How many diagonal lines in a octagon

A hexagon is a 6-sided polygon. An octagon is an eight-sided polygon. A circle or parabola or anything with a curved side is not a polygon. A point where two of the sides of a polygon meet is called a vertex. The number of vertices a polygon has is always equal to the number of sides it has. Another important polygon fact concerns the sum of angles. This pattern generalizes. The sum of all n angle in any n-sided polygon is:.

A shape is regular if and only if it is both equilateral and equiangular—that is, if and only if all the sides have the same length and all the angles are equal.

For higher polygons, you are most likely to see the regular version on the GRE, because the test like all mathematicians loves symmetry.

Now, we can talk about diagonals. A diagonal is any line through the interior of a polygon that connects two non-adjacent vertices. What does this mean? Well, first of all, starting from any vertex, an adjacent vertex is either vertex connected to the starting vertex by one side of the polygon. The only vertex not connected to A by a side of the quadrilateral is C. Thus, one diagonal goes from A to C.

Any quadrilateral has just two diagonals. Notice that triangles NEVER have diagonals : if we start from any vertex in a triangle, the other two vertices are adjacent. There simply are no non-adjacent vertices in a triangle, so diagonals are not possible.

Among quadrilaterals, there are special rules for the diagonals of a parallelogram and the categories within parallelograms:. The diagonals of a parallelogram bisect each other: that is to say, the intersection point of the two diagonals is the midpoint of each one. A rhombus is a parallelogram with four equal sides. The diagonals of a rhombus bisect each other and are perpendicular. The rectangle of a rhombus bisect each other and have equal length. This is related to an old trick among carpenters.

When a carpenter cuts two pairs of equal lengths to make the sides of a door or window frame, he knows he has a parallelogram because of the equal lengths, but how does he know whether he has a rectangle? A square is a parallelogram, a rectangle, and a rhombus. The diagonals of a square bisect each other, have equal length, and are perpendicular. The only pentagon you are likely to meet on the GRE is the most symmetrical, the regular pentagon.

Since the angles are equal, we can divide the sum of the angles by five. How many diagonals does a pentagon have? Any pentagon has exactly five diagonals. These diagonals trace out the shape of a classic five-pointed star, such as that those on the Flag of the United States of America.

How would we find any angles in this shape? Look, for example, triangle ABC. The other two angles must be equal: call them x. From that, we could find many other angles inside the shape. We could use analogous means to find angles involving the diagonals of any higher polygon. In a regular hexagon,. How many diagonals does a hexagon have?

Starting from one vertex, two other vertices are adjacent, so 3 vertices are non-adjacent, making possible three diagonals from one vertex. From A, we can draw diagonals to C, D, and E. Draw some other polygons and count the number of diagonals. The polygon does not have to be symmetric for this method to work. In the case of a concave polygon, you may have to draw some of the diagonals outside the actual polygon. A octagon has 20 diagonals. Method 2. Define the formula. You may see it either way, both equations are identical.

This equation can be used to find the number of diagonals of any polygon. Note that the triangle is an exception to this rule. Due to the shape of the triangle, it does not have any diagonals. Identify the number of sides in the polygon. To use this formula, you must identify the number of sides that the polygon has. The number of sides is given in the name of the polygon, you just need to know what each name means. Here are some of common prefixes you will see in polygons: [10] X Research source Tetra 4 , penta 5 , hexa 6 , hepta 7 , octa 8 , ennea 9 , deca 10 , hendeca 11 , dodeca 12 , trideca 13 , tetradeca 14 , pentadeca 15 , etc.

For example, a sided polygon would be written as gon. If you are given a picture of the polygon, you can simply count the number of sides.

Plug the number of sides into the equation. Once you know how many sides the polygon has, you just need to plug that number into the equation and solve. Solve the equation. Finish by solving the equation using the proper order of operations. Start by solving the subtraction, then multiply, then divide. The final answer is the number of diagonals the polygon has.

Practice with more examples. The more practice you have with a math concept, the better you will be at using it. Doing lots of examples will also help you memorize the formula in case you need it for a quiz, test, or exam. Remember, this formula works for a polygon of any number of sides greater than 3. Did you know you can get expert answers for this article?

Unlock expert answers by supporting wikiHow. Jake Adams. Support wikiHow by unlocking this expert answer. Not Helpful 0 Helpful 4. You can draw six, one for each of the vertices, except for the vertex you're drawing from, and the two adjacent vertices. Not Helpful 5 Helpful What is the relationship between the number of sides in an icosikaipentagon and the number of diagonals?

Not Helpful 4 Helpful Not Helpful 8 Helpful It depends on the shape of the base. A square or rectangular base has two diagonals, a pentagonal base has three diagonals, etc. The slanted surfaces are always triangles and have no diagonals. Not Helpful 4 Helpful 9. If the polygon is not regular, there is no formula available to calculate the interior angles. Not Helpful 4 Helpful 2. Substitute 11 for n in either of the formulas shown above.

Not Helpful 5 Helpful 2. Not Helpful 2 Helpful 4. Not Helpful 4 Helpful 0. The n-3 comes from picking any of the n vertices and subtracting the 3 vertices we can't draw a diagonal to, itself and the neighbouring vertices on each side. Dividing by 2 counts corrects this overcount and counts every diagonal once. Not Helpful 2 Helpful 2. Include your email address to get a message when this question is answered.

Related wikiHows How to. How to. Expert Interview. More References 4. About This Article. Co-authored by:. Co-authors: Updated: September 21, Categories: Geometry Mathematics. Article Summary X To find out how many diagonals a polygon has, first count the number of sides, or straight lines, that make up the polygon. Italiano: Trovare il Numero di Diagonali in un Poligono.

Deutsch: Herausfinden wie viele Diagonalen ein Polygon hat. Nederlands: Het aantal diagonalen in een veelhoek bepalen. Thanks to all authors for creating a page that has been read , times. Great help explaining the topic to children. More reader stories Hide reader stories.