How to Identify and Calculate Diagonals in Different Shapes
What is Diagonal?
A diagonal, in geometry, is a line segment joining two vertices of a polygon or polyhedron, if the vertices present are not present on the same edge. In general, any sloping line is called a diagonal.
Here you can read about the basic properties of diagonals, different types of diagonals, and some easy examples.
Key Definitions to Remember While learning Diagonals
Polygon: A polygon can be defined as a two-dimensional closed shape with straight sides. It is also defined as a flat or plane.
There are two types of Polygons are Regular polygon and irregular polygon
Vertex: (Plural-vertices) it is described as the corner of any shape.
Number of Diagonals in a Polygon
The number of diagonals in a polygon is based on the number of sides of that particular polygon. We can easily calculate the number of diagonals in a polygon by using a simple formula.
Number of diagonals = (n(n-3))/2, where n is the number of vertices in the polygon
For Example- Calculate the number of diagonals in a 12-sided regular polygon.
According to the formula, number of diagonals in a polygon = (n (n-3))/ 2.
So, the 12-sided polygon will contain 12(12-3)/2 = 54 diagonals.
Diagonal of a Polygon Formula
If any vertex is connected by sides to two other vertices those connections cannot count as diagonals. That vertex cannot connect to itself either.
n: the number of sides
sum of diagonal = [(5-3) *5] / 2 = 5
Diagonal of a Square
A square has two diagonals. The line segment joining two corners of a square is known as its diagonal.
Properties of the Diagonals in a Square
The diagonals of a square bisect each other.
Each diagonal of the square divides the square in such a way that it becomes an isosceles triangle. The isosceles triangles formed are congruent to each other.
The diagonals of a square are parallel and perpendicular to each other.
In case a circumcircle is drawn, the diameter of the circumcircle is equal to the length of the diagonal.
The diagonals of a square are also called congruent diagonals as the length of the diagonals of the square is the same.
If you have the length of the side of a diagonal then you can easily find out the length of the diagonal of the square. You have to use the given formula to find out the length of the diagonal of a square.
The length of the diagonal of a square = a/sqrt(2) where a is the length of any one of the sides of the square.
Diagonal of a Rectangle
A special type of parallelogram whose angles are at ninety degrees is called a rectangle. The diagonals of a rectangle are congruent to each other and they bisect each other. A rectangle is one of the Quadrilaterals with congruent diagonals. The opposite angles formed at the point where diagonals meet are also congruent.
You can calculate the length of the diagonal of a rectangle easily if you are provided with the dimensions of its sides. The formula given here is the way you can calculate the length of the diagonal in a rectangle.
The length of the diagonal of a rectangle = sqrt (a² + b²), where ‘a’ is the length of the rectangle and ‘b’ is the breadth of the rectangle.
Rectangle and Formulas
Area= Length * Width
Perimeter= 2L + 2W
Diagonal = \[\sqrt{L^{2}+W^{2}}\]
Diagonal of a Rhombus
In a rhombus the angles which are opposite to each other are equal. These angles formed are bisected by the diagonals of a rhombus. A rhombus has four sides with two Diagonals. When all the angles of the rhombus turn 90 degrees it becomes a square.
You can also calculate the length of the diagonal of a rhombus if you are provided with the area.
A= (PQ)/2, where p and q are the lengths of diagonals of the rhombus.
The formula for the length of one diagonal of a rhombus is = p = 2a/q
Area of Rhombus
Area of Rhombus = \[\frac{(d1\ast d2)}{2}\]
Diagonal d2= \[\sqrt{4a^{2}+d1^{2}}\]
Diagonal d1 = \[\sqrt{4a^{2}+d2^{2}}\]
Diagonal of a Parallelogram
A parallelogram is a type of rectangle whose angle is not equal to 90 degrees. The two diagonals of a parallelogram bisect each other, and the opposite sides and angles of any parallelogram are congruent.
You can also calculate the length of one diagonal of a parallelogram if the dimensions of its sides and the length of the other diagonal is provided to you.
Length of one diagonal of a Parallelogram = p = sqrt (2a² + 2b² – (q)²) where p is the diagonal length which you don't know, q is the diagonal length provided to you, a
and b is the length and breadth of the parallelogram, respectively.
Diagonal of Parallelogram
Relation between sides and diagonals of a parallelogram
Type equation here.
d12 + d22 = 2(a2+b2)
FAQs on Diagonals in Geometry: Definition, Formula & Key Concepts
1. What is a diagonal in geometry?
In geometry, a diagonal is a straight line segment that connects two non-consecutive or non-adjacent vertices of a polygon or a polyhedron. In simple terms, it's a line that goes from one corner to another, but not along an edge. For example, in a square, a line drawn from the top-left corner to the bottom-right corner is a diagonal.
2. What is the main difference between a side and a diagonal of a polygon?
The key difference lies in the vertices they connect. A side of a polygon connects two adjacent (or consecutive) vertices, forming the boundary of the shape. A diagonal, on the other hand, connects two non-adjacent vertices, passing through the interior of the polygon.
3. How do you calculate the length of a diagonal in common geometric shapes?
The formula for a diagonal's length depends on the shape, often using the Pythagorean theorem. Here are some common examples:
- Square: The length of a diagonal is d = s√2, where 's' is the length of one side.
- Rectangle: The length of a diagonal is d = √(l² + w²), where 'l' is the length and 'w' is the width.
- Cube: The length of a space diagonal is d = s√3, where 's' is the length of one edge.
4. Why is the Pythagorean theorem so important for finding diagonal lengths?
The Pythagorean theorem (a² + b² = c²) is fundamental because a diagonal often acts as the hypotenuse of a right-angled triangle formed by the sides of the shape. For instance, in a rectangle, the length, width, and diagonal form a right-angled triangle. By knowing the side lengths, we can use the theorem to directly calculate the diagonal's length.
5. Are there different types of diagonals in geometry?
Yes, especially when discussing three-dimensional shapes. The two main types are:
- Face Diagonal: A diagonal that lies on one of the flat faces of a polyhedron. For example, a diagonal across one of the square faces of a cube.
- Space Diagonal: A diagonal that passes through the interior of a polyhedron, connecting two vertices that are not on the same face. For example, a line connecting the bottom-front-left corner to the top-back-right corner of a cube.
6. Is there a general formula to find the number of diagonals in any polygon?
Yes, there is a universal formula to calculate the number of diagonals for any polygon with 'n' sides. The formula is: Number of Diagonals = n(n-3)/2. This formula works because from each of the 'n' vertices, you can draw a diagonal to (n-3) other vertices (excluding the vertex itself and its two adjacent vertices). The result is divided by 2 to avoid counting each diagonal twice (once from each endpoint).
7. Why does a triangle have no diagonals?
A triangle has no diagonals because the definition of a diagonal requires it to connect two non-adjacent vertices. In a triangle, every vertex is adjacent to the other two vertices. There are no pairs of non-adjacent vertices to connect, so it is impossible to draw a diagonal. Using the formula n(n-3)/2, for a triangle (n=3), we get 3(3-3)/2 = 0.
