Shapes are typically identified by their corners, sides, and diagonals. A diagonal is a line connecting two corners of a polygon that do not share a side. Every polygon with more than 3 sides will have diagonals.
This topic today will delve into “what is a diagonal line” in geometry as well as explain the mathematical methods for counting diagonals.
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What is a diagonal line?
The term “diagonal” is derived from the Greek word diagnosis, which means “from angle to angle.” It was used by Euclid and Strabo to denote a line connecting two vertices of a cuboid or a rhombus; it was eventually known in Latin as diagonus (slanting line).
A diagonal refers to a straight line that runs across the vertices of a polygon and links its opposing corners. In other terms, a polygon’s diagonal is a line segment that connects any two non-adjacent corners.
A diagonal, for example, will connect opposing corners of a square that would otherwise be separated by another point. Lines in geometry extend in both directions indefinitely; a diagonal line is not a line since it is enclosed between two points.
Depending on how many sides a polygon has, distinct polygons will have a different number of diagonals.
Diagonals of 2D shapes
A polygon’s diagonal is a line segment that links any two non-adjacent vertices. The number of diagonal lines and their properties varies depending on the kind of polygon and the number of sides.
Let’s learn more about the diagonals of several polygons now.
Diagonal of a Square
A closed, two-dimensional shape with four sides and four corners is referred to as a square. All of the sides are the same length and the opposite sides are parallel to one another. A line segment that connects any two of a square’s opposing vertices is said to be its diagonal.
Diagonal of a Rectangle
A line segment connecting any two of the non-adjacent vertices of a rectangle is the rectangle’s diagonal.
Diagonal of a Rhombus
The line segments connecting the opposing vertices of a rhombus are known as the diagonals. Because these line segments bisect one another at a 90° angle, all diagonals have equal lengths on both sides. A quadrilateral with four equal sides that has a diamond-like form is known as a rhombus.
Diagonal of a Pentagon
A pentagon is a 2-dimensional closed shape having 5 sides and 5 corners. A regular pentagon has five sides that are of the same length.
Diagonal of a Hexagon
A hexagon is a 2-dimensional closed shape having 6 sides and 6 corners. A regular hexagon has six sides that are of the same length.
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Diagonals of 3D shapes
The diagonals in 3D shapes are similar to those in polygons. They feature body diagonals (space diagonals) plus face diagonals since they are three-dimensional figures. Based on the number of edges, various solids have varying numbers of diagonals and varied characteristics.
Diagonal of a Cube
A cube is a 3-dimensional solid shape with the same length edges as a square solid. This implies that a cube’s length, breadth, and height are all the same, and all of its faces are squares.
A cube’s body diagonal is the line segment that runs through its center, connecting the opposing vertices. The face diagonal, on the other hand, is the one that connects the opposing vertices on each face.
Diagonal of a Cuboid
A cuboid is a rectangle in three dimensions. A cuboid’s body diagonal is the diagonal that runs through its center, whereas the face diagonal connects the opposing vertices on each face.
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The formula for the Number of Diagonals
The number of diagonals in a polygon varies depending on its kind and the number of sides.
The following formula is used to calculate the number of diagonals in a polygon:
The number of diagonals = n (n-3)2
where n is the number of sides of the polygon
We can employ this formula to determine the number of diagonals in any polygon without sketching it.
For instance, count the diagonals in a decagon.
Answer:
n = 10 makes up the number of sides of a decagon. The following formula can be used to determine a decagon’s number of diagonals:
n (n-3)2=10 (10-3)2= 35
This means that a decagon has 35 diagonals.
| Shape | Number of Diagonals | |||
|---|---|---|---|---|
| Triangle | 0 | |||
| Quadrilateral | 2 | |||
| Pentagon | 5 | |||
| Hexagon | 9 | |||
| Heptagon | 14 | |||
| Octagon | 20 | |||
| Nonagon | 27 | |||
| Decagon | 35 | |||
| Hendecagon | 44 | |||
| Dodecagon | 54 |
Diagonal Length
The length of a diagonal in any polygon is determined by the kind of polygon. There is no generic formula for calculating the diagonal length. Rather, the formula for calculating the length of the diagonal can be determined from the dimensions of the specific polygon.
This part will go over the formula for calculating the diagonal length of a square and a rectangle.
The diagonal length of a square
The Pythagorean theorem may be used to calculate the length of a square’s diagonal. A square’s diagonal divides it into 2 right-angled triangles.
We can determine the length of the diagonal (denoted by d) of a square with a side (denoted by a) using Pythagoras’ theorem.
d = a2
The diagonal Length of a rectangle
A rectangular diagonal divides it into 2 right-angled triangles. We can determine the length of the diagonal of a rectangle with length (denoted by l) and width (denoted by b) using Pythagoras’ theorem.
d2 =l2+b2
As a result, d2 =l2+b2, where d is diagonal, l is the length, and b is the width of the rectangle.
FAQs
Why is there no diagonal in the triangle?
The diagonal of a polygon is a line that connects a vertex to a non-adjacent vertex. As a result, the most basic polygon, a triangle, seems to lack diagonals. We cannot link a line from one internal angle to another that is not also a triangle side.
Which four-sided shapes have diagonals that cross one another?
It is referred to as bisecting when one diagonal of a polygon splits the other diagonal into two equal halves. The parallelogram, rhombus, square, and rectangle are the quadrilaterals that have bisecting diagonals.
Summary
It is critical for youngsters to understand what is a diagonal line. This geometrical word can be used to define a shape. Children will learn about horizontal, vertical, parallel, and perpendicular lines as part of their shape work. If you are planning to teach your kids about this essential topic, you can make your own collections of diagonal line worksheets using our worksheet maker.
