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
Formula of Tangential Quadrilateral
Let a convex quadrilateral with sides a, b, c, d, then the area of a Tangential quadrilateral is, a + c = b + d
Examples of Tangential Quadrilateral Formula
Example: Find the measure of the fourth side of a quadrilateral circumscribed about a circle, if three other sides have the measures of 5 cm, 6 cm and 4 cm listed consecutively.
Solution
Let x be the measure of the fourth side of our quadrilateral. Since the quadrilateral is circumscribed about a circle, the sums of the measures of its opposite sides are equal. Thus, you can write the equation:
5 + 4 = 6 + x
From this equation, x = 5 + 4 – 6 = 3
Answer: The fourth side of the quadrilateral is of 3 cm long.
Tangential Quadrilateral
A quadrilateral which has an incircle, i.e., one for which a single circle can be constructed which is tangent to all four sides. Opposite sides of such a quadrilateral satisfy
where 
is the inradius. Using Bretschneider’s formula together with (1) and (3) then gives the beautiful formula
where 
and 
are the diagonal lengths.
A rhombus is a special case of a tangential quadrilateral.
Characteristics of Tangential Quadrilateral
- All the four sides of the quadrilateral must touch the circle.
- The circle should be fully contained within the quadrilateral. No part of the circle must protrude outside the quadrilateral.
- The angle bisectors of four sides of quadrilateral meet at the centre of the inscribed circle.
- The sum of length of two opposite sides must be equal. For example if the four sides of a tangential quadrilateral are as a,b,c and d. Then, a+c = b+d.
- Any quadrilateral can become tangential quadrilateral if it satisfies all the above conditions.
Area Formula of Tangential Quadrilateral
Area of a Tangential Quadrilateral can be derived from the semi-perimeter and radius of the inscribed circle. Take the following suppositions:
Four sides are a,b,c and d.
Semi-Perimeter (Half Perimeter) or s = (a+b+c+d)/ 2
Radius of the incircle or inscribed circle = r
Thus, area of the quadrilateral, A = r.s
Another formula for area of quadrilateral is, A= √abcd
But before calculating make sure you check whether a+c = b+d.
Construction of Tangential Quadrilateral
- Create the angle bisector of the four angles in the corner.
- Meet the angle bisectors at a point inside the quadrilateral. This point is called the incentre of the inscribed circle.
- From that point draw a circle which touches all sides of the quadrilateral. The sides of the quadrilateral touching the circle are tangents to it. Thus, a tangential quadrilateral is made.
- The distance between the centre of the circle and point of tangent is the radius of the inscribed circle.
Sample Questions
Ques: Which parameters of the tangential quadrilateral are needed to find the area? (2 Marks)
Ans: As we know, the formula to find the area of the tangential quadrilateral is A = r.s where, r is the radius of the inscribed circle and s is the semi-perimeter of the quadrilateral. Sometimes in problems, you need to find out the semi-perimeter of the quadrilateral and length of the sides are given. In that case you need to have length of all four sides, unless the area is given.
Ques: Solve the following problem:
The area of a tangential quadrilateral needs to be found. Semi-Perimeter of the quadrilateral is 16 cm, radius of the circle is 8 cm. (2 Marks)
Ans. The formula of the area of tangential quadrilateral says, A (area) = r (radius) * s(semi-perimeter).
Thus, Area = 8*16 = 128 cm2.
Ques: Four sides of a tangential quadrilateral are 4 cm, 3 cm, 5 cm and 6 cm. The distance between the incentre and point of contact on the quadrilateral is 5 cm. Find the area of the quadrilateral. (2 Marks)
Ans: First of all, calculate the semi-perimeter.
Semi-perimeter (s) = (4+3+5+6)/2 = (18/2) = 9 cm.
The distance between the incentre and point of contact on the quadrilateral is the radius of the inscribed circle, which is = 5 cm.
Hence, Area = 9*5 = 45 cm2.
Ques: The square of the product of length of four sides is 81. Find the area of the tangential quadrilateral. (2 Mark)
Ans: If four sides of the quadrilateral are a,b,c and d then abcd = 81
=> We know that, A= √abcd
=> A = √81 = 9 cm2
Thus, Area of the tangential quadrilateral is 9 cm2.
Ques: If thrice the area of a tangential quadrilateral is 54cm2 and double the perimeter is 24 cm. Find radius of the inscribed circle. (3 Marks)
Ans: Let us take Area = a, perimeter = p, radius = r.
Given, 3*a = 54 cm2, 2*p = 24 cm.
Thus, a = 54/3 = 18 cm2 and p = 24/2 = 12 cm.
Semi-perimeter (s) = 12/2 = 6 cm.
As per the formula of area, A = r.s
=> 18 = r*6
=> 18/6 = r
Therefore, r = 3 cm.
Ques: If the perimeter of a tangential quadrilateral is 1/4th of its area, which is 88 cm2. Find the radius of the inscribed circle. (3 Marks)
Ans: Given, area = 88 cm2 and perimeter = 1/4th of area
→ perimeter = 1/4 * 88 = 22 cm.
→ Semi-perimeter = 22/2 = 11 cm.
→ Area = Radius*Semi-Perimeter
→ 88 = Radius*11
Thus, Radius = 88/11 = 8 cm
Ques: Diameter of the inscribed circle in a tangential quadrilateral is 14 cm. Sum of four sides of the quadrilateral is 32 cm. Calculate the area of the tangential quadrilateral. (3 Marks)
Ans: Given, diameter = 14 cm and sum of four sides = 32 cm.
→ Radius (r) = diameter/2 = 14/2 = 7 cm.
→ Sum of four sides means perimeter (p), which is 32 cm.
→ Semi-perimeter (s) = 32/2 = 16 cm.
→Area = r.s
Thus, Area = 7*16 = 102 cm2.
Ques: The area of a tangential quadrilateral is 48 cm2 and radius of inscribed circle is 6 cm. Length of three sides of the quadrilateral are given as 6 cm, 4 cm, 3 cm, length of one side is unknown. Find the length of the unknown side. (5 Marks)
Ans: Given data, area = 48 cm2 and radius = 6 cm.
Let us assume the length of the unknown side as x cm.
Now, the semi-perimeter (s) = (6+4+3+x)/ 2 = (13+x)/ 2 cm
We know that, Area = r.s
→ 48 = 6*(13+x)/ 2
→ 48*2 = 6*(13+x)
→ 96/6 = 13+x
→ 16 – 13 = 3 = x
Therefore, the length of the unknown side is 3 cm.
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