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Diagonals are different types of straight lines that connect a polygon opposite corners by its vertices. In other words, the diagonal of a polygon is a line segment that joins two non-adjacent corners.
- Different polygons can have different numbers of diagonals depending on the number of sides.
- The number of vertices is not present on the same edge.
- The shape of a diagonal is that of a straight line.
- They are bigger than the sides of the polygon.
- Diagonals of a particular figure bisect each other at 90 degrees.
- It is derived from a Greek word named diagonios, which means angle to angle.
- Depending on the type of polygon and the number of sides, the number of diagonals and their properties vary.
Read More: Centroid Formula
Key Terms: Diagonals, Polygon, Vertices, Straight Line, Angle, Sides, Triangle, Square, Rectangle, Rhombus, Parallelogram, Hexagon, Cube
What are Diagonals?
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A diagonal line is a line segment that connects two vertices of a shape which are not already connected by an edge. It doesn't go straight up, down or across. The shape of the diagonals is always a straight line.
- Diagonals are straight line that connects the opposite corners of a polygon or a polyhedron by its vertex.
- It is defined as shapes that have corners or lateral shapes.
- It depends upon the number of sides of a figure.
- This means a polygon consists of more than one diagonal.
- Diagonal lashing and diagonal pliers are some common examples of diagonals.
In the above figure, AC and AC' are the diagonals of the shape.
Solved Example of DiagonalsExample 1: Find the diagonal of a square whose side measure is 3 cm Solution: Given:
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Read More: Quadrilateral angle sum property
Diagonals Formula
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If “n” is the number of vertices of a polygon, then the number of diagonals of a polygon can be found using the formula:
Number of diagonals of a polygon with “n” vertices = \(\frac{n(n-3)}{2}\)
Solved Example of Diagonals FormulaExample 1: Find the number of diagonals in a octagon. Solution: The number of sides in a decagon is n = 8. The number of diagonals of a octagon is calculated using the formula: n(n - 3)/2 = 8(8 - 3)/2 = 8(5)/2 = 40/2 = 20 Therefore, the number of diagonals of a octagon = 20 |
Read More: Understanding Quadrilaterals
Diagonals of Shape
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Different shapes have different numbers of diagonals of different lengths. Here, we discuss the diagonals of various shapes in geometry.
Diagonals of Triangle
A triangle is a 3-sided enclosed polygon that has three vertices. No vertices of the triangle are non-adjacent. Hence, a triangle does not have any diagonal. The number of diagonals for a triangle is determined to be 0.
Diagonals of a Triangle = 0
Read More: Areas Related to Circles
Diagonals of Square
The diagonals of a square are the line segments that connect opposite vertices of the square. A square has two diagonals. The two diagonals of the square are congruent. The diagonals of a square divide into two. Each diagonal bisects the square in two congruent isosceles right triangles.
- The formula to find the length of the diagonal of a square is:
Diagonals of a Square = a√2
- Where "a" is the length of any side of a square.
Solved Example of Diagonals of SquareExample 1: Find the diagonal of a square whose side measure is 9 cm Solution: Given:
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Diagonals of Rectangle
A rectangle has two diagonals as it has four sides. Like a square, the diagonals of a rectangle are congruent and bisected. If a diagonal bisects a rectangle, we get two congruent right triangles.
- The formula to find the length of the diagonal of a rectangle is:
Diagonals of a Rectangle = √l2+b2
- Where "l" and "b" are the length and breadth of the rectangle, respectively.
Solved Example of Diagonals of RectangleExample 1: Find the diagonal of a rectangle with a side of 6 cm and 6 cm respectively. Solution: Given, Length of Rectangle(l) = 6 cm
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Read More: Difference between Area and Volume
Diagonals of Rhombus
A rhombus has four sides, and its two diagonals bisect each other at right angles. If all the angles of a rhombus are 90 degrees, a rhombus is a square or a rectangle. Since all the sides of the rhombus are congruent, the opposite angles are parallel.
- Area of a rhombus, A = (½) pq square units
- Where "p" and "q" are the two diagonals of the rhombus.
- Thus, the formula to find the length of the diagonal of the rhombus is:
Diagonals of a Rhombus, p = 2(A)/q and q = 2(A)/p
Solved Example of Diagonals of RhombusExample 1: Find the length of the diagonal of a rhombus if the area is 54 units2, and one diagonal is 9 units. Solution: The area of the rhombus = 54 square units; one diagonal (q) = 9 units We will use the formula for the diagonal of rhombus, p = (2 × Area)/q, where 'p' and 'q' are the two diagonals of the rhombus. Substituting the values, p = (2 × 54)/9 = 12 units. |
Diagonals of Parallelogram
A parallelogram is a quadrilateral. A parallelogram's opposite sides and corners are congruent, and the diagonals are bisected. The length of the diagonals of the parallelogram is determined using the formula:
The diagonal of a parallelogram:
Diagonals of Parallelogram, d1 = p = \(\sqrt{2a^2+2b^2 – q^2}\)
Diagonals of Parallelogram, d2 = q = \(\sqrt{2a^2+2b^2 – p^2}\)
Solved Example of Diagonals of ParallelogramExample: Calculate the length of the diagonal of a parallelogram with sides 2 units, 3 units and an interior angle A which is equal to 60 degrees. Solution: Given, a = 2 units, b = 3 units, angle A = 60°
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Read More: Area of Parallelogram
Diagonals of Pentagon
A pentagon is a closed five-sided shape or polygon with five vertices. A regular polygon has all of its sides the same length. A pentagon has a total of five diagonals, which are connected by opposite non adjacent vertices.
Diagonals of Pentagon = 5
Diagonals of Hexagon
A hexagon is a six-sided closed shape that has five vertices. It is a polygon that has a total of nine diagonals when the nonadjacent corners are joined together.
Diagonals of Hexagon = 9
Diagonals of Cube
A cube is a three-dimensional shape with six square faces of equal size. It has 12 edges and eight vertices. The primary diagonals of the cube are the straight lines that pass through the centre of the cube and join the opposite vertices. The diagonals of the faces of the cube are the straight lines that join the opposite vertices on each face. Therefore;
- Number of primary diagonals of cube = 4
- The number of diagonals on the faces of the cube = 12
- Total diagonals of the cube = 12 + 4 = 16
Solved Examples of CubeExample: Calculate the length of the body diagonal of a cube whose each side measures 3 units. Solution: Given, length of each side of the cube (a) = 3 units; body diagonal = ? So, let us substitute the given values in the formula, Length of body diagonal of a cube = √3a = √3 × 3 = 5.19 units |
Diagonals of Cuboid
A cuboid is also a three-dimensional shape which has six rectangular faces. Similar to a cube, it has 12 edges and eight vertices. It is also called a rectangular prism. Since the cube and the cuboid structure are similar, the number of diagonals of the two shapes will also be the same.
- Number of primary diagonals of cuboid = 4
- The number of diagonals on the faces of the cuboid = 12
- Total diagonals of the cuboid = 12 + 4 = 16
Solved Example of Diagonals of CuboidExample: Find the length of the diagonal of cuboid with dimensions 2 × 3 × 4. Solution: We have l = 2, b = 3 and h = 4. So, using the diagonal of cuboid formula, the length of the diagonal is given by, Body diagonal = √(l2 + b2 + h2) units = √(22 + 32 + 42) units = √(16 + 9 + 4) = √(29) units |
Read More: Area of Square using Diagonals
Number of Diagonals
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Given below is the summary of the number of diagonals of different polygons:
| Polygon | Number of Vertices | Calculation | Number of Diagonals |
|---|---|---|---|
| 3 | \(\frac{[3(3-3)]}{2}\) | 0 | |
| Quadrilateral | 4 | \(\frac{[4(4-3)]}{2}\) | 2 |
| 5 | \(\frac{[5(5-3)]}{2}\) | 5 | |
| Hexagon | 6 | \(\frac{[6(6-3)]}{2}\) | 9 |
| 7 | \(\frac{[7(7-3)]}{2}\) | 14 | |
| 8 | \(\frac{[8(8-3)]}{2}\) | 20 |
Length of Diagonals
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The length of diagonals of different shapes depends on their dimensions.
| Length of Diagonal | Formulas |
|---|---|
| Length of diagonal of Square | a√2 where a is the length of side of square |
| Length of diagonal of Rectangle | √l2+b2 where l and b are the length and breadth of the rectangle respectively |
| Length of diagonal of Cube | a√3 where a is the length of edges of the cube |
| Length of diagonal of Cuboid | √(l2 + b2 + h2) where l, b and h are length, breadth and height of cuboid |
Read More: Types of Polygon
Things to Remember
- Diagonals are defined as line segments that join one vertex of a polygon with another.
- It can be calculated with the formula \(\frac{n(n-3)}{2}\)where 'n' represents the number of facets in the polygon.
- Diagonals of a rectangle is calculated with the formula, diagonal (d) = \(\sqrt{l^2+b^2}\)
- A triangle has no diagonals.
- The number of diagonals of a cube is 16.
- Length of the diagonals of a cuboid is calculated with the formula, d = √(l2 + b2 + h2)
Read More: Cyclic Quadrilateral
Sample Questions
Ques. If a polygon has 15 sides, how many diagonals does it have? (2 marks)
Ans. Let us assume that the number of sides of the given polygon is 'n'. We can find the number of diagonals of the given polygon with the formula, Number of diagonals = n(n-3)/2
Substituting the value of n = 15
n(n-3)/2 = 15(15-3)/2 = (15 × 12)/2 = 90. Therefore, the polygon has 90 diagonals.
Ques. Determine the diagonal of a rhombus whose area is 50cm2 and one of its diagonal measures is 7cm. (5 marks)
Ans. Area of a rhombus = 50cm2
One of the diagonal of a rhombus, say q = 7cm
Thus, the formula to find the diagonal, p is given as:
Diagonal, p = 2(A)/q
Now, substitute the given values in the formula, we get:
p = 2(50)/7
p = 100/7
p= 14.28, which is approximately equal to 14.3
Hence, the other diagonal of a rhombus is 14.3 cm.
Ques. Which polygon has equal diagonals? (1 marks)
Ans. A rhombus is a parallelogram with four equal sides. We know that the diagonals of a rhombus divide in two and are perpendicular. A rectangle is a parallelogram with four 90° angles. The rectangle of a given rhombus bisects and has equal diagonals.
Ques. The angles of a quadrilateral are in the ratio of 2 : 3 : 5 : 8. Find the measure of each angle. (2 marks)
Ans. Sum of all interior angles of a quadrilateral = 360°
Let the angles of the quadrilateral be 2x°, 3x°, 5x° and 8x°.
2x + 3x + 5x + 8x = 360°
⇒ 18x = 360°
⇒ x = 20°
Hence the angles are
2 × 20 = 40°,
3 × 20 = 60°,
5 × 20 = 100°
and 8 × 20 = 160°.
Ques. In the given figure, ABCD is a rhombus. Find the values of x, y and z. (2 marks)
Ans. AB = BC (Sides of a rhombus)
x = 13 cm.
Since the diagonals of a rhombus bisect each other
z = 5 and y = 12
Hence, x = 13 cm, y = 12 cm and z = 5 cm.
Ques. If AM and CN are perpendiculars on the diagonal BD of a parallelogram ABCD, Is \( \bigtriangleup\)AMD≅ \( \bigtriangleup\)CNB? Give a reason. (2 marks)
Ans. In triangles AMD and CNB,
AD = BC (opposite sides of parallelogram)
∠AMB = ∠CNB = 90°
∠ADM = ∠NBC (AD || BC and BD are transversal.)
So, \( \bigtriangleup\)AMD≅ \( \bigtriangleup\)CNB by AAS
Ques. The diagonal of a rectangle is thrice its smaller side. Find the ratio of its sides. (3 marks)
Ans. Let AD = x cm
diagonal BD = 3x cm
In right-angled triangle DAB,
AD2 + AB2 = BD2 (Using Pythagoras Theorem)
x2 + AB2 = (3x)2
⇒ x2 + AB2 = 9x2
⇒ AB2 = 9x2 – x2
⇒ AB2 = 8x2
⇒ AB = √8x = 2√2x
Required ratio of AB : AD = 2√2x : x = 2√2 : 1
Ques. Length and breadth of a rectangular wire are 9 cm and 7 cm respectively. If the wire is bent into a square, find the length of its side. (2 marks)
Ans. Perimeter of the rectangle = 2 [length + breadth]
= 2[9 + 7] = 2 × 16 = 32 cm.
Now perimeter of the square = Perimeter of rectangle = 32 cm.
Side of the square = 32/4
= 8 cm.
Hence, the length of the side of the square = 8 cm.
Ques. In the given figure ABCD, find the value of x. (2 marks)
Ans. Sum of all the exterior angles of a polygon = 360°
x + 70° + 80° + 70° = 360°
⇒ x + 220° = 360°
⇒ x = 360° – 220° = 140°
Ques. What is a regular polygon? (2 marks)
State the name of a regular polygon of
(A) 3 sides (C) 4 sides
(B) 6 sides
Ans. A polygon with equal sides and equal angles is called regular polygon.
(A) Equilateral triangle is a figure that has three sides.
(B) Square is a figure that has four sides.
(C) Regular hexagon is a figure that has six sides.
Ques. Determine the diagonal of a rhombus whose area is 60cm2 and one of its diagonal measures is 8 cm. (5 marks)
Ans. Area of a rhombus = 60cm2
One of the diagonal of a rhombus, say q = 8cm
Thus, the formula to find the diagonal, p is given as:
Diagonal, p = 2(A)/q
Now, substitute the given values in the formula, we get:
p = 2(60)/8
p = 120/8
p= 15 cm
Ques. Length and breadth of a rectangular wire are 10 cm and 9 cm respectively. If the wire is bent into a square, find the length of its side. (2 marks)
Ans. Perimeter of the rectangle = 2 [length + breadth]
= 2[10 + 9] = 2 × 19 = 38 cm.
Now perimeter of the square = Perimeter of rectangle = 38 cm.
Side of the square = 38/4
= 9.5 cm.
Hence, the length of the side of the square = 9.5 cm.
Ques. The angles of a quadrilateral are in the ratio of 2 : 2 : 6 : 8. Find the measure of each angle. (2 marks)
Ans. Sum of all interior angles of a quadrilateral = 360°
Let the angles of the quadrilateral be 2x°, 2x°, 6x° and 8x°.
2x + 2x + 6x + 8x = 360°
⇒ 18x = 360°
⇒ x = 20°
Hence the angles are
2 × 20 = 40°,
2 × 20 = 40°,
6 × 20 = 120°
and 8 × 20 = 160°.
Ques. Find the length of the diagonal of cuboid with dimensions 2 × 3 × 5. (2 marks)
Ans. We have l = 2, b = 3 and h = 5. So, using the diagonal of cuboid formula, the length of the diagonal is given by,
Body diagonal = √(l2 + b2 + h2) units
= √(22 + 32 + 52) units
= √(16 + 9 + 25)
= √(50) units
Ques. Calculate the length of the diagonal of a parallelogram with sides 7 units, 8 units and an interior angle A which is equal to 60 degrees. (2 marks)
Ans. Given, a = 7 units, b = 8 units, angle A = 60°
- Using diagonal of parallelogram formula,
- p=√x2+y2−2xycosA
- Putting the values in the formula for p:
- p=√72+82−28
- p=√49 + 64 – 6
- p=√107
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