_6, \ \ldots \ A_{n-1}$ to get triangles with only one side common. Is it not just $ ^{n}C_3?$ ..and why so many views? The formula that is used to find the number of diagonals in any polygon is, Number of diagonals = n(n-3)/2; where 'n' represents the number of sides of the polygon. There is more triangle to the other side of the last of those diagonals. The sum of all interior angles of a triangle will always add up to 180 degrees. C. This way, we have 4 triangles for each side of the octagon. The answer is 3/4, that is, approximately, 0.433. No, an octagon is not a quadrilateral. As shown in attachment if we a diagonals from one vertex then only 3 diagonals are drawn which results into 4 triangles. A regular hexagon is composed of 12 congruent { 30^o,60^o,90^o } triangles. Area of octagon = 2a2(1 + 2), Substituting the value of 'a' = 6, Area of octagon = 2 (62) (1 + 2) = 72 (1 + 2) = 173.8 square units. All the interior angles are of different measure, but their sum is always 1080. Bubbles present an interesting way of visualizing the benefits of a hexagon over other shapes, but it's not the only way. A fascinating example in this video is that of the soap bubbles. How many vertices does a right triangle have? Using this, we can start with the maths: Where A means the area of each of the equilateral triangles in which we have divided the hexagon. but also in many other places in nature. The length of the sides can vary even within the same hexagon, except when it comes to the regular hexagon, in which all sides must have equal length. There are five arrangements of three diagonals to consider. Therefore, there are 20 diagonals in an octagon. No, all octagons need not have equal sides. A quadrilateral is a closed shape with four vertices and four sides and an octagon has 8 sides and 8 vertices. Convex octagons are those in which all the angles point outwards. 5 How many triangles can be formed by joining the vertices of a regular octagon such that at least one side of the triangle is same as the side of the octagon? What am I doing wrong here in the PlotLegends specification? How many triangles can be made with 13 toothpicks? In an equilateral triangle, each vertex is 60. Does a barbarian benefit from the fast movement ability while wearing medium armor? We sometimes define a regular hexagon using equilateral triangles, or triangles in which all of the sides have equal length. Here we explain not only why the 6-sided polygon is so popular but also how to draw hexagon sides correctly. How many triangles can be formed by joining the vertices of Heptagonal? It is expressed in square units like inches2, cm2, and so on. There are 3 diagonals, so 3 triangles counted in 35 are actually a LINE.. Total left 35-3=32. Here, n = 8, so after substituting the value of n = 8 in this formula, we get, 1/2 n (n - 3) = 1/2 8 (8 - 3) = 20. The number of vertices in a triangle is 3 . If you draw all diagonals of a regular hexagon you have $3 \cdot 6 = 18$ possible triangles, but 3 of those are the same (the equilateral triangles) so we have $18 - 3 = 15$ possible triangles. The perimeter of an octagon is the total length of its boundary. 3 More answers below . In a regular hexagon, four triangles can be created using diagonals of the hexagon from a common vertex. The above formula $(N_0)$ is valid for polygon having $n$ no. The angle bisectors create two half angles which measure 60: mOAB=mOBA=60. . In a regular hexagon, four triangles can be created using diagonals of the hexagon from a common vertex. Learn the hexagon definition and hexagon shape. This website uses cookies to improve your experience while you navigate through the website. If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? The next case is common to all polygons, but it is still interesting to see. Observe the question carefully and find out the length of side of a regular hexagon. Math is a subject that can be difficult for some students to grasp. We know that in a regular octagon, all the sides are of equal length. The number of triangles that make a hexagon depends on the type of hexagon and how we Our experts can answer your tough homework and study questions. Complete step by step solution: The number of vertices in a hexagon is 6 . The sum of all the interior angles in an octagon is always 1080. These cookies ensure basic functionalities and security features of the website, anonymously. Polygon No. Find the value of $\frac{N}{100}$. Clear up mathematic problems How many congruent sides does an equilateral triangle have? How many segments do a 7 sided figure have joined the midpoints of the sides? As for the angles, a regular hexagon requires that all angles are equal and sum up to 720, which means that each individual angle must be 120. The octagon in which each interior angle is less than 180 is a convex octagon. The answer is not from geometry it's from combinations. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. This fact is true for all hexagons since it is their defining feature. We cannot go over all of them in detail, unfortunately. We can find the area of a regular hexagon with Therefore, the formula to find the area of 357+ PhD Experts 4.5/5 Quality score 49073 Clients Get Homework Help Example 2: Find the length of each side of a regular octagon if the perimeter of the octagon is 160 units. This is called the angle sum property of triangle. How many maximum number of isosceles triangle are possible in a regular polygon of $n$ sides? The result is that we get a tiny amount of energy with a longer wavelength than we would like. How many signals does a polygon with 32 sides have? This honeycomb pattern appears not only in honeycombs (surprise!) case II, 3) triangles with no side common There 6 equilateral triangles in a regular hexagon. How many triangles can be constructed with sides measuring 6 cm, 2 cm, and 7 cm? This is a significant advantage that hexagons have. An alternated hexagon, h{6}, is an equilateral triangle, {3}. $\mathrm{A_1, \ A_2,\ A_3, \ A_3, \ldots , A_{n-1}}$, $$N=\text{number of ways of selecting 3 vertices out of n}=\color{}{\binom{n}{3}}$$, $$N_1=\text{(No. We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. The 120 angle is the most mechanically stable of all, and coincidentally it is also the angle at which the sides meet at the vertices when we line up hexagons side by side. 9514 1404 393. If all of the diagonals are drawn from a vertex of a quadrilateral, how many triangles are formed? The most unexpected one is the shape of very bright (point-like) objects due to the effect called diffraction grating, and it is illustrated in the picture above. If $N_0$ is the number of triangles having no side common with that of the polygon then we have $$N=N_0+N_1+N_2$$ $$N_0=N-N_1-N_2$$ $$=\binom{n}{3}-(n-4)n-n$$ $$=\color{}{\frac{n(n-1)(n-2)}{6}-n^2+3n}$$ The word 'Octagon' is derived from the Greek word, 'oktgnon' which means eight angles. Here, the perimeter is given as 160 units. Yes you create 4 triangles with a sum of 720, but you would have to subtract the 360 that are in the middle of the quadrilateral and that would get you back to 360. The side length of an octagon can be calculated if the perimeter and the other sides are given. 10 triangles made of 2 shapes. Easy Solution Verified by Toppr There are 6 vertices of a hexagon. [ n C r = n! The area of an octagon is the total space occupied by it. 1 See answer Advertisement Edufirst Quadrilateral: two (you can only trace one diagonal and it forms two triangles) Hexagon: four (you can trace thre diagonals and four triangles are formed) Octagon: six (you can trace five diagonals and six triangles are formed) Degagon: eight (you can trace seven diagonals and eight triangles are formed) hexagon = 6 sides, 9 diagonal formed, ????????? regular octagon regular hexagon regular decagon |regular dodecagon mber of triangles ed in 4 O prior angle sum is 1.800 amber of triangles O ned is 6 2. With our hexagon calculator, you can explore many geometrical properties and calculations, including how to find the area of a hexagon, as well as teach you how to use the calculator to simplify any analysis involving this 6-sided shape. How many triangles can be formed by joining the vertices of a hexagon ? After multiplying this area by six (because we have 6 triangles), we get the hexagon area formula: We hope you can see how we arrive at the same hexagon area formula we mentioned before. Analytical cookies are used to understand how visitors interact with the website. https://www.youtube.com/watch?v=MGZLkU96ETY. Draw a circle, and, with the same radius, start making marks along it. $\implies$ can also be written as sum of no of triangles formed in the following three cases, 1) no of triangles with only one side common with polygon, The easiest way to find a hexagon side, area Hexagon tiles and real-world uses of the 6-sided polygon, Honeycomb pattern why the 6-sided shape is so prevalent in nature. Step-by-step explanation: Given a hexagon that can be divided into triangles by drawing all of the diagonals from one vertex. Necessary cookies are absolutely essential for the website to function properly. The sum of the exterior angles. Formula : Here number of vertical parts " n" and horizontal parts "m" then possible triangles is Figure - 11: Triangle counting in Fig - 11 = 30 Solution : Here number of vertical parts " 4 and horizontal parts "3" then possible triangles is 4 x 3 x 5 /2 = 30 Figure - 12: Triangle counting in Fig - 12 = 45 Number of triangles contained in a hexagon = 6 - 2 = 4. A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. THE SUM OF THE INTERIOR ANGLES OF A TRIANGLE IS 180. How many lines of symmetry does a triangle have? This is interesting, @Andre considering the type of question I guess it should be convex-regular. We will now have a look at how to find the area of a hexagon using different tricks. Is a PhD visitor considered as a visiting scholar. Since the interior angles of each triangle totals 180, the hexagons interior angles will total 4(180), or 720. To determine the area of a hexagon with perimeter P: You could also go directly from P to the area by using the formula area = 3 P / 24. 2. Thus there are $(n-4)$ different triangles with each of $n$ sides common. a) n - 2 b) n - 1 c) n d) n + 1. $$= \frac{n(n-1)(n-2)}{6}$$ The honeycomb pattern is composed of regular hexagons arranged side by side. The sum of all the exterior angles in an octagon is always 360. The cookie is used to store the user consent for the cookies in the category "Other. In order to calculate the perimeter of an octagon, the length of all the sides should be known. We will dive a bit deeper into such shape later on when we deal with how to find the area of a hexagon. Since a regular hexagon is comprised of six equilateral triangles, the . Also, a triangle has many properties. This pattern repeats within the regular triangular tiling. Choosing the vertices of a regular hexagon, how many ways are there to form four triangles such that any two triangles share exactly one vertex? This cookie is set by GDPR Cookie Consent plugin. An octagon consists of 8 interior angles and 8 exterior angles. There are six equilateral triangles in a regular hexagon. It will also be helpful when we explain how to find the area of a regular hexagon. In a hexagon there are six sides. Sides No. Joining each vertex with its opposite, the regular hexagon is divided into six equilateral triangles. For the sides, any value is accepted as long as they are all the same. How many degrees are in each angle of a regular hexagon and a regular octagon? I can see 35 in a pentagon, by organising my triangles by the quantity of shapes each is constructed of: 10 triangles made of 1 shape. we will count the number of triangles formed by each part and by taking two or more such parts together. [We are choosing the vertex common to the two common sides,which can be done in $nC1$ ways. total no of triangles formed by joining vertices of n-sided polygon If you are having trouble with maths I really suggest you to get this app, used this several times, and can officially say it's a lifesaver. Discover more with Omni's hexagon quilt calculator! However, if we consider all the vertices independently, we would have a total of 632 triangles. Puzzling Pentacle. Octagon is an eight-sided two-dimensional geometrical figure which consists of 8 interior angles and 8 exterior angles. So 7C3= 7! We can do this by $nC1$ ways . It should be no surprise that the hexagon (also known as the "6-sided polygon") has precisely six sides. :)) Share Cite Follow answered Mar 6, 2013 at 19:45 user65382 1 Add a comment 0 $$= \text{total - (Case I + Case II)}$$ How many obtuse angles can a isosceles triangle have?