A convex polygon has interior angles that are all less than 180 degrees, whereas a concave polygon has **at least one interior angle** that is higher than 180 degrees. These special polygons can be formed by connecting **any number** of points on the boundary of a circle.

Concave-convex polygons play **an important role** in geometry. They arise as projections of certain surfaces in three dimensions. A famous example is the paraboloid of revolution: a plane passes through the center of curvature of **its projection** onto a plane.

There are also applications in physics and math. For example, they show up in the study of elasticity, where they are called "elasticae." In mathematics, they have important roles to play in differential geometry and geometric analysis.

Concave-convex polygons may look complicated, but they are not that hard to draw! Here's how: Start with a circle and connect any number of its points. You will get a collection of closed curves, some smooth and some not. Within these curves, find the places where two different curves meet; these are the vertices. The rest is just smoothing out the corners a bit.

- What are concave convex polygons?
- What is the difference between concave and convex polyhedrons?
- What is a polygon with curved sides called?
- Do convex polygons have dents?
- What is the interior angle of a convex polygon?
- Is an arrow concave or convex?
- Does the formula for the sum of interior angles work for concave polygons?

Every polygon has a convex or concave shape. The angle measurements are what distinguishes convex and concave polygons. To be convex, a polygon's inner angles must all be smaller than 180 degrees. The polygon is otherwise concave. Convex polygons include triangles, squares, and regular n-gons with n greater than 4.

Concave polygons include circles and ellipses. There is no simple way to describe the shapes that are not convex; they can only be shown by drawing them. Some common concave polygons are pentagons and hexagons.

Concavity cannot be removed from the definition of a polygon, but it is possible to make some shapes look concave by adding **extra sides** to them. These additional sides do not change **the overall conicity** of the polygon.

There are many ways to describe the shape of a polygon. A good description should be accurate and easy to understand for those who are not familiar with **polygon terminology**. In mathematics, a polygon is called concave if its interior angles are all less than 180 degrees. This means that any side of the polygon faces an angle that is less than 90 degrees. All other polygons are convex.

It is important to understand that this article is only describing the general shape of a polygon.

Concave polygons have at least one interior angle that is more than 180 degrees. A "caved-in" side of a polygon is a typical technique to identify a concave polygon. Convex polygons have no interior angles higher than 180 degrees. The main techniques for identifying **convex polygons** are by drawing lines from the center to **each vertex** and not crossing **any edges** more than once.

Concave quadrilaterals are polygons with four sides. They are commonly referred to as trapezoids when two opposite sides are parallel and rectangles when all four sides are equal.

Concave pentagons have five sides. They are also known as tripods, trigons, or five-sided figures with three equal sides and two opposite angles greater than 90 degrees. There are two ways to identify a concave pentagon: either by using caved-in corners or by drawing lines from the center to each vertex and crossing at least one edge twice.

Concave hexagons have six sides. They are also called hexipedes or six-legged figures with three unequal sides and three equal angles. There are many techniques for identifying a concave hexagon: you can use **caved-in corners**, draw lines from the center to each vertex and cross at least one edge twice, or even trace over its outline with your finger!

Its internal angles are all pointing outward. A convex polygon's internal angles are all smaller than 180 degrees. A concave polygon has at least one greater-than-180-degree angle. There is no clear line between having some angles that are greater than 180 degrees and having **many great angles**. All the interior angles of a concave polygon are greater than 180 degrees.

All interior angles in a convex polygon are less than or equal to **180 degrees**, but all interior angles in **a strictly convex polygon** are strictly less than 180 degrees. That is, any interior angle of a convex polygon is less than or equal to 180 degrees, but not all angles are equal to 180 degrees.

Hint: Consider two adjacent sides of the polygon.

Example: Let us consider a convex polygon with 4 sides. Then the sum of the internal angles of the polygon is 360 degrees. Since each internal angle is less than or equal to 180 degrees, we can conclude that the polygon is symmetrical about each of its sides.

Thus, the polygon is symmetrical about both the vertical and horizontal axes.

Now, let us consider another example. Suppose we take **a convex polygon** with 8 sides. Then the sum of the internal angles of the polygon is 720 degrees.

A concave polygon is a non-convex polygon. This polygon is the inverse of a convex polygon.... Is an arrow a type of polygon?

MATHS Related Links | |
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Table Of 25 | Logarithmic Functions |

Exterior Angles Of A Regular Polygon | Geometry Symbols |

If a polygon has an angle pointing inward, it is said to be concave, and this theorem does not apply. In other words, for this theorem to operate, all of the inner angles of the polygon must have a measure of no more than 180 degrees. If even one angle is greater than 180 degrees, then the polygon is not convex.

The formula for the sum of **its interior angles** is valid for **any convex polygon**, but it may not give **the actual total number** of degrees if there are any concave angles. For example, the polygon below has a total of 360 degrees, but because there is one inner angle whose measure is greater than 180 degrees, it cannot be calculated using this formula.

The formula works by adding up each angle in the sequence until all of the angles are smaller than or equal to 180 degrees. There are two cases to consider: when an angle is less than 90 degrees and when it's greater than 180 degrees.

If an angle is less than 90 degrees, we know that it can be divided into two smaller angles which are both less than 90 degrees. So we can replace it with **these new angles** and add them to the total. This process will continue until no angles are left over from this step.