Open In App

Tangential Quadrilateral Formula

Improve
Improve
Like Article
Like
Save
Share
Report

In mensuration, a tangential quadrilateral is defined as a convex quadrilateral whose all sides are tangent to a single circle within itself. This quadrilateral is also known by the name circumscribable quadrilateral or circumscribing quadrilateral, as it is drawn by encircling or circumscribing its incircles. This circle is known as the quadrilateral’s incircle or inscribed circle, and its centre and radius are known as the incenter and the inradius respectively.

 

Tangential Quadrilateral Formula

In a tangential quadrilateral, the sum of lengths of a pair of opposite sides is equal to the other. In other words, if a tangential quadrilateral has side lengths as a, b, c and d then we can say that a + c = b + d. Also, it can be said that sum of one pair of opposing sides is equal to the semi-perimeter of the quadrilateral. The area of a tangential quadrilateral is equal to the square root of the product of its sides. 

A = √(abcd)

where p, q, r and s are the side lengths of the tangential quadrilateral.

The formula can also be written as,

A = rS

where,

r is the radius of inscribed circle,

S = (a + b + c + d)/2 is the semi-perimeter of the quadrilateral.

Sample Problems

Problem 1. Find the missing length of a tangential quadrilateral if three of its sides are, 10 cm, 15 cm and 21 cm.

Solution:

We have,

a = 10, b = 15 and c = 21.

Using the opposite side property of tangential quadrilateral we get,

a + c = b + d

=> 10 + 21 = 15 + d

=> 31 = 15 + d

=> d = 16 cm

Problem 2. Find the area of a tangential quadrilateral if its side lengths are, 7 cm, 11 cm, 14 cm and 10 cm.

Solution:

We have,

a = 7, b = 11, c = 14 and d = 10.

Using the formula we get,

A = √(abcd)

= √(7 × 11 × 14 × 10)

= √10780

= 103.82 sq. cm

Problem 3. Find the area of a tangential quadrilateral if its side lengths are, 9 cm, 13 cm, 16 cm and 12 cm.

Solution:

We have,

a = 9, b = 13, c = 16 and d = 12.

Using the formula we get,

A = √(abcd)

= √(9 × 13 × 16 × 12)

= √22464

= 149.87 sq. cm

Problem 4. Find the area of a tangential quadrilateral if the radius of the inscribed circle is 5 cm and its semi-perimeter is 80 cm.

Solution:

We have,

r = 5 and S = 80.

Using the formula we get,

A = rS

= 5 (80)

= 400 sq. cm

Problem 5. Find the radius of the inscribed circle of a tangential quadrilateral if its semi perimeter is 40 cm and the area is 120 sq. cm.

Solution:

We have,

S = 40 and A = 120.

Using the formula we get,

A = rS

=> r = A/S

=> r = 120/40

=> r = 3 cm 

Problem 6. Find the radius of the inscribed circle of a tangential quadrilateral if its perimeter is 200 cm and the area is 845 sq. cm.

Solution:

We have,

P = 200 and A = 845.

We know, P = 2S

=> S = 200/2

= 100 cm

Using the formula we get,

A = rS

=> r = A/S

=> r = 845/100

=> r = 84.5 cm

Problem 7. Find the perimeter of a tangential quadrilateral if the radius of the inscribed circle is 10 cm and the area is 280 sq. cm.

Solution:

We have,

r = 10 and A = 280.

Using the formula we get,

A = rS

=> S = A/r

=> S = 280/10

=> S = 28 cm

We know, P = 2S.

P = 2 (28)

= 56 cm


Last Updated : 21 Mar, 2024
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads