How Many Diagonals Does A Convex Quadrilateral Have

Introduction

A convex quadrilateral has 2 diagonals. A convex quadrilateral is a four-sided polygon. All interior angles in this polygon measure less than 180 degrees. Shapes like Squares, Rectangles, Rhombus, Kites and Parallelograms are examples of convex quadrilaterals. In this article, we will learn about a convex quadrilateral's diagonals. We will learn about their limitations and conditions and the formulas to find the number of diagonals.

How Many Diagonals Does A Convex Quadrilateral Have?

A convex quadrilateral has two diagonals. A diagonal is a line segment connecting two points not connected by the side in the polygon. As we know that all the interior angles in the convex quadrilateral are less than 180, And also it has 4 vertices and 4 sides, Thus, there are only two line segments possible in the polygon between the points which are not connected to a side.

For example, we have a convex quadrilateral ABCD. We notice that any side does not connect (A, C) and (B, D). Hence we have two diagonals in a convex quadrilateral.

Formula Method

We can also find the number of diagonals by the formula below.

=\frac{\mathbf{n}(\mathbf{n}-\mathbf{3})}{\mathbf{2}}

In this formula, n represents the number of sides the polygon (a convex quadrilateral in this case) has. We input the number in the formula, and after solving the equation, we have the number of diagonals a convex quadrilateral has.

For example, in the above convex quadrilateral, we have 4 sides (A, B)(B, C)(C, D)(D, A). Thus n=4, inputting the value in the formula:-

The number of diagonals in a convex quadrilateral

\begin{aligned}

&=\frac{4(4-3)}{2} \\

&=2 \times 1 \\

&=2

\end{aligned}

So, there are 2 diagonals in the ABCD convex quadrilateral.

Important Facts About Convex Quadrilateral

• Both diagonals in a convex quadrilateral lie inside the polygon.

• All 4 interior angles must be less than 180 degrees for a quadrilateral to be called a Convex.

• A diagonal is a line segment joining two points which are not connected to any side.

Notes

• A convex quadrilateral has 2 diagonals, as it has only two line segments which lie inside and which join points which are not connected by sides.

• A concave quadrilateral has 2 diagonals. One diagonal lies inside the quadrilateral, while the other lies outside the quadrilateral.

• A convex quadrilateral is called a four-sided polygon with all its interior angles less than 180 degrees.

• All convex quadrilaterals have four sides and interior angles of less than 180 degrees. All convex quadrilaterals will have only two diagonals.

• Squares, Rectangles, Rhombus, Kites and Parallelogram are some types of convex quadrilaterals.