Diagonal of Parallelogram Formula

A parallelogram is defined as a quadrilateral that has equal pairs of opposite sides and angles. One of its pairs of opposite sides is parallel to each other. The interior angles lying on the supplementary in nature, that is, their sum is 180 degrees. The diagonals of a parallelogram bisect each other, that is, they divide each other in two equal parts. The sum of all interior angles of a parallelogram is 360 degrees.

The above figure depicts a parallelogram ABCD with sides AB, BC, CD and AD and diagonals AC and BD. Here the lengths of opposite sides AB and CD are equal to each other. Similarly, the lengths of BC and AD are the same. The pairs of opposite angles, that is, ∠A and ∠C and ∠B and ∠D are equal to each other.

Diagonal of Parallelogram Formula

The formula for the length of a diagonal of a parallelogram is equal to the magnitude of the resultant of any two adjacent sides.

x = √(a² + b² – 2ab cos A) = √(a² + b² + 2ab cos B)

y = √(a² + b² + 2ab cos A) = √(a² + b² – 2ab cos B)

where,

x and y are the lengths of diagonals,

a and b are adjacent side lengths,

A and B are the angles formed between the sides.

The diagonal lengths and sides of a parallelogram have a relation between each other. The sum of squares of diagonals is equal to twice the sum of squares of two adjacent sides.

x² + y² = 2(a² + b²)

where,

x and y are diagonal lengths,

a and b are adjacent side lengths.

Sample Problems

Problem 1. Calculate the length of the diagonals of a parallelogram of side lengths 5 m and 10 m, if one of the interior angles is 60°.

Solution:

We have,

a = 5

b = 10

∠A = 60°

∠B = 120°

We have to find the diagonal lengths x and y.

Using the formula we get,

x = √(a² + b² – 2ab cos A)

= √(5² + 10² – 2 (5) (10) cos 60°)

= √75

= 8.66 m

y = √(a² + b² + 2ab cos A)

= √(5² + 10² + 2 (5) (10) cos 60°)

= √175

= 13.22 m

Problem 2. Calculate the length of the diagonals of a parallelogram of side lengths 4 m and 7 m, if one of the interior angles is 30°.

Solution:

We have,

a = 4

b = 7

∠A = 30°

We have to find the diagonal lengths x and y.

Using the formula we get,

x = √(a² + b² – 2ab cos A)

= √(4² + 7² – 2 (4) (7) cos 30°)

= √16.48

= 4.06 m

y = √(a² + b² + 2ab cos A)

= √(4² + 7² + 2 (4) (7) cos 30°)

= √73.63

= 8.5 m

Problem 3. Calculate the length of one of the diagonals of a parallelogram of side lengths 5 m and 9 m, if one of the interior angles is 25°.

Solution:

We have,

a = 5

b = 9

∠A = 25°

We have to find the diagonal length.

Using the formula we get,

x = √(a² + b² – 2ab cos A)

= √(5² + 9² – 2 (5) (9) cos 25°)

= √24.40

= 4.06 m

Problem 4. Calculate the length of one of the diagonals of a parallelogram of side lengths 12 m and 16 m, if one of the interior angles is 37°.

Solution:

We have,

a = 12

b = 16

∠A = 37°

We have to find the diagonal length.

Using the formula we get,

x = √(a² + b² – 2ab cos A)

= √(12² + 16² – 2 (12) (16) cos 37°)

= √93.315

= 9.66 m

Problem 5. Calculate the length of one of the diagonals of a parallelogram of side lengths 4 m and 6 m, if the other diagonal is 8 m.

Solution:

We have,

a = 4

b = 6

x = 8

Using the formula we get,

x² + y² = 2(a² + b²)

=> 8² + y² = 2 (4² + 6²)

=> 64 + y² = 2 (16 + 36)

=> 64 + y² = 104

=> y² = 40

=> y = 6.32 m

Problem 6. Calculate the length of one of the diagonals of a parallelogram of side lengths 8 m and 12 m, if the other diagonal is 14 m.

Solution:

We have,

a = 8

b = 12

x = 14

Using the formula we get,

x² + y² = 2(a² + b²)

=> 14² + y² = 2 (8² + 12²)

=> 196 + y² = 2 (16 + 144)

=> 196 + y² = 320

=> y² = 124

=> y = 11.13 m

Problem 7. Calculate the length of one of the diagonals of a parallelogram of side lengths 7 m and 9 m, if the other diagonal is 11 m.

Solution:

We have,

a = 7

b = 9

x = 11

Using the formula we get,

x² + y² = 2(a² + b²)

=> 11² + y² = 2 (7² + 9²)

=> 121 + y² = 2 (49 + 81)

=> 121 + y² = 260

=> y² = 139

=> y = 11.78 m