Based on the measure of their interior angles, all quadrilaterals are classified into two groups: convex and concave.
Convex and Concave Quadrilateral
What is a Convex Quadrilateral
A convex quadrilateral is a quadrilateral having all its interior angles measuring less than 180°. Both the diagonals of a convex quadrilateral lie inside the closed figure.
In the given convex quadrilateral ABCD, all its four interior angles ∠ABC, ∠BCD, ∠CDA, and ∠DAB measure less than 180°. Also the diagonals AC and DB are found to reside inside the closed figure.
Given is a quadrilateral ABCD, having ∠ABC = 80°, ∠BCD = 70°, ∠CDA = 110°. Find the measure of the ∠DAB and state whether it is a convex quadrilateral.
Solution:
As we know, Sum of the interior angles in a convex quadrilateral = 360° Thus, in quadrilateral ABCD, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360° => 80° + 70° + 110° + ∠DAB = 360° => 280° + ∠DAB = 360° => ∠DAB = 360° – 260° => ∠DAB = 100°
Since all the interior angles of the given quadrilateral ABCD measure less than 180°, it is a convex quadrilateral.
What is a Concave Quadrilateral
A concave quadrilateral is a quadrilateral having at least one of its interior angles measuring more than 180°. One of the diagonals of a concave quadrilateral lies outside the closed figure. Any quadrilateral-shape that is not convex is thus a concave quadrilateral.
Example: A dart or an arrowhead.
Concave Quadrilateral
In the given concave quadrilateral ABCD, ∠BCD measures more than 180°, and the diagonal BD lies outside the closed figure.
Given is a quadrilateral ABCD, having ∠ABC = 30°, ∠DAB = 80°, and ∠CDA = 25°. Find the measure of the ∠BCD and state whether it is a concave quadrilateral.
Solution:
As we know, Sum of the interior angles in a concave quadrilateral = 360° Thus, in quadrilateral ABCD, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360° => 30° + ∠BCD + 25° + 80° = 360° => ∠BCD + 135° = 360° => ∠BCD = 360° – 135° => ∠BCD = 225°
Since one of the interior angles of the given quadrilateral ABCD measure more than 180°, it is a concave quadrilateral.
FAQs
Q1. What is the difference between a convex and a concave quadrilateral?
Ans. A convex quadrilateral has all its interior angles measuring less than 180° and has both the diagonals lying inside the closed figure. In contrast, a concave quadrilateral has one of its interior angle measuring more than 180° and one of its diagonals lie outside the closed figure.
Q2. Why rectangle is a convex quadrilateral?
Ans. Since all the four interior angles of a rectangle measure less than 180°and both diagonals lie wholly inside the closed shape, it is a convex quadrilateral.
Q3. Can a concave quadrilateral be regular?
Ans. Since all the interior angles in a concave quadrilateral are not equal (one of its interior angles measuring more than 180°) it can never be a regular quadrilateral where all the interior angles are of equal measure.
Last modified on August 3rd, 2023
One thought on “Convex and Concave Quadrilaterals”
Excellent explanation