**Concave Quadrilaterals** are those with **one angle greater than 180°**. They also have general properties like, one of their diagonals being outside the quadrilateral. The most common example of a concave quadrilateral is *the dart*.

Index

**History**

The idea of a concave quadrilateral was first proposed in the 10^{th }century by * Aryabhata – II*. he mentioned these in his work the

*. Even mathematicians like*

**Mahasiddantah***Aryabhata*and

*Bhaskhara*who came before him only mentioned

*regular*and

*cyclic quadrilaterals*in their works.

**Properties of Concave Quadrilateral**

- One of the internal angles is a
**reflex angle**, i.e, it is between**180°**and**360°**. - One of the diagonals lies
**outside**the bounds of the quadrilateral. - The sum of internal angles is 360°, same as the convex quadrilaterals.
- It is
. This is because an equilateral quadrilateral has its opposite sides parallel to each other and also its side lengths must be equal.*not possible*to draw*equilateral concave quadrilaterals*

## Applications **of Concave Quadrilateral**

- The dart shape is used in aeroplane wings and even an aeroplane itself is made in the shape of a dart.

- This shape is also found in nature when moths fold their wings.

- This shape is also found in boomerangs.

**Example Problems**

**Question 1.** If the internal angles of a dart are 50°, 30° and 30°, then find the fourth angle.

**Solution.** We know that dart is also a quadrilateral, so it must follow the rule that its sum of internal angles is 360°.

From using this we can say that,

∠A + ∠B + ∠C + ∠D = 360°

50 + 30 + ∠C + 30° = 360°

∠C = 360° – 110°

∠C = 250°

Hence the reflex angle of the given dart ABCD is 250°.

**Question 2.** Show that the sum of internal angles of a concave quadrilateral is 360°.

**Solution.** We know that the sum of internal angles of any polygon can be found by using the formula,

Sum of internal angles = 180°(n – 2)

Here; ‘n’ is number of sides of the polygon

So, by substituting 4 in this formula we get the value of the sum of all internal angles of a this quadrilateral.

Sum of internal angles = 180(4 – 2) = 180(2) = 360°.

Hence, the sum of internal angles is 360°.

**FAQs**

**What is a concave quadrilateral?**A quadrilateral one of whose angle is a reflex angle is called a concave quadrilateral.

**Define a concave quadrilateral.**A concave quadrilateral is defined as, a four sided polygon with one of its angles greater than 180°, and its diagonals partially outside the bounds of the polygon.

**Can an equilateral concave quadrilateral be drawn?**No, an equilateral concave quadrilateral cannot be drawn, as one of its requirements is that opposite sides must be parallel.

**Are non-convex and concave quadrilaterals the same?**No, non-convex quadrilaterals are not only concave quadrilaterals but also complex quadrilaterals which crossover on themselves.