The answer is not from geometry it's from combinations. After substituting the value of n = 8 in this formula, we get, (8 - 2) 180 = 1080. This website uses cookies to improve your experience while you navigate through the website. This is because of the relationship apothem = 3 side. There are 8 interior angles and 8 respective exterior angles in an octagon. There are 20 diagonals in an octagon. A quadrilateral is a closed shape with four vertices and four sides and an octagon has 8 sides and 8 vertices. On the circumference there were 6 and then 12 on the second one. Where A means the area of each of the equilateral triangles in which we have divided the hexagon. They completely fill the entire surface they span, so there aren't any holes in between them. With Cuemath, you will learn visually and be surprised by the outcomes. Example 1: How many triangles can be formed by joining the vertices of an octagon? 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. A truncated hexagon, t{6}, is a dodecagon, {12}, alternating two types (colors) of edges. In the adjoining figure of a hexagon ABCDEF, on joining AC, An equilateral hexagon can be divided into 6 equilateral triangles of side length 6. In a hexagon there are six sides. Three sprinters A, B, and C begin running from points A 1 , B 1 and C 1 respectively. we will count the number of triangles formed by each part and by taking two or more such parts together. Step-by-step explanation: For the first vertex of the triangle, there are 8 choice possibilities, for the second vertex, there are 7 possibilities and for the third vertex, there are 6 choice possibilities. What is the point of Thrower's Bandolier. but also in many other places in nature. How many diagonals can be formed by joining the vertices of the polygon having 5 sides? 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. There 6 equilateral triangles in a regular hexagon. Thus there are $n$ pairs of alternate & consecutive vertices to get $n$ different triangles with two sides common (Above fig-2 shows $n$ st. lines of different colors to join alternate & consecutive vertices). This can be done in 6 C 3 ways. Does a barbarian benefit from the fast movement ability while wearing medium armor? We will directly count the number of triangles with 3, 4 and 5 endpoints (top three figures). 1.) 55 ways. rev2023.3.3.43278. However, you may visit "Cookie Settings" to provide a controlled consent. There are three paths formed by the triangles A 1 A 2 A 3, B 1 B 2 B 3, and C 1 C 2 C 3, , as shown. Then, after calculating the area of all the triangles, we add their areas to get the area of the octagon. How many distinct diagonals does a hexagon have? How many non-congruent triangles can be formed by the vertices of a regular polygon of $n$ sides. This part of the camera is called the aperture and dictates many properties and features of the pictures produced by a camera. Since each of the six interior angles in a regular hexagon are equal in measure, each interior angle measures 720/6 = 120, as shown below. How many axes of symmetry does an equilateral triangle have? Here is how you calculate the two types of diagonals: Long diagonals They always cross the central point of the hexagon. We also answer the question "what is a hexagon?" You have 2 angles on each vertex, and they are all 45, so 45 8 = 360. Hexa means six, so therefore 6 triangles. For the regular hexagon, these triangles are equilateral triangles. We are, of course, talking of our almighty hexagon. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. There are 6 vertices of a hexagon. How many triangles are there in a nonagon? How many obtuse angles are in a triangle? A regular octagon has 4 pairs of parallel sides (parallel lines). How many triangles can be formed by joining the vertices of Heptagonal? The perimeter of an octagon is the total length of its boundary. No triangle. r! This cookie is set by GDPR Cookie Consent plugin. , What are examples of venial and mortal sins? How many equilateral triangles are there in a regular hexagon? Octagon is an eight-sided two-dimensional geometrical figure. We can obtain four triangles, specifically two equilaterals ABG and ECG, one isosceles triangle EFD and one right angle triangle ABC. Minimising the environmental effects of my dyson brain. 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?" Polygon No. We know that in a regular octagon, all the sides are of equal length. Using a very simple formula, you can calculate the number of diagonals in any polygon, whether it has 4 sides or 4,000 sides. Interesting. In this case, there are 8 sides in an octagon. Check out our online resources for a great way to brush up on your skills. If all of the diagonals are drawn from a vertex of a pentagon, how many triangles are formed? One C. Two D. Three. Since the interior angles of each triangle totals. i.e. Answer: C. This effect is called the red shift. For example, if the perimeter of a regular octagon is 96 units, then the length of one side = Perimeter 8 = 96/8 = 12 units. How are probability distributions determined? 4! Equivalent Fractions in Hexagon Drawing a line to each vertex creates six equilateral triangles, which is six equal areas. Let us choose triangles with $1$ side common with the polygon. The side length of an octagon can be calculated if the perimeter and the other sides are given. The total number of hexagon diagonals is equal to 9 three of these are long diagonals that cross the central point, and the other six are the so-called "height" of the hexagon. 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. How many sides does an equilateral triangle have? ABC, ACD and ADE. After multiplying this area by six (because we have 6 triangles), we get the hexagon area formula: A = 6 A = 6 3/4 a A = 3 3/2 a = (3/2 a) (6 a) /2 = apothem perimeter /2 The sum of the given sides can be reduced from the perimeter to get the value of the unknown side. How many sides does a regular polygon have? Then, the numbers of triangles that can be formed by joining the vertices of a hexagon can be calculated by applying the concept of combination. Triangular Hexagons. Pentagon = 5 sides, 5 diagonal formed, 40 triangles formed 4.) The number of triangles with no side common with regular polygon having $n$ number of sides $$=^nC_3-n-n(n-4)$$. The interior angle at each vertex of a regular octagon is 135. How many triangles can be made with 13 toothpicks? We sometimes define a regular hexagon. These restrictions mean that, for a regular hexagon, calculating the perimeter is so easy that you don't even need to use the perimeter of a polygon calculator if you know a bit of math. How Many Equilateral Triangles are there in a Regular Hexagon? What is a reasonable budget for Facebook ads? Observe the figure given below to see the regular hexagon with 6 equilateral triangles. How many lines of symmetry does an equilateral triangle have? Get access to this video and our entire Q&A library, What is a Hexagon? When all these eight sides are equal in length, it is known as a regular octagon, whereas when even at least one of the sides is different in measurement, it is known as an irregular octagon. All triangles are formed by the intersection of three diagonals at three different points. How many triangles exist in the diagonals intersections of an heptagon? 3 How many triangles can be formed by joining the vertices of Heptagonal? As the name suggests, a "triangle" is a three-sided polygon having three angles. 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. A: 209 diagonals So, a polygon with 22 sides has 209 diagonals. 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. (33 s2)/2 where 's' is the side length. How many angles are on a square-based pyramid? six a) 1 b) 2 c) 3 d) 4. 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. Hexa means six, so therefore 6 triangles. Similarly, join alternate vertices $A_2$ & $A_4$ to get another triangle $A_2A_3A_4$ with two sides $A_2A_3$ & $A_3A_4$ common & so on (as shown in above figure-2). The formula to calculate the area of a regular hexagon with side length s: (3 3 s^2)/2. 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 . :)) Share Cite Follow answered Mar 6, 2013 at 19:45 user65382 1 Add a comment 0 The answer is 3, that is, approximately 1.73. For example, if one side of a regular octagon is 6 units, let us find the area of the octagon. For the sides, any value is accepted as long as they are all the same. How many angles does a rectangular-based pyramid have? How many acute angles are in a right triangle? What is the number of triangles that can be formed whose vertices are the vertices of an octagon? 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. Sum of interior angles of a polygon = (n - 2) 180 = (8 - 2) 180 = 1080. $$=\left[\frac{n(n-1)(n-2)}{6}\right]-\left[n(n-4) + n\right]$$ non-isosceles triangles with vertices in a 20-sided regular polygon. Challenge Level. Why are physically impossible and logically impossible concepts considered separate in terms of probability? Very great, it helps me with my math assignments. Here, the side length, a = 5 units. Analytical cookies are used to understand how visitors interact with the website. Q: In a convex 22-gon, how many diagonals can be drawn from one vertex? Before using counting tools, we need to know what we are counting. Now, the 11 vertices can be joined with each other by 11C2 ways i.e. So, the total diagonals will be 6(6-3)/2 = 9. How many exterior angles does a triangle have? 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. However, when we lay the bubbles together on a flat surface, the sphere loses its efficiency advantage since the section of a sphere cannot completely cover a 2D space. Well it all started by drawing some equilateral triangles so that they made a regular hexagon: Then we made a bigger one: Well there was the thought about how many dots there were in various places. Let us learn more about the octagon shape in this article. However, if we consider all the vertices independently, we would have a total of 632 triangles. Circumradius: to find the radius of a circle circumscribed on the regular hexagon, you need to determine the distance between the central point of the hexagon (that is also the center of the circle) and any of the vertices. The octagon in which at least one of its angles points inwards is a concave octagon. of sides)}=\color{blue}{(n-4)n}$$, Now, join the alternate vertices $A_1$ & $A_3$ by a straight (blue) line to get a triangle $A_1A_2A_3$ with two sides $A_1A_2$ & $A_2A_3$ common. The angle bisectors create two half angles which measure 60: mOAB=mOBA=60. Here is one interpretation (which is probably not the one intended, but who knows? In a regular hexagon three diagonals pass through the centre. If the triangle's area is 4, what is the area of the hexagon? ], So if we subtract the part $2$ and $3$ from part $1$ we will get our desired result. After substituting the value of 'n' = 8 in the formula, we get, Number of diagonals = n(n-3)/2 = 8(8 - 3)/2 = (8 5)/2 = 20. There are a total of 8 sides in an octagon, and those eight sides are parallel to their respective opposite side in the case of a regular octagon. This pattern repeats within the regular triangular tiling. It is expressed in square units like inches2, cm2, and so on. Log in, WhatsApp Guess the Toothpaste brand names puzzle, Guess Marwadi Names from whatsapp emoticons. 3 More answers below How to show that an expression of a finite type must be one of the finitely many possible values? The sum of its interior angles is 1080 and the sum of its exterior angles is 360. Jamila has 5 sticks of lengths 2,4,6,8, and 10 inches. A regular octagon is an example of a convex octagon. Here, n = 8, so after substituting the value of n = 8 in this formula, we get, 1/2 n (n - 3) = 1/2 8 (8 - 3) = 20. Just calculate: where side refers to the length of any one side. A: The net of a pentagonal pyramid consists of two pentagons and five rectangles . THE SUM OF THE INTERIOR ANGLES OF A TRIANGLE IS 180. And how many if no side of the polygon is to be a side of any triangle ? Consider a regular polygon with $n$ number of vertices $\mathrm{A_1, \ A_2,\ A_3, \ A_3, \ldots , A_{n-1}}$ & $\mathrm{A_{n}}$, Total number of triangles formed by joining the vertices of n-sided regular polygon $$N=\text{number of ways of selecting 3 vertices out of n}=\color{}{\binom{n}{3}}$$ $$N=\color{red}{\frac{n(n-1)(n-2)}{6}}$$ The perimeter of an octagon is expressed in linear units like inches, cm, and so on. :/), We've added a "Necessary cookies only" option to the cookie consent popup. High School Math : How to find the area of a hexagon 1.Write down the formula for finding the area of a hexagon if you know the side length. This means the length of the diagonal can be calculated if the side length of the regular hexagon is known. $$N_0=\color{red}{\frac{n(n-4)(n-5)}{6}}$$ You also have the option to opt-out of these cookies. How many degrees are in each angle of an equilateral triangle? I can see 35 in a pentagon, by organising my triangles by the quantity of shapes each is constructed of: 10 triangles made of 1 shape. You may need to first identify how many sides are present in the polygon. For now, it suffices to say that the regular hexagon is the most common way to represent a 6-sided polygon and the one most often found in nature. How many distinct equilateral triangles exist with a perimeter of 60? 6 How many diagonals can be drawn by joining the vertices? If $N_0$ is the number of triangles having no side common with that of the polygon then we have $$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 interior angles add up to 1080 and the exterior angles add up to 360. Can archive.org's Wayback Machine ignore some query terms? Solution: Since it is a regular hexagon, we know that 6 equilateral triangles can be formed inside it. a) 5 b) 6 c) 7 d) 8. To solve this lets break this problem into $3$ parts: Total number of triangles that can form without any restrictions$=nC3$. Why are Suriname, Belize, and Guinea-Bissau classified as "Small Island Developing States"? How many triangles can be formed by the vertices of a regular polygon of $n$ sides? Where does this (supposedly) Gibson quote come from? Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet. That is because despite being very bright objects, they are so very far away that only a tiny fraction of their light reaches us; you can learn more about that in our luminosity calculator. Convex or not? Using that, you get (n choose 3) as the number of possible triangles that can be formed by the vertices of a regular polygon of n sides. It is an octagon with unequal sides and angles. 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. And there is a reason for that: the hexagon angles. It does not store any personal data. Another important property of regular hexagons is that they can fill a surface with no gaps between them (along with regular triangles and squares). When we plug in side = 2, we obtain apothem = 3, as claimed. When you create a bubble using water, soap, and some of your own breath, it always has a spherical shape. The hexagon calculator allows you to calculate several interesting parameters of the 6-sided shape that we usually call a hexagon. Did any DOS compatibility layers exist for any UNIX-like systems before DOS started to become outmoded? What is the sum of the interior angles of a hexagon? We have,. Necessary cookies are absolutely essential for the website to function properly. In a regular hexagon, however, all the hexagon sides and angles must have the same value. A regular hexagon has a perimeter of 30 m. What is the area of the hexagon? =20 The number of vertices in a triangle is 3 . . 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. The area of an octagon is the total space occupied by it. ABC=PQR x-10o= In a regular hexagon, four triangles can be created using diagonals of the hexagon from a common vertex. We've added a "Necessary cookies only" option to the cookie consent popup. How many triangles can be formed using 10 points located in each of the sides (but not vertices) of a square? What sort of strategies would a medieval military use against a fantasy giant? Thus, for each of the 8 vertices you can draw 5 diagonals and hence there can be 5 8 = 40 diagonals. Here we are choosing triangles with two sides common to the polygon. By drawing a line to every other vertex, you create half as many equal areas (3 equal areas). Another way to find the number of triangles that can be formed in an octagon is by using the formula, (n - 2), where n = number of sides of the polygon. How many diagonals does a 20 sided polygon have? They are constructed by joining two vertices, leaving exactly one in between them. The sum of the exterior angles. In a convex 22-gon, how many. How many triangles can we form if we draw all the diagonals . 820 Math Experts 92% Recurring customers 101064 Orders Deliver Get Homework Help How many triangles can be formed with the side lengths of 12,15, and 18? Here are a few properties of an octagon that can help to identify it easily. On top of that, the regular 6-sided shape has the smallest perimeter for the biggest area among these surface-filling polygons, which makes it very efficient. A regular hexagon is made from equilateral triangle by cutting along the dotted lines and removing the three smaller triangles. The next simplest shape after the three and four sided polygon is the five sided polygon: the pentagon. (cont) [4 distinct ones by 2D rotation, 3 distinct ones by 3D rotation] To prove there are only 6 triangles, when drawing all the diagonals (lines going through the centre of mass) of a regular hexagon, I am not quite sure how to proceed. Since the interior angles of each triangle totals 180, the hexagon's interior angles will total 4(180), or 720. 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 . How many diagonals are in a 100-sided shape? Math is a subject that can be difficult for some students to grasp. Therefore, the formula that is used to find its perimeter is, Perimeter of an octagon = Sum of all its sides, Perimeter of a regular octagon = 8a (Where 'a' is the length of one side of the octagon). $\forall \ \ \color{blue}{n\geq 3}$, Consider a side $\mathrm{A_1A_2}$ of regular n-polygon. The area of an octagon is the total space occupied by it. We need to form triangles by joining the vertices of a hexagon To form a triangle we require 3 vertices. In a regular octagon, by joining one vertex to the remaining non-adjacent vertices, 6 triangles can be formed. satisfaction rating 4.7/5. The sum of all interior angles of a triangle will always add up to 180 degrees. How many right angles does a triangle have? Find the value of $\frac{N}{100}$. What is a hexagon? Fill order form Confidentiality Hexagon Calculator. = 20 So, 20 triangles are possible inside a hexagon. Discover more with Omni's hexagon quilt calculator! The circumradius is the radius of the circumference that contains all the vertices of the regular hexagon. Similarly, there are $(n-4)$ different triangles with only one side $A_2A_3$ common & so on. This is called the angle sum property of triangle. There are five arrangements of three diagonals to consider. Pentagon 5 sides 3 triangles 180 = 540 Hexagon 6 sides 4 triangles 180 = 720 Heptagon 7 sides 5 triangles 180 = 900 Octagon 8 sides 6 triangles 180 = 1080. b. Now by subtracting n with nC2 ways, the formula obtained is n(n-3)/2. How to react to a students panic attack in an oral exam? C. The next case is common to all polygons, but it is still interesting to see. The hexagon shape is one of the most popular shapes in nature, from honeycomb patterns to hexagon tiles for mirrors its uses are almost endless. All rights reserved. How many diagonals can be formed by joining the vertices of hexagon? The sum of the interior angles of an octagon can be calculated with the help of the following formula where 'n' represents the number of sides (8) in an octagon. How many different triangles, if any, can be drawn with one 90 degrees angle and side lengths of 5 cm and 12 cm? Minimising the environmental effects of my dyson brain. How many diagonals can be drawn by joining the vertices? Styling contours by colour and by line thickness in QGIS. Substituting the value of 'a' in the formula, we get, Area of a Regular Octagon = 2a2(1 + 2) = 2 (5)2 (1 + 2) = 50 (1 + 2) = 120.71 square units. The number of triangles that can be formed by joining them is C n 3. $$= \text{total - (Case I + Case II)}$$ For the regular triangle, all sides are of the same length, which is the length of the side of the hexagon they form. An octagon in which the sides and angles are not congruent is an irregular octagon. Thus there are $(n-4)$ different triangles with only one side $A_1A_2$ common. How many right triangles can be constructed? a. 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 number of quadrilaterals that can be formed by joining them is C n 4. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. 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. Here, n = 8, so after substituting the value of n = 8 in the formula, Number of triangles that can be formed in a polygon = (n - 2), we get, (8 - 2) = 6. Using this calculator is as simple as it can possibly get with only one of the parameters needed to calculate all others and includes a built-in length conversion tool for each of them. One triangle is formed by selecting a group of 3 vertices from given 6 vertices. A regular hexagon, which means a hexagon with equal sides and equal interior angles, is the shape that has 3 pairs of parallel sides. Looking for a little arithmetic help? I thought that the answer is $\binom{6}{3}=20$ but this is not the right answer, why? We cannot go over all of them in detail, unfortunately. Convex octagons are those in which all the angles point outwards. The angles of an arbitrary hexagon can have any value, but they all must sum up to 720 (you can easily convert them to other units using our angle conversion calculator). for 1 side we get (n-4) triangles $\implies$ n (n-4) triangles for n sides. In a regular octagon, all the interior angles are of equal measure and each interior angle measures 135. All the interior angles are of different measure, but their sum is always 1080. However, if you . How many triangles do you get from six non-parallel lines? Therefore, 8*9*7= 336 there are possible triangles inside the octagon. How many obtuse angles does a rhombus have. Most of the entries in the NAME column of the output from lsof +D /tmp do not begin with /tmp. 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? there are 7 points and we have to choose three to form a triangle, Learn Sentence Correction Strategies with 780 Scorer. Observe the figure given below to see what an octagon looks like. A regular hexagon is a hexagon in which all of its sides have equal length. 3. Total number of triangles formed by joining the vertices of regular polygon having $n$ number of sides $$=^{n}C_3$$ Easy Solution Verified by Toppr There are 6 vertices of a hexagon. But the DIAGONAL too is made from 3 points : 2vertices and 1 centre.. And here we make a line and not a triangle.. It solves everything I put in, efficiently, quickly, and hassle free. Therefor the interior angles of the polygon must be the sum of all the triangles' interior angles, or 180 (n-2). What am I doing wrong here in the PlotLegends specification? An alternated hexagon, h{6}, is an equilateral triangle, {3}. No, an octagon is not a quadrilateral. The cookie is used to store the user consent for the cookies in the category "Analytics". In case of a regular octagon, we use the formula, Perimeter of regular octagon = 8 Side length, because all the sides are of equal length. How many triangles can be drawn in a heptagon? Why are Suriname, Belize, and Guinea-Bissau classified as "Small Island Developing States"? Createyouraccount. The perimeter of a hexagon can be calculated Passing Rate Deal with math problem Solve math equation . Since the interior angles of each triangle totals 180, the hexagons interior angles will total 4(180), or 720. One of the biggest problems we experience when observing distant stars is how faint they are in the night sky. In triangle TAG, angle A = 70 degrees, a = 19, g = 26 A. We use cookies on our website to give you the most relevant experience by remembering your preferences and repeat visits. This honeycomb pattern appears not only in honeycombs (surprise!) How about an isosceles triangle which is not equilateral? Their length is equal to d = 3 a. If all of the diagonals are drawn from a vertex of a hexagon, how many triangles are formed? To get a triangle with only one side $A_1A_2$ common (As shown in figure-1 below), Join the vertices $A_1$ & $A_2$ to any of $(n-4)$ vertices i.e. - Definition, Area & Angles. 2) no of triangles with two sides common, You will end up with 6 marks, and if you join them with the straight lines, you will have yourself a regular hexagon. How many lines of symmetry does a triangle have? A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. If all of the diagonals are drawn from a vertex of a quadrilateral, how many triangles are formed? Must the vertices of the triangles coincide with vertices of the hexagon? It will also be helpful when we explain how to find the area of a regular hexagon. [We are choosing the vertex common to the two common sides,which can be done in $nC1$ ways. Since a regular hexagon is comprised of six equilateral triangles, the. One triangle is formed by selecting a group of 3 vertices from the given 6 vertices. The step by step can be a little confusing at times but still extremely useful especially for test where you must show your work. Starting at a random point and then making the next mark using the previous one as the anchor point, draw a circle with the compass. Learn more about Stack Overflow the company, and our products. This same approach can be taken in an irregular hexagon.In a regular hexagonregular hexagonFor a regular n-gon, the sum . Advertisement cookies are used to provide visitors with relevant ads and marketing campaigns. Is it possible to rotate a window 90 degrees if it has the same length and width? Convex octagons bulge outwards, whereas concave octagons have indentations (a deep recess).
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