Convex And Concave Octagon

What Are Convex and Concave Octagons?

A convex octagon is an eight-sided polygon where all interior angles measure less than ${180}^{\circ }$ and all vertices point outwards, away from the center of the shape. In contrast, a concave octagon is an eight-sided polygon that has at least one interior angle greater than ${180}^{\circ }$, which creates an inward-pointing vertex that looks like a 'dent' or a 'cave' in the shape.

Every polygon you encounter can be classified as either convex or concave. Think of a standard stop sign; that's a perfect example of a convex octagon. It has no dents, and its sides bulge outwards. Now, imagine someone pushed one of its corners inward. That new shape would be a concave octagon. The key difference lies in the angles and the direction of the vertices (the corners).

In simpler terms:

• Convex: All corners point out. No 'caves'.
• Concave: At least one corner points in, creating a 'cave'.

This single difference leads to several other unique properties related to their diagonals and how we work with them, which we will explore in detail.

What Are the Defining Properties of a Convex Octagon?

Convex octagons are the most familiar type of octagon. They are predictable and follow a clear set of rules. Understanding these properties is essential for identifying them and solving geometry problems.

Here are the three main characteristics of any convex octagon:

1. All Interior Angles are Less Than ${180}^{\circ }$: Every single one of the eight interior angles is a reflex angle. This means they are all smaller than a straight line. If you were to stand at any vertex inside the shape and look at the angle, it would be less than ${180}^{\circ }$. A regular octagon, like a stop sign, has eight equal interior angles of ${135}^{\circ }$, all of which are clearly less than ${180}^{\circ }$.
2. All Vertices Point Outwards: There are no 'dents' or 'indentations' in a convex octagon. Every vertex points away from the interior of the polygon. This gives the shape a rounded, bulging appearance with no parts caving in on themselves.
3. All Diagonals are Inside the Polygon: A diagonal is a line segment that connects two non-adjacent vertices. In a convex octagon, if you draw a line connecting any two corners that aren't right next to each other, that entire line will lie completely inside the boundaries of the shape. None of it will cross into the outside space.

These three properties are interconnected. Because all vertices point outwards, all interior angles must be less than ${180}^{\circ }$, and because of that, all diagonals must be contained within the shape.

What Makes an Octagon Concave?

Concave octagons are less common in everyday examples but are fascinating from a mathematical perspective. They are defined by what they *don't* do compared to their convex cousins. The presence of just one 'caved-in' point changes everything.

Here are the defining properties of a concave octagon:

1. At Least One Interior Angle is Greater Than ${180}^{\circ }$: This is the most important rule. A concave octagon must have at least one reflex angle—an angle that is larger than a straight line (${180}^{\circ }$). This reflex angle occurs at the vertex that points inward. An octagon can have one, two, or even three such reflex angles.
2. At Least One Vertex Points Inwards: This is the visual clue. The vertex where the reflex angle is located points toward the center of the polygon instead of away from it. This creates the characteristic 'dent' or 'cave' that gives the shape its name.
3. At Least One Diagonal Lies Outside the Polygon: Because one vertex is pushed inward, it's now possible to draw a diagonal that travels outside the shape's boundary. If you connect two vertices and the line segment passes through the exterior of the octagon, you have definitive proof that the polygon is concave.

Think of it like a deflated balloon. A fully inflated balloon is convex. If you poke a finger into it, creating a dent, it becomes concave at that point. The same principle applies to polygons.

How Can You Tell if an Octagon is Convex or Concave?

Sometimes, just looking at a shape is enough. But for more complex or subtly drawn octagons, you need a reliable method. Here are two simple tests to determine if an octagon is convex or concave.

The Diagonal Test

This is the most definitive test. A diagonal is a line segment connecting two vertices that are not next to each other.

1. Pick a vertex on the octagon.
2. Draw all possible diagonals from that single vertex to every other non-adjacent vertex.
3. Observe the paths of these lines.

• If all the diagonals you drew are completely inside the octagon, the shape is convex.
• If even one diagonal goes partially or fully outside the octagon, the shape is concave.

The Side Extension Test

This test involves extending the sides of the polygon into straight lines.

1. Pick any side of the octagon.
2. Using a ruler, extend that side into an infinitely long straight line in both directions.
3. Observe where the line goes.

• If the line does not pass through the interior of the polygon, the shape is likely convex. (You must check every side to be sure).
• If the line cuts through the inside of the polygon at any point, the shape is definitively concave.

How Do You Calculate Interior Angles in Octagons?

One of the most surprising facts in geometry is that the sum of the interior angles is the same for ALL octagons, whether they are convex or concave. The formula depends only on the number of sides, which is always eight for an octagon.

The formula for the sum of the interior angles of any polygon is:

Where $n$ is the number of sides.

For an octagon, $n=8$. Let's plug it in:

Sum = $(8-2)\times {180}^{\circ }$ = $6\times {180}^{\circ }$ = ${1080}^{\circ }$

This means if you add up all eight interior angles of any octagon, you will always get ${1080}^{\circ }$. This is true for a regular stop sign and for a strangely shaped concave octagon.

• For a regular convex octagon, all angles are equal. So, each angle is ${1080}^{\circ }/8={135}^{\circ }$.
• For an irregular or concave octagon, the angles will be different, but their sum will still be ${1080}^{\circ }$. A concave octagon will have at least one of these angles be greater than ${180}^{\circ }$.

How Many Diagonals Does an Octagon Have?

Both convex and concave octagons have the exact same number of diagonals. A diagonal is a line segment that connects two non-consecutive (not next to each other) vertices. The formula to calculate the number of diagonals in any polygon is:

Where $n$ is the number of sides.

For an octagon, we use $n=8$:

$$D=\frac{8(8-3)}{2}=\frac{8(5)}{2}=\frac{40}{2}=20$$

So, every octagon has exactly $20$ diagonals. The critical difference, as we've discussed, is their location:

• In a convex octagon, all $20$ diagonals are located entirely inside the polygon.
• In a concave octagon, some of the $20$ diagonals will be partially or fully outside the polygon.

Common Mistakes to Avoid

When working with octagons, students often fall into a few common traps. Being aware of these can help you avoid them.

• The 'Stop Sign' Assumption: Many people automatically picture a regular octagon (equal sides, equal angles) when they hear the word 'octagon'. Remember that an octagon is any $8$-sided shape. They can be irregular, stretched, and concave.
• Angle Sum Confusion: A frequent mistake is thinking that the interior angle sum formula, $(n-2)\times {180}^{\circ }$, only works for convex polygons. This is incorrect! The sum of ${1080}^{\circ }$ is a universal property for all octagons.
• Misidentifying a Reflex Angle: When measuring angles in a concave polygon, it's easy to accidentally measure the smaller, exterior angle at an inward-pointing vertex instead of the large interior reflex angle. Always measure the angle on the inside of the polygon.
• Mixing Up Convex and Concave: The terms can be confusing. Use the mnemonic: 'Concave' has a 'cave' or a dent in it. 'Convex' flexes outwards.

Quick Summary: Convex vs. Concave Octagon

Use this table as a quick reference guide to remember the key differences between convex and concave octagons.