Question

How many diagonals does a convex quadrilateral have?

Answer 1

Step 1: Define convex quadrilateral and formula used

1. A convex quadrilateral is a four-sided polygon that has interior angles that measure less than $180°$ each.
2. Whereas, a quadrilateral having one of its interior angles more than $180°$ is called a concave quadrilateral.
3. Examples of a convex quadrilateral include square, rectangle, rhombus, kite, parallelogram etc.

The number of diagonals in an $n$ sided polygon is given as,

${\mathit{D}}_{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{3}\mathbf{)}}{\mathbf{2}}$

Step 2: Calculate the number of diagonals in a convex quadrilateral

A convex quadrilateral has $4$ sides, i.e., $n=4$

Thus, the number of diagonals in a convex quadrilateral is,

$\begin{array}{cl}& D_4=\frac{4(4-3)}{2}\\ \Rightarrow & D_4=2\times 1\\ \Rightarrow & D_4=2\end{array}$

Therefore, a convex quadrilateral has $2$ diagonals.