Diagonals

A diagonal is a straight line that runs through the vertex of a polygon and connects its opposite corners. To learn about diagonals, we must first understand the following:

• It’s a line segment (diagonal).

• Polygons are plane figures with at least three sides and angles, and they’re frequently used to designate figures with five or more.

• A vertex is a shape’s corner.

• A diagonal is a line segment that connects two non-adjacent vertices of a polygon. It joins a polygon’s vertices while disregarding the figure’s edges.

Formula to calculate the diagonal

Diagonals can be formed for polygons of different shapes and sizes, and there is a formula for calculating the number of diagonals.

In a polygon with n vertices, the number of diagonals is equal to n(n-3)/2

We can simply compute the number of diagonals in a polygon using this formula.

Diagonals of different geometrical shapes

Polygons: A polygon is a plane figure characterized by a finite number of straight-line segments joined together to form a closed polygonal chain in geometry (or polygonal circuit). A polygon can be a bounded planar region, a bounding circuit, or both at the same time. The edges or sides of a polygonal circuit are the segments that make up the circuit. The vertices (singular: vertex) or corners of a polygon are the spots where two edges meet.

Diagonals of a triangle

A triangle is a closed shape with three sides, three angles, and three vertices. It’s the most basic sort of polygon. In a triangle, there are no non-adjacent vertices. It signifies that no diagonals can be formed using line segments. As a result, the number of diagonals in a triangle equals zero.

Diagonals of a square

A square is a four-sided, four-cornered, closed two-dimensional figure. The opposite sides are parallel to one another, and all sides are the same length. The diagonal is a line segment that connects any two of a square’s opposing vertices. To connect the facts below, look at the following square and diagonals:

• A square has two diagonals.

• Two diagonals, AC and BD, result from linking the opposite vertices.

• The diagonals AC and BD are both the same length in the provided square.

• A square’s diagonal divides it into two equal right triangles, with the diagonal serving as the hypotenuse of the right-angled triangles created.

Diagonal of a square = a√2 

Diagonals of a rectangle

The diagonal of a rectangle is a line segment that connects any two of its non-adjacent vertices. Look at the rectangle below, where the diagonals are AC and BD. We can observe that both AC and BD have the same length. A diagonal divides a rectangle into two right triangles, with the diagonals serving as the hypotenuse.

Diagonal of a rectangle = √(l²+b² )

Diagonals of a rhombus

The diagonals of a rhombus are the line segments that unite the opposite vertices and bisect them at a 90° angle, ensuring that the two halves of every diagonal are equal in length. A rhombus is a quadrilateral with four equal sides that is fashioned like a diamond.

Diagonals of 3D shapes

3D shapes feature diagonals, just like polygons. They have body diagonals (space diagonals) and face diagonals because they are three-dimensional figures. The number of diagonals and their attributes change depending on the number of edges in a solid. Let’s study more about a cube’s diagonal and a cuboid’s diagonal.

Diagonals of a cube

A cube is a three-dimensional solid shape with equal-length edges, commonly known as a square solid. A cube’s length, breadth, and height are all the same, and all of its faces are squares. The line segment that goes through the center of a cube and connects the opposing vertices is called the body diagonal. The face diagonal, on the other hand, is the line that connects the opposing vertices of all the faces. There are 16 diagonals in a cube. Each of the six sides of a cube is a square. Each face, therefore, has two diagonals. Consider the length x units of a cube. The length of each faces is diagonal of the cube Equals √(2x) units, according to Pythagoras’ theorem. There are four more diagonals (body diagonals) that cross through the center of the square in addition to the diagonals on the faces. The formula for the length of a cube’s diagonal is the same as the formula for the length of a square’s diagonal. The length of a cube’s body diagonal is √(3x) units.

Diagonals of a cuboid

A rectangle in three-dimensional form is called a cuboid. A cuboid’s body diagonal is the line that runs through its center, while the face diagonal is the line that connects the opposing vertices on each face. There are 16 diagonals in a cuboid. Consider a cuboid with dimensions of l, w, and h. Assume that the body diagonal that goes through the cuboid’s center is ‘d.’ √(l² + w² + h²) is the length of a cuboid’s body diagonal.