Table of Contents
Last modified on August 3rd, 2023
Based on their interior angles, all polygons are broadly classified into two groups: convex and concave.
A convex polygon is a polygon that has all its interior angles less than 180°. All the diagonals of a convex polygon lie inside the closed figure. A convex polygon can be both regular and irregular. Regular convex polygons have all sides of the same length and all interior angles of the same measure (less than 180°). Convex polygons that are not regular are called irregular convex polygons.
Examples: Triangles, all convex quadrilaterals, regular pentagon, and regular hexagon are all convex polygons. A square is a regular convex polygon.
In the given pentagon ABCDE all interior angles are of equal measure. State whether it is a convex polygon.
As we know,
Interior angle of a Regular Polygon = (n-2) × 180° /n, here n = 5
= (5-2) x 180° /5
= 3 x 180° /5
= 108°
Since all the five interior angles in the given pentagon measure less than 180°, it is a convex polygon.
It is the total space enclosed within the polygon. The formula is given below:
Area (A) = ½|(x1y2 – x2y1)+(x2y3– x3y2)+……..+(xny1– x1yn)|, here (x1, y1), (x2, y2), (x3, y3),…….., (xn, yn) are the vertices of the polygon on the coordinate plane
Find the area of a convex polygon with vertices (5, 2) (12, 8) and (5, -2)
As we know,
Area (A) = ½|(x1y2 – x2y1)+(x2y3– x3y2)+(x3y1 – x1y3)|,
here (x1, y1) = (5, 2), (x2, y2) = (12, 8), (x3, y3) = (5, -2)
= ½|(40-24)+ (-24-40)+ (10)-10)|
= ½|16-64|
= 48/2
The interior angles of a polygon are the angles found inside the polygon. A polygon has the same number of interior angles as the number of sides.
Sum of the Interior Angles
It is the total measure of all the interior angles combined in the polygon. The formula to determine the sum of all angles in any convex regular polygon is given below:
Sum of the measure of interior angles = (n-2) × 180°, here n = total number of sides of the polygon
Calculate the sum of the interior angles in a 14-sided convex regular polygon.
As we know,
Sum of the measure of interior angles = (n-2) × 180°, here n = 22
= (14-2) x 180°
= 2160°
One Interior Angle
Each interior angle measures the same in a regular convex polygon and can be obtained by dividing the sum of the interior angles by the total number of sides. The formula is given below:
Interior angle of a Convex Polygon = (n-2) × 180° /n, here n = total number of sides of the polygon
Find the measure of the interior angles of a convex regular polygon having 12 sides.
As we know,
Interior angle of a Regular Polygon = (n-2) × 180° /n, here n = 18
= (18-2) x 180° /18
= 16 x 180° /18
= 160°
The exterior angle of a polygon is the angle formed by one side and the extension of the adjacent side.
The exterior angle sum theorem states that the sum of the exterior angles of any convex polygon is 360°. For regular convex polygons having n sides, each exterior angle is determined using the following formula:
Exterior angle of a Convex Regular Polygon = 360°/n, here n = total number of sides of the polygon
Calculate the exterior angle in a 15-sided regular convex polygon.
As we know,
Exterior angle of a Regular Polygon = 360°/n, here n =15
= 360°/15
= 24°
A concave polygon, also known as a non-convex polygon, has at least one of its interior angles measuring more than 180°. Some diagonals of a concave polygon lie outside the closed figure.
All concave polygons are irregular since all the interior angles are of different measures. Thus concave polygons are never regular.
Examples: A dart or an arrowhead in quadrilaterals, some irregular pentagon, and hexagon
The sum of interior angles of the given hexagon ABCDEF is 720°. If ∠ABC = 78°, ∠BCD = 140°, ∠CDE = 80°, ∠EFA =88°, ∠FAB = 130°. Find ∠DEF and state whether the polygon ABCDEF is concave.
As we know,
The sum of the angles in hexagon ABCDEF = 720°
So, ∠ABC + ∠BCD + ∠CDE +∠DEF + ∠EFA + ∠FAB =
78° + 140° + 80° +∠DEF + 88° + 130° = 720°
=> ∠DEF = 720° – 516°
=> ∠DEF = 204°
Since ∠DEF is greater than 180°, polygon ABCDEF is a concave polygon. Also if we draw a line joining points D and F, we find that the diagonal DF lies outside the closed figure, which further proves that ABCDEF is a concave polygon.
Since all the sides and interior angles of a concave polygon are not equal there is no standard formula to determine their area. To determine the area of a concave polygon, we have to split the polygon into shapes such as triangle, rectangle, parallelogram, or other shapes, find the area of each such shapes, and finally adding them to get the total area of the polygon. Thus the area of a concave polygon is:
Area (A) = Sum of the area of all the shapes available within the polygon
It is the total distance covered around the boundary of the polygon. The perimeter of a concave polygon is thus obtained by simply adding together the length of all sides. Thus the perimeter is:
Perimeter (P) = Sum of all the sides of the polygon
Let’s understand these concepts using examples.
Find the area and perimeter of the concave polygon with the given side measurements.
The given polygon is divided into three rectangles. Finding the area of each rectangle and adding them will give the area of the polygon.
Thus, area of the polygon = Area of rectangle ABCD + Area of rectangle GDHF + Area of rectangle EFIJ
= (14 x 4) + (6 x 4) + (14 x 4)
= 56 + 24 + 56
= 136 square units
Perimeter of the polygon = AB + BD + DF +FJ + IJ+ EI + EH + GH + CG + AC
= (14 + 4 + 6 + 4 +14 + 4 +10 + 6 +10+ 4)
= 76 units
Find all the sides of the given concave polygon. Then find its area and perimeter. (Trick: Divide the given irregular polygon into two regular polygons)
The given polygon is divided into one rectangle and one square. Finding the area of the rectangle and the square and adding them will give the area of the polygon.
Thus, area of the polygon = Area of rectangle ABGF + Area of square CDEG
= (22 x 12) + (8x 8)
= 264 + 64
= 328 square units
Perimeter of the polygon = AB + BC + CD + DE + FE + AF
= (12 + 14 + 8 + 8 + 20 + 22)
= 84 units
To determine the sum of the interior angles we use the following formula:
Sum of interior angles = (n-2) × 180°, here n = total number of sides of the polygon
Like all other polygons, the exterior angle sum theorem stating that all exterior angles add up to 360° is true for all concave polygons.
Ans. A polygon should have at least three sides and can have any maximum number of sides.
Ans. An equiangular triangle has all three sides of equal length and all angles of equal measure. Since the sum of the angles in a triangle is 180° each of the angles measure 60°. Thus an equiangular triangle is always a regular convex polygon.
Ans. Diagonal is the line segment that joins two nonconsecutive vertices of a convex polygon
Last modified on August 3rd, 2023