Tangential Quadrilateral Formula: Characteristics & Examples

Tangential Quadrilateral is a quadrilateral in which a circle is inscribed and all the sides of the quadrilateral act as tangents to the circle. The enclosed circle is known as an incircle or inscribed circle with respect to the quadrilateral. Some synonyms of Tangential Quadrilateral are circumscribable quadrilateral, circumscriptive quadrilateral and circumscribing quadrilaterals.

Read More: Inradius of Incircle

Key Terms: Tangential Quadrilateral, Formula, Area, Incircle

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Tangential Quadrilateral

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Common quadrilaterals are square, rectangle, parallelogram, kite, diamond, rhombus etc. But the Tangential Quadrilateral is a little different from the mentioned quadrilaterals. Tangential Quadrilaterals refer to those quadrilaterals that have a well-defined circle inscribed in it.

Tangential Quadrilateral

The four respective sides of the quadrilateral turn out to be tangents for the circle. The sides of the Tangential Quadrilateral may or may not be equal. 

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Characteristics of Tangential Quadrilateral

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• 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. 

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Area Formula of Tangential Quadrilateral

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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.

Also Read: Section Formula

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Construction of Tangential Quadrilateral

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1. Create the angle bisector of the four angles in the corner. 
2. Meet the angle bisectors at a point inside the quadrilateral. This point is called the incentre of the inscribed circle. 
3. 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.
4. The distance between the centre of the circle and point of tangent is the radius of the inscribed circle. 

Construction of Tangential Quadrilateral

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Things to Remember

• Before considering any quadrilateral as a tangential quadrilateral, first verify whether pairs of opposite sides are equal or not i.e., a+c = b+d. 
• Tangential quadrilaterals come under the category of convex quadrilaterals. Convex quadrilaterals are those quadrilaterals whose interior angles are less than 180o and diagonals lie inside the perimeter. 
• While drawing the inscribed circle all the angle bisectors should meet at a single point inside the circle (incentre). 

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