Diagonals in shapes: General Data Protection Regulation(GDPR) Guidelines BYJU’S

What is Diagonal? — Definition Facts & Example

Diagonals in Geometry

A polygon is defined as a flat or plane, two-dimensional closed shape bounded with straight sides. A diagonal is a line segment connecting the opposite vertices (or corners) of a polygon. In other words, a diagonal is a line segment connecting two non-adjacent vertices of a polygon. It joins the vertices of a polygon, excluding the edges of the figure. The following shapes have a diagonal drawn on them:

History of the Diagonal

The word diagonal comes from the ancient Greek word diagonios, which means “from angle to angle.” Both Euclid and Strabo used it to describe a line that connects two vertices of a cuboid or a rhombus; later, it became known in Latin as diagonus (slanting line).

Diagonals of Polygon

Diagonal Formula

Diagonals for polygons of all shapes and sizes can be made and for every shape; there is a formula to determine the number of diagonals.

The number of diagonals in a polygon with n vertices = $\frac{n(n-3)}{2}$

So, from this formula, we can easily calculate the number of diagonals in a polygon.

The given table shows the number of diagonals in different polygons:

Diagonals of Solid Shapes

Just like polygons, solid or 3D shapes also have diagonals. Based on the number of edges, the number and properties of diagonals vary for different solids. The following solids have some diagonals drawn on them:

Length of a Diagonal

The length of diagonals of any shape depends on the dimensions of its sides.

Length of Diagonal of Square

The length of the diagonal of a square can be derived using the Pythagoras theorem. A diagonal of a square divides it into two right-angled triangles. Applying the Pythagoras theorem, we can find the length of the diagonal (d) of a square with side (a) as a$\sqrt{2}$.

Diagonal length of a square with each side a units  = a$\sqrt{2}$ units

Length of Diagonal of Rectangle

A diagonal of a rectangle divides it into two right-angled triangles. {2}}$, where d is diagonal, l is length, and b is the breadth of the rectangle. 

Solved Examples

1. What is the total number of diagonals in a polygon of 12 sides?

Solution:

The number of diagonals in a polygon with n vertices = $\frac{n(n-3)}{2}$

Therefore, the number of diagonals in a polygon with 12 sides  = $\frac{12(12-3)}{2}$ = 54

2. What is the length of the diagonal of a square with each side 6 cm long?

Solution:

Side, a = 6 cm

Length of the diagonal = a $\times \sqrt{2}$

                             = 6 $\times \sqrt{2}$

                             =  6$\sqrt{2}$ cm

3. Rahul is strolling across a rectangular park that is 20 meters long and 15 meters wide. Determine the diagonal of the rectangular park.

Solution:

Length of the rectangular park = 20 m, Breadth of the rectangular park = 15 m

Length of the diagonal = $\sqrt{l^{2} + b^{2}}$

                                           = $\sqrt{20^{2} + 15^{2}}$

                                           = $\sqrt{400 + 225}$

                                      = $\sqrt{625}$

                                           = 25 m

Practice Problems

5

6

8

9

Correct answer is: 9
Number of diagonals in hexagon (6 vertices) = $\frac{6(6-3)}{2}$ = 9

$\sqrt{10}$ cm

2$\sqrt{10}$ cm

10 cm

20 cm

Correct answer is: 2$\sqrt{10}$ cm
Breadth and Length of the rectangle is 2 cm and 6 cm respectively. {2}}$ = $\sqrt{4+36}$ = $\sqrt{40}$ = $2\sqrt{10}$ cm

6 cm

24 cm

3$\sqrt{2}$ cm

24$\sqrt{2}$ cm

Correct answer is: 24 cm
Diagonal of a square with side length a is a$\sqrt{2}$ . Since a$\sqrt{2}$ = 6$\sqrt{2}$, a must be 6 cm.
Therefore, the perimeter of the square must be 4 × 6 cm or 24 cm.

Frequently Asked Questions

Which polygon has an equal number of sides and diagonals?

A pentagon has five sides and five diagonals. <br>

Number of diagonals in a polygon with n vertices = $\frac{n(n-3)}{2}$ <br>
Number of diagonals in a pentagon = $\frac{5(5-3)}{2}$ = 5

Why does the triangle not have any diagonal?

A polygon’s diagonal is a line connecting a vertex to a non-adjacent vertex. As a result, the simplest polygon, a triangle, seems to have no diagonals. We can’t connect a line from one internal angle to another that isn’t also a side of the triangle.

Which quadrilaterals have diagonals that bisect each other?

When each diagonal of a polygon cuts the other diagonal into two equal parts, they are said to bisect each other. The quadrilaterals with bisecting diagonals are rectangle, square, parallelogram, and rhombus.

Number of Diagonals in a Polygon — Concept

A diagonal is a segment that connects two non-consecutive vertices in a polygon. The number of diagonals in a polygon that can be drawn from any vertex in a polygon is three less than the number of sides. To find the total number of diagonals in a polygon, multiply the number of diagonals per vertex (n — 3) by the number of vertices, n, and divide by 2 (otherwise each diagonal is counted twice).

vertex

diagonal

non consecutive vertices

polygon

How many diagonals are a polygon with 300 sides? Well wouldn’t make a whole lot of sense to draw a polygon with 300 sides and draw on all those diagonals. There has to be a shortcut or a formula.
Well, first let’s back up. What is a diagonal? A diagonal is any line segment that connects two non-consecutive vertices. So if we look at a triangle. If I look at every single vertex, again the vertex is where two ends meet two sides meet. There’s no way for me to draw on a diagonal here because for this vertex both of these sides are consecutive. So there’s no way for us to have any diagonals.
If I look at a square however, I can see that there is one non-consecutive vertex if I look at this vertex. I look at another vertex there’s only one non-consecutive vertex. So let’s see if we can figure out the pattern. To do that, we’re going to use this table over here where I have three columns; one for the number of vertices, one for the number of diagonals per vertex and the total number of diagonals that we see in a polygon.
So we’ve already started with two different polygons. We’ve talked about a triangle. So, number of vertices in a triangle well, that’s just three. The number of diagonals we said was zero because there is no way for us to draw in a diagonal. Which means our total diagonals is still zero. Okay?
Let’s go back and look at the square. The square we said, there are 1, 2, 3, 4 vertices. This vertex right here has only one diagonal, this vertex right here has only one diagonal so we’re four vertices, each vertex has one diagonal but we only see two of them. So we see that there’s going to be some sort of division that’s going to have to go on here.
Last, let’s look at a pentagon. If I look at this vertex, I can draw in one, two diagonals. And I’m going to see that for every vertex, I’m going to be able to draw in, two different diagonals. So number of vertices here is five, number of diagonals per vertex is two and the total of diagonals here we have a little star so we have five diagonals. So I want to know first for n vertices because I’m going to draw dot dot dot for n vertices, what will be the total number?
Well I see that if I multiply 3 times 0, so we make a dot here. 3 times 0 is 0 so we’re okay there. Here we have 4 times 1, but that does not equal 2. So what we’re going to have to do is I’m going to have to take 4 times 1 and divide that in half. 5 times 2 divided in half is equal to 5. So I look at the number of vertices we have three so we’re going to call that n. Here we have number of diagonals per vertex, here we have 0, 1 and 2 and I see that to get from 3 to 0 I’m going to subtract 3 to get from 5 to 1, I subtract 3 from 5 to 2 I subtract 3. So we have n times the quantity of n minus 3 all divided by 2.
So two key things about this formula right here which tells you the number of diagonals and I’m going to abbreviate DIAG.
So the number of diagonals, there’s two key things I want to point out. The first is this n-3. Where is n-3 come from? Well if we have five vertices here. We’re not going to count the vertex that’s itself because you can’t draw a vertex to itself plus there are two more consecutive vertices for total of three vertices in this polygon that we’re not really counting.
Second key part here is this divide by 2. Why do we have to divide this by 2? If I go back to this to the square, if I look at this vertex, I’ve drawn in one diagonal. From this vertice’s perspective, I’ve only drawn one. From this vertex’s perspective I’ve drawn another diagonal. But it’s the same diagonal. So every vertex and every diagonal that we draw from a vertex will be counted twice which is why we have to divide our formula by two.

Mathematics Properties of the diagonals of a rectangle. Properties of the diagonals of a square

9. Properties of the diagonals of a rectangle. Properties of the diagonals of a square

Organizational stage You know that one person can be bored and uninterested. More fun together, faster together, stronger together. This applies not only to the game, but also to work. Today our lesson will be held under the motto: «Only together we are strong. » In the lesson we will remember the names of geometric shapes, find out what the diagonal of a rectangle is, we will learn how to build diagonals, get acquainted with the properties of the diagonals of a rectangle and a square.

Stage of preparing students for active conscious assimilation of knowledge Oral account Reference Calculate in a convenient way. 23+123+67 110+28+390 928+22+126 333+222+777 Test yourself. 23+123+67 = 23+(123+67)=213 23+123+67 = 123+(23+67)=213 110+28+390 = (110+390)+28=528 928+22+126 = (928+22)+126=1076 333+222+777 = (333+777)+222=1332 Discuss tasks in groups. Reference Petya thought of one of the numbers (10, 20, 30 or 40). Ask him only two questions to find out what number he has in mind. 1st question: Is it 10 or 20? If “yes”, then the 2nd question: Is it the number 10? If «no», then Petya conceived the number 20. If the answer to the 1st question is “no”, then the 2nd question is: Is it the number 30? If «no», then Petya conceived the number 40.

Reference Ivanov, Smirnov and Belov parked cars in the yard. Ivanov does not have the smallest car, and Smirnov did not park the car next to Ivanov’s car. Where is whose car? From left to right: cars of Ivanov, Belov, Smirnov.

Acquisition of new knowledge Work in pairs, consult, check. Consider geometric shapes. What is the common name for them? Squares, rectangles, pentagons, hexagons, squares. These are polygons. What groups can all polygons be divided into? Can be divided into quadrangles and non-quadrangles. Let’s look at the quadrilaterals. What do they have in common? All quadrilaterals have four sides, four vertices, four corners. Draw a rectangle with sides 4 cm and 2 cm. Divide it into 4 equal parts. This can be done in several ways by connecting the midpoints of opposite sides or the vertices of opposite sides. Who knows the name of the segment connecting the opposite vertices of the rectangle? The segment connecting the opposite vertices of the rectangle is called diagonal . Draw a square with a side of 4 cm. Draw diagonals in it. Let’s find out the properties of the diagonals of a rectangle and a square. To do this, we need a compass and a right triangle. Diagonal properties Rectangle Square Diagonals are drawn in the FDSA rectangle. Let’s put one leg of the compass at vertex F, the other one — at the opposite vertex S. Without changing the distance between the legs, we move one leg of the compass to vertex A. We see that the second leg of the compass is at vertex D. What conclusion can be drawn? The diagonals of the rectangle are equal. Write to the table. Denote the square by the letters MNKL. Check the lengths of the diagonals. The diagonals of the square are equal. Write to the table. Let’s denote by the letter O the point of intersection of the diagonals in the rectangle. Let’s check the length of the segments obtained by crossing the diagonals with a compass. We see that the segments FO, OS, DO and OA are equal. Make a conclusion. The segments obtained by crossing the diagonals of a rectangle are equal. Write to the table. Let’s check the segments formed at the intersection of diagonals in a square. The segments obtained by crossing the diagonals of the square are equal. Write to the table. What angles are obtained when the diagonals intersect? When the diagonals of a rectangle intersect, sharp and obtuse angles are obtained. When the diagonals of a square intersect, 4 right angles are obtained. Write in the table in the properties of the square. What properties of the diagonals of a rectangle and a square are similar? First property: the diagonals are equal. Second property: the segments obtained by crossing the diagonals are equal. Rectangle Square 1) The diagonals are equal. 2) The segments obtained by crossing the diagonals are equal. 3) When the diagonals intersect, 4 right angles are obtained.

Material fixing Draw a rectangle ABCD with sides 6 cm and 3 cm in a notebook. Draw diagonals, measure them and draw a conclusion. Diagonal AC is 6 cm 7 mm, diagonal BD is 6 cm 7 mm. The diagonals of the rectangle AC and BD are equal. Now do the same with the 3 cm square ZXEP. Diagonal ZE is 4 cm 2 mm and diagonal XP is 4 cm 2 mm. The diagonals of the square ZE and XP are equal. Check the second property of the rectangle and square diagonals. The segments obtained by crossing the diagonals are equal.

What is the area of ​​a rectangle with sides 5 cm and 3 cm? Solution: 5 3 = 15 (cm²) — area of ​​the rectangle Answer: 15 cm². The area of ​​a rectangle is 15 cm². The area of ​​a rectangle is 48 cm² and the length of one side is 12 cm. What is the perimeter of the rectangle? Check your solution. Solution: 1) 48 : 12 = 4 (cm) — length of the second side of the rectangle 2) (12 + 4) 2 = 32 (cm) — perimeter of rectangle Answer: 32 centimeters.

Debriefing What questions would you like to ask each other about the topic of the lesson? Formulate them.

Reflection When is it easier and more comfortable for you to work: in pairs, in a group or alone?

calculation formulas, properties and finding through diameter

Geometry

11/12/21

7 min.

When solving problems in physical and mathematical disciplines, sometimes it is necessary to find the diagonals of a rectangle. The formula on the Internet is not always reliable. It is very important at the initial stages of calculations to correctly identify the figure in order to apply the necessary properties and relationships to it. Experts recommend not to start immediately to practice, but to deal with the theory.

Contents:

• General information
• Formulas and ratios
• Calculation example

A rectangle is a geometric plane figure consisting of four pairwise parallel sides, between which right angles are formed. It can be confused with a square, which has similar properties and identities. When solving a problem, it is very important to correctly find a figure that has certain signs of definition. Some students confuse the latter with properties. These two terms are different.

Identification Method

Feature — a set of some criteria that allow you to correctly distinguish figures. Rectangle can be identified by the following rules:

1. Inequality of sides that are adjacent.
2. Diagonals do not form a 90 degree angle when they intersect.
3. Diagonals are not angle bisectors of large triangles obtained by intersection.
4. A circle can only be described, not inscribed.

If at least one of the signs is applicable for the desired figure, then it can be classified as a rectangle.

After successful identification, it is necessary to proceed to the consideration of properties that are recommended by experts to use in calculating parameters and proving statements (identities and theorems).

Important properties of

Properties is a set or list of assertions and identities used in computing required quantities and for proving theorems, and 91/2), i.e. the quadratic value of the diagonal is equal to the sum of the sides, each of which is multiplied by the equivalent value.

Similarity of small and large triangles formed by diagonals.

There is only a circumscribed circle whose diameter is equivalent to the diagonal of the rectangle.

When drawing a diagonal, two equal triangles are formed, which are right-angled.

It should be noted that the above properties are the required minimum, which is not enough to perform calculations and prove other identities.

Formulas and ratios

To navigate the formulas, you need to enter some notation. These include the following:

1. Diagonal — m.
2. Sides — k and l.
3. Perimeter — P.
4. Semiperimeter — p.
5. Area — S.
6. An acute angle formed by two diagonals is Z and an obtuse angle is Y.
7. Diameter — D.

After that it is necessary to consider the main identities. They are recommended to be used when calculating various parameters of a figure. 9(1/2) * 0.5. The values ​​»(2k + 2l)» can be replaced by the perimeter P when it is known. 2. 92-14l+48=0.

Calculate its roots: l1=6 and l2=8.

Substitute in the fourth paragraph and count the sides: l=6 and k=8.

It should be noted that the calculation of the roots is performed by substitution, in which there are duplicate solutions. Among them, you need to choose any two pairs. Based on the ninth point, you can calculate the value of the area, knowing the two sides. Using formulas, you can find other parameters. For example, calculate the value of an acute angle.

Thus, before solving problems in geometry, mathematicians recommend correctly identifying a geometric figure using signs, and then using any ratios.

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