Shape diagonals: What is diagonal? | TheSchoolRun
Posted onPolygons Diagonals — Formula, Diagonals, Solved Examples, and FAQs
baseline;white-space:pre;white-space:pre-wrap;»>The diagonal of a polygon is a line segment that connects any two non-adjacent vertices. The number of diagonals and their attributes vary depending on the type of polygon and the number of sides. Let’s review what a polygon is and what a diagonal is before learning the diagonal of a polygon formula.
A closed shape made up of three or more line segments is called a polygon. A line segment generated by joining any two non-adjacent vertices forms the diagonal of a polygon. Let’s look at the formula for a polygon’s diagonal, as well as some examples of solved problems. You can quickly count all of the possible diagonals of a basic polygon with a few sides. Counting polygons can be difficult when they become more intricate.
Fortunately, there is a straightforward formula for calculating the number of diagonals in a polygon. Because any vertex (corner) is connected to two other vertices by sides, those connections cannot be considered diagonals. That vertex, too, is unable to make a connection with itself. As a result, ‘n’ we’ll immediately lower the number of viable diagonals by three.
That vertex, too, is unable to make a connection with itself. For example, our door only has two diagonals if you don’t include moving from the top hinge to the bottom opposite and back. Any solution will have to be divided by two.
Diagonal- Polygons Diagonals
A diagonal is a segment of a polygon that connects two non-consecutive vertices. In a polygon, the number of diagonals that can be drawn from any vertex is three less than the number of sides. Multiply the number into totaling of diagonals per vertex (n — 3) by the number of vertices, n, then divide by 2 to get the total number of diagonals in a polygon (otherwise each diagonal is counted twice).
Number of Diagonals = \[\frac{n(n-3)}{2}\]
Simply subtract the total sides from the diagonals given by each vertex to another vertex to arrive at this formula.
To put it another way, an n-sided polygon has n-vertices that can be connected in nC2 ways.
The formula obtained by subtracting n using nC2 methods is \[\frac{n(n-3)}{2}\].
The total sides of a hexagon, for example, are six. As a result, the total diagonals are 6(6-3)/2 = 9
baseline;white-space:pre;white-space:pre-wrap;»>Let’s know what a diagonal is. A diagonal of a polygon can be defined as a line segment joining two vertices. From any given vertex, there are no diagonals to the vertex on either side of it, since that would lay on top of the side. Also, remember that there is obviously no diagonal from a vertex back to itself. This means there are three less diagonals than the number of vertices. (We do not count diagonals to itself and one either side). This is a diagonal definition.
Here, we are going to discuss the number of diagonals in a polygon, diagonal definition.
Formula for the Number of Diagonals
As described above, the number of diagonals from a single vertex is three less than the number of vertices or sides, or (n-3).
There are a total number of N vertices, which gives us n(n-3) diagonals.
But each diagonal of the polygon has two ends, so this would count each one twice. So as a final step we need to divide by 2, for the final formula:
Number of distinct diagonals = \[\frac{n(n-3)}{2}\]
where,
n is the number of sides (or vertices).
Diagonals of Polygon
The diagonals of a polygon is a segment line in which the ends are non-adjacent vertices of a polygon.
How many diagonals does n-polygon have? Let’s see the diagonals of a polygon and the no. of diagonals in a polygon.
For n = 3 we have a triangle. We can clearly see the triangle has no diagonals because each vertex has only adjacent vertices. Therefore, the number of diagonals in a polygon triangle is 0.
For n = 4 we have quadrilateral. It has 2 diagonals. Therefore, the number of diagonals in a polygon quadrilateral is 2.
For n = 5, we have a pentagon with 5 diagonals. Therefore, the number of diagonals in a polygon pentagon is 5.
For n = 6, n-polygon is called hexagon and it has 9 diagonals. Therefore, the number of diagonals in a polygon hexagon is 9.
Since n was a lower number we could easily draw the diagonals of n-polygons and then count the number of diagonals in a polygon.
Diagonals in Real Life
Diagonals in rectangles, as well as diagonals in squares, add toughness to construction, whether for a house wall, bridge, or tall building. You may have seen diagonal wires used to keep the bridges steady. When houses are being built, look for the diagonal braces that tend to hold the walls straight and true.
Bookshelves and scaffolding are braced with diagonals. For a catcher in softball or for a catcher in baseball to throw out a runner at second base, the catcher throws along a diagonal from home plate to second.
{2}}\]
For cubes, s is the side of a cube
Important Note
The above-given formula gives us the number of distinct diagonals — that is, the number of actual line segments. At times it is easy to miscount the diagonals of a polygon when doing it by eye.
If you glance quickly at the pentagon given below, you may be tempted to say that the number of diagonals is 10. After all, there are 2 at each vertex and 5 vertices. Few people watch them making 3 triangles, for 6 diagonals. But there are only 5 diagonals. You need to count them carefully.
Solved Examples
Question 1) Find the total number of diagonals contained in an 11-sided regular polygon.
Solution) In an 11-sided polygon, total vertices are 11. Now, the 11 vertices can be joined with each other by C211 ways i.e. 55 ways.
Now, there are a total of 55 diagonals possible for an 11-sided polygon which includes its sides also.
So, subtracting the sides will give the total number of diagonals contained by the polygon.
So, total diagonals contained within an 11-sided polygon = 55 -11 which is equal to 44.
Formula Method:
According to the formula, the number of diagonals equals \[\frac{n(n-3)}{2}\].
So, 11-sided polygon will contain 11(11-3)/2 = 44 diagonals.
Diagonals of Parallelograms, Rectangles, and Rhombi (Video & Practice)
TranscriptPractice
Hi, and welcome to this video on diagonals! Today we’re going to explore the diagonals of parallelograms, rectangles, rhombi, and squares, and see how these shapes relate to each other overall. Let’s get started!
All of the shapes that we are going to look at today are quadrilaterals, meaning they are all four-sided polygons.
A polygon is a shape with multiple sides.
Types of Quadrilaterals
Properties of a Parallelogram
A parallelogram is a quadrilateral that has two sets of parallel sides. The blue arrows denote which sides are parallel to each other. The opposite sides of a parallelogram, marked by green tick marks, are congruent, which means they have the same measure.
Properties of a Rectangle
A rectangle is a shape we all know well since it appears in the real world so often. But one thing we don’t usually think about is that rectangles are just a special kind of parallelogram. That means that the opposite sides are both parallel and congruent, just like our first parallelogram. What makes a rectangle a special parallelogram is that its interior angles are the same. In fact, they are all right angles, which means they measure 90 degrees.
Properties of a Rhombus
Another special parallelogram is a rhombus.
It has all the properties of the parallelogram but all of the sides of a rhombus are congruent. Note that the tick marks are all the same, which tells us that all the sides are the same length.
Properties of a Square
Our last special parallelogram is a rectangle and a rhombus at the same time, so it has the right angles of the rectangle and the congruent sides of the rhombus. It’s another shape we all know well: the square!
As you can see, the opposite sides are parallel, all the angles are right angles, and all the sides are congruent.
So let’s recap our quadrilaterals before we start exploring their diagonals:
Here, we’ve arranged our quadrilaterals into a sort of “tree”, starting with the square as the trunk. Every square is a rhombus and every rhombus is a parallelogram. Every square is also a rectangle and every rectangle is a parallelogram. That’s how it works going up the tree. But it doesn’t work going down. Not every parallelogram is a rectangle and not every rhombus is a square.
Okay, now let’s explore the diagonals of these four quadrilaterals. A diagonal is a line segment connecting opposite vertices, or corners, of a quadrilateral.
Diagonals of a Parallelogram
Here’s our regular, non-special parallelogram with the diagonals shown:
As you can see, the diagonals of a parallelogram bisect each other. In other words, they cut each other in half. We could add some numbers to illustrate this better…
or we could use tick marks to show that the diagonals have been bisected:
Since the three special quadrilaterals we’re talking about are all parallelograms, they will all have bisecting diagonals.
Diagonals of a Rectangle
Now let’s look at the rectangle. Here’s a rectangle with its diagonals:
The diagonals of a rectangle are congruent, and, again, since a rectangle is a parallelogram, the diagonals bisect each other, making each half the same length:
Each diagonal of a rectangle also divides the rectangle into two congruent right triangles:
This means that, if we wanted to, we could calculate the length of the diagonal using the Pythagorean Theorem.
Diagonals of a Rhombus
Now that we’ve got rectangles figured out, let’s see what happens with another special parallelogram, the rhombus:
Once again, since every rhombus is a parallelogram the diagonals bisect each other. The diagonals are NOT the same size though, so what’s special about this one? Take a look at the angles at which the diagonals intersect. These angles look like they could all be the same, and since there are four angles there it must mean…
That each angle is 90 degrees! This means that the diagonals of a rhombus are perpendicular to each other in addition to bisecting each other.
Diagonals of a Square
Okay, only one quadrilateral left, the square. Remember, the square is a parallelogram, a rectangle, and a rhombus, so it should have all the properties of those shapes:
- The diagonals will bisect each other.
- The diagonals will be the same length.
- The diagonals will be perpendicular to each other.
Let’s see if we’re right:
We were right! If we look closely, we can also see that the two diagonals cut the square into four congruent isosceles right triangles.
Review
Now that we’ve looked at our four parallelograms and their diagonals, let’s finish with a review to see what all you can remember.
Which of the following statements is false?
- Every rectangle is a parallelogram.
- Every rhombus is a square.
- Every square is a rectangle.
- The diagonals of a parallelogram bisect each other.
- The diagonals of a rhombus intersect at right angles.
- A diagonal of a rectangle divides it into two congruent right triangles.
- The diagonals of a rectangle are the same length.
- A quadrilateral whose diagonals bisect each other, intersect at right angles, and are congruent must be a square.
Thanks for watching, and happy studying!
Practice Questions
Question #1:
Which statement is false, regarding the diagonal lines drawn through a square?
The diagonal lines will be the same length.
Four congruent isosceles right triangles will be formed.
The diagonal lines will bisect each other.
Four scalene triangles will be formed.
Show Answer
Answer:
When diagonal lines are drawn through a square, three things will be true: the diagonal lines will bisect each other at their midpoints, the diagonal lines will be the same length, and the diagonal lines will create four congruent triangles that are all right isosceles triangles. Four scalene triangles will not be a result of diagonal lines drawn through a square.
Hide Answer
Question #2:
Which statement is correct?
A parallelogram is always a rectangle.
A rectangle is always a square.
A rhombus is always a parallelogram.
A parallelogram is always a rhombus.
Show Answer
Answer:
It is true that a rhombus is always a parallelogram.
A rhombus has four congruent side lengths, with diagonal lines that are perpendicular to each other. Every square is a rhombus, but not every rhombus is square.
Hide Answer
Question #3:
Which definition is not accurate for the shape below?
Rectangle
Quadrilateral
Rhombus
Parallelogram
Show Answer
Answer:
This shape can be classified as a rectangle, quadrilateral, or parallelogram. However, this is not a rhombus. A rhombus has all side lengths congruent.
Hide Answer
Question #4:
Sir John Boys’ House is a historical building in Canterbury. The building is now famous for its uniquely slanted red front door. A carpenter wants to document some of the measurements of the door.
Which statement will be true about two diagonal lines measured across from corner to corner?
Both of the diagonal lines will be the same length.
The diagonal lines will bisect each other at the midpoint of each line.
The diagonal lines will bisect to form 90 degree angles.
The diagonal lines will create four right triangles.
Show Answer
Answer:
The diagonal lines of a parallelogram will bisect each other at the midpoint of each line. This means that each line will be cut in half by the other line.
Hide Answer
Question #5:
A hail storm has damaged the rectangular shutters on Sean’s house, so he wants to make some repairs. Before he begins the repair job, he wants to make a few measurements. Which statement about the diagonal lines drawn across the windows is incorrect?
The diagonal lines will create four 90 degree angles in the center of the window where they bisect.
Both diagonal lines will be congruent.
The diagonal lines will bisect each other at the midpoint of each line.
Each of the diagonal lines divides the rectangle into two congruent right triangles.
Show Answer
Answer:
If the window frame was a square or a rhombus, the bisecting diagonals would create four 90 degree angles in the center of the window. However, this particular window frame is a rectangular parallelogram, so the bisecting diagonals will not create 90 degree angles.
Hide Answer
Return to Geometry Videos
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Diagonal
Diagonal — dense fabric with raised ribs on the front side.
To the edge of the fabric, they are located at an angle of 45 ° and outwardly resemble a diagonal rectangle — hence the name. Such a pattern is obtained as a result of a certain selection of the thickness of the threads for the weft and warp, and then their special weave. The yarn for the manufacture of the material is strong, in most cases pure cotton, although small additions are sometimes possible. The fabric is dense, the density value is in the range of 180-250 grams per square meter. Due to the natural component, it has hygroscopicity, the ability to pass air currents, which makes things comfortable to wear. Products sewn from it are unique: they protect quite well from the cold, and in hot weather they provide freshness, and at the same time you can not be afraid of allergic reactions even in children.
Fabric diagonal is a dense cotton, or with small additives, a material made of durable yarn. On the surface of such a fabric, there are clearly visible scars that stand out brightly, inclined at an angle of 45 ° relative to the edge of the material.
They look like some kind of diagonal rectangle, it is this distinctive feature that contributed to the formation of the name. Such scars are on the fabric for a reason. They are created after a preliminary selection of the total thickness of the base, the required density and with a special type of weaving of threads. The density of this fabric is from 180 to 250 grams per square meter. Has a width of 80 centimeters. Supplied in rolls of 60-80 running meters. The color range is quite wide: black , dark blue , orange , white color On request, manufacturers are ready to offer other colors and shades. But when ordering an individual color, it is necessary to take into account the rise in the cost of the material, and the minimum order quantity is 5000 running meters. Such a fabric is also sold in a harsh (unfinished) form. Severe fabric diagonal has a light beige color. It is mainly used for sewing work gloves.
Since the fabric in the manufacture is based on natural ingredients, this means that such a fabric is hygroscopic and can freely pass air. All things made from such raw materials bring only comfort and practicality to their owner. Products do not cause any allergic reactions. Separately, I would like to highlight the thermal insulation properties of the material: in cold weather, clothes made of it warm and retain heat, and at elevated temperatures — bring coolness and freshness.
Depending on the fibers used, the following types of diagonals are distinguished: pure wool, wool blend ( wool in combination with cotton), cotton, silk ( natural silk in combination with viscose) and staple (from viscose fiber). To increase durability, polyester is added in various ratios.
The method of weaving refers to twill, as a result of which a scar moving along an oblique line is formed — “ diagonal «.
It is possible both to increase the rapport shift, and to add two rapports of different twill weaves. This allows you to get a more prominent line of scars, located at an angle of at least 45 degrees to the edge of the fabric.
Weaving threads of different colors in a certain order allows you to get a camouflage material. Thickness combinations for the warp and weft make it possible to obtain a material with a certain density, which varies from 190 to 260 g per square meter. At the final stage, the fabric is treated with a water-repellent composition to improve performance. The rubberized version is impregnated with polyvinyl chloride.
Available in pure wool, wool blend and cotton. The pure wool diagonal is made from fine twisted yarn. Half-woolen is produced either from half-woolen twisted yarn in the warp and in the weft, or from woolen yarn in the warp and cotton yarn in the weft. Width of wool blend and wool diagonal 124-139 cm . Cotton diagonal is produced from medium quality twisted yarn, width 142-147 cm .
Some silk and staple fabrics are also given the name «diagonal». Silk diagonal is produced from silk yarn of natural silk and viscose silk threads, width 70-103 see . Staple is produced from twisted staple viscose yarn and is subjected to waterproof impregnation, width 83-132 cm . All types of diagonal are produced one-colored. This is a beautiful, durable fabric, mainly men’s suits and uniforms are sewn from it.
Heavy fabric applications
diagonal
From a dense diagonal woven from woolen warp and cotton weft threads, military uniforms are sewn. Coats and jackets in military style, equipment for tourists, fishermen and hunters, uniforms made of this material have the stiffness and strength required for fabrics for this category of wardrobe items.
Summer overalls for protection against general industrial pollution ( suits , overalls and dressing gowns ) are sewn from a cotton diagonal, the addition of polyester fiber to which reduces the amount of shrinkage of the product after washing.
Raincoats, Aprons and mitten handguards are usually made of a rubberized variety. The wear resistance of this dense diagonal fabric has found its application in the production of bags, backpacks, bags, furniture upholstery.
Diagonal can be called a universal material. Everything that requires special strength is sewn from it, for example, army uniforms. This fabric makes excellent overalls and even shoes .
A small amount of additives further improves the quality of the diagonal, making it stronger and more durable. This modified fabric is used to make suits for heavy industry workers, furniture upholstery. The so-called «light» diagonal is made from pure natural cotton, from which wonderful dresses and suits are obtained.
The geometric concept «diagonal» is a dense fabric that has characteristic oblique small scars on the surface.
Such textile , which may include wool , cotton , silk, synthetics, is particularly durable, which is why it is used for military uniforms, camping equipment, work uniforms and various accessories.
Diagonal , falling under the «suit» category, is used for casual and business wear, whose owners value durability and comfort.
Diagonal features
This fabric is woven in a twill weave in which the warp and weft threads overlap by one pitch. As a result, diagonal scars are formed on the surface of the canvas, which is why it acquires increased strength, and at the same time elasticity. Strengthening the strength is also facilitated by the fact that this fabric, like crepe, is often made from reinforced twisted threads.
A real boom in the use of this material occurred at the end of the century before last — the beginning of the last century, and is associated with the name of the French general Galiffe.
Due to the effects of the wound, the general had to wear trousers of a special cut, subsequently named after him. The convenience of riding breeches, especially for riding, was appreciated by the military, and the most suitable fabric for them was the diagonal, the composition of which at that time included only wool . Diagonal riding breeches were iconic menswear during the First World War, and in the USSR the fashion for them continued until the fifties . The very same woolen and cotton diagonal is still widely used in sewing military uniforms.
The most common diagonal composition at present is cotton . This textile has a number of advantages:
- abrasion and tear resistance;
- low price;
- hygiene, good air exchange;
- durability and no deformation;
- environmental friendliness and hypoallergenicity.
This fabric is used to make durable and reliable travel and work clothing.
Also, this material, especially camouflage, is well suited for the current military style. When adding synthetic fibers, primarily elastane, diagonal fabric can be used as a practical suit and material for durable casual trousers and jackets
.
A special place in the assortment of fabrics is occupied by diagonal crepe, which can be either synthetic or silk.
Silk crepe diagonal is an elite textile , which is used for elegant men’s and women’s clothing.
Like any crepe, this fabric is different:
- high strength and durability;
- no shrinkage;
- elasticity and resilience;
- good thermoregulation in both hot and cold weather;
- is stain resistant.
Thus, both the composition and the structure of any kind of diagonal make this textile a good choice for clothing and accessories that are able to maintain their appearance for a long time and will not let you down in conditions of increased loads and not particularly favorable weather conditions.
What is sewn diagonally?
The high reputation of this fabric as a material for military uniforms of various types makes it popular in the creation of special and tourist clothing. To improve the protective properties of this textile, is treated with water-repellent impregnations, it is treated to protect it from dust and dirt. Diagonal is used not only for clothes ( jackets , trousers , overalls ), but also for shoes, various accessories and devices (bags, mittens, awnings, protective covers), as well as in the production of furniture, especially folding furniture.
A special place is occupied by a diagonal weave fabric (including crepe), which includes wool and silk. This textile is used for the manufacture of business style suits, elegant and durable trousers, a variety of outerwear, sports and leisure shoes.
Fabric diagonal is widely used in tailoring uniforms for the army, as well as for the manufacture of overalls and unique footwear.
But casual clothes , bags , bags and furniture did not pass by such a wonderful material. It is simply indispensable in the manufacture of those fabric products, which are subject to increased requirements regarding strength characteristics and density. Also, they cannot do without a diagonal during repair, construction and other various works. Depending on the required volume of manufactured products, such material is ordered either in bulk or in small bulk (from one roll).
The cost of a diagonal is much lower than, for example, the price of a tarpaulin fabric. Such a difference is quite understandable and predictable, because the manufacture of such a material requires much less raw materials, as well as less labor costs. That is why industrial companies and, first of all, sewing shops for large volumes of workwear are more likely to buy a diagonal, however, ordinary buyers are often also interested in purchasing such a durable and high-quality fabric.
Despite the fact that the diagonal fabric has a completely natural basis, at the moment, manufacturers are increasingly using various chemical compounds that give such a material even more improved strength or special properties. Such additives can strengthen fibers, increase wear resistance, and improve physical characteristics. It should be noted that the modified diagonal is used in various areas of heavy industry. And the so-called «easy» Diagonal fabric is made from 100% cotton fiber. That is why when buying the type of diagonal you need, you need to monitor the composition of which the fabric is made.
Diagonal fabric has huge advantages in the market of materials that are used for sewing various workwear: wear resistance, strength, density, reliability. Technologists have long noticed a special way of weaving, which distinguishes it into a separate category of fabrics that can withstand significant loads. The inherent durability and accessibility of the diagonal made it one of the most popular fabrics, which is very firmly rooted in almost all spheres of life of a modern person.
Such a fabric as diagonal fabric will never lose its popularity due to its relative cheapness, and, of course, naturalness. This fabric is one of the classic representatives of the cotton industry.
Diagonal care
Caring for products made from this material is easy. All manipulations — washing, spinning, drying can be easily carried out in the usual way, however, referring to the attached label. Usually such things are looked after in almost the same or exactly the same way as cotton ones. The care of ribbed fabric depends on the fibers that make up its composition, and can be both simple and gentle.
Diagonal products are recommended to be dry-cleaned; cotton and staple diagonal can be washed respectively as cotton fabrics or as rayon fabrics
The only condition is the prohibition of bleaches for colored fabrics.
Cotton textile tolerates all modes of automatic washing and ironing at high temperatures, however, if impregnated, it must be cared for in accordance with the manufacturer’s instructions.
