# Octagon shape

Here you will learn about the octagon shape, including how to identify an octagon, how to find the perimeter of an octagon and how to find the area of an octagon.

Students will first learn about the octagon as a part of geometry in 2 nd and 3 rd grade. Students will learn about the area of the octagon shape as a part of geometry in 7 th grade.

## What is the octagon shape?

An octagon shape is an eight-sided polygon with eight angles. An octagon shape is a closed shape. There are different types of octagons.

Regular octagons have 8 equal sides and 8 equal angles.

Irregular octagons have 8 sides and 8 angles, with unequal sides and unequal angles.

Convex and concave octagons are other types of octagons.

A concave octagon is an octagon with at least one of the interior angles greater than 180^{\circ}, causing the shape to appear to “cave in”, having one angle pointing inward.

A convex octagon is an octagon where every interior angle is less than 180^{\circ}.

### Other properties of an octagon shape

Octagon shapes have both interior and exterior angles. A pair of interior and exterior angles of all polygon shapes add to 180^{\circ} and form a straight line because they are supplementary angles.

Interior angles of an octagon are the angles inside the 2D shape, formed when two sides of the shape meet. The sum of the interior angles of a regular octagon is 1080^{\circ}.

Exterior angles of an octagon are the angles between the octagon and the extended line from the next side. The sum of the exterior angles of any polygon is always 360^{\circ}.

A regular octagon has 8 lines of symmetry, dividing the octagon into two equal halves.

### Perimeter of an octagon shape

The perimeter of an octagon is the sum of the lengths of all the sides of an octagon.

To find the perimeter of any octagon, you can use the following formula:

P=s+s+s+s+s+s+s+s

Or for regular octagons, you can use:

P=s\times{8}

For example, find the perimeter of the octagon.

\begin{aligned}& P=s \times 8 \\\\ & P=5 \times 8 \\\\ & P=40 \text { inches } \end{aligned}

### Area of a regular octagon shape

When you find the area of an octagon, you are finding the amount of space inside the shape’s 8 sides.

To find the area of a regular octagon, you will use the formula A=2(1+\sqrt{2}) \times s^2, where s is the length of the side of the octagon.

For example,

Find the area of the octagon shape. Round the area to two decimal places.

\begin{aligned}& A=2(1+\sqrt{2}) \times s^2 \\\\ & A=2 \times(1+\sqrt{2}) \times 3^2 \\\\ & A=43.45584 \\\\ & A=43.46 \text { meters } \end{aligned}

## Common Core State Standards

How does this relate to 2 nd grade math, 3 rd grade math, and 7 th grade math?

• Grade 2: Geometry (2.G.A.1)
Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

• Grade 3: Measurement and Data (3.MD.D.8)
Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.

• Grade 7: Geometry (7.G.B.6)
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

## How to identify an octagon shape

In order to identify an octagon, you need to:

1. Look for the characteristics of an octagon.
2. State whether or not the shape is an octagon.
3. If the shape is not an octagon, explain what characteristics are different.

## Octagon shape examples

### Example 1: identifying octagons

Look at the image below and determine if it’s an octagon or not.

1. Look for the characteristics of an octagon.

An octagon is an 8 sided shape with 8 vertices and 8 angles.

This shape has 7 sides, vertices and angles.

2State whether or not the shape is an octagon.

This shape is not an octagon.

3If the shape is not an octagon, explain what characteristics are different.

This shape is not an octagon, but is a heptagon. The shape only has 7 sides.

### Example 2: real-world example

Look at the image below and determine if it’s an octagon or not.

Look for the characteristics of an octagon.

State whether or not the shape is an octagon.

If the shape is not an octagon, explain what characteristics are different.

## How to find the perimeter of an octagon shape

In order to find the perimeter of an octagon:

1. Identify the length of each side.
2. Find the sum of all sides.
3. State your answer with the correct units.

### Example 3: perimeter of a regular octagon

Find the perimeter of the octagon below.

Identify the length of each side.

Find the sum of all sides.

State your answer with the correct units.

### Example 4: perimeter of an irregular octagon

Find the perimeter of the octagon below.

Identify the length of each side.

Find the sum of all sides.

State your answer with the correct units.

## How to find the area of an octagon shape

In order to find the area of an octagon:

1. Substitute the length of the side into the area of an octagon formula.
2. Solve the formula.
3. State your answer with the correct units.

### Example 5: area of an octagon

Find the area of the given octagon. Round the answer to two decimal places.

Substitute the length of the side into the area of an octagon formula.

Solve the formula.

State your answer with the correct units.

### Example 6: area of an octagon

Find the area of the given octagon. Round the answer to two decimal places.

Substitute the length of the side into the area of an octagon formula.

Solve the formula.

State your answer with the correct units.

### Teaching tips for octagon shape

• When introducing octagon shapes to students, use real-life examples, such as a stop sign, spider web, or umbrella to allow students to make connections within their own lives.

• While worksheets have their place in the classroom, consider providing students with tasks that require problem-solving and the use of technology tools to reinforce the learning of octagon shapes.

### Easy mistakes to make

• Confusing regular and irregular octagons
Regular octagons have eight sides and angles of equal measures, where irregular octagons have eight sides and angles that are not equal in measure.

• Misidentifying the diagonals
Students can sometimes mix up short diagonals and long diagonals. A short diagonal connects vertices that are two sides apart, while a long diagonal connects vertices that are further apart.

• Identifying lines of symmetry
Regular octagons have 8 lines of symmetry, but irregular octagons can have fewer or sometimes no lines of symmetry. It’s important to identify the type of octagon before looking for lines of symmetry.

### Practice octagon shape questions

1. Which of the following shapes is an octagon shape?

An octagon shape is an 8 sided shape with 8 vertices.

This shape has 8 sides and 8 vertices.

2. Which of the following shapes is an octagon shape?

An octagon shape is an 8 sided shape with 8 vertices.

This shape has 8 sides and 8 vertices.

3. What is the perimeter of the octagon shape below?

257 \, inches

272 \, inches

189 \, inches

221 \, inches

The octagon is a regular octagon, therefore you can use the formula P=s\times{8}.

\begin{aligned}& P=s \times 8 \\\\ & P=34 \times 8 \\\\ & P=272 \text { inches } \end{aligned}

4. What is the perimeter of the octagon shape below?

96 \, cm

120 \, cm

102 \, cm

105 \, cm

Because it is an irregular octagon and the sides of the polygon are not equal, we can use the formula, P=s+s+s+s+s+s+s+s.

\begin{aligned}& P=s+s+s+s+s+s+s+s \\\\ & P=12+12+12+12+12+12+15+15 \\\\ & P=102 \mathrm{~cm} \end{aligned}

5. Find the area of the given octagon and round to two decimal places.

1,043.37 \, meters

216.09 \, meters

1,004.90 \, meters

1,103.38 \, meters

Substitute the length of the side into the area of an octagon formula, then solve.

\begin{aligned}& A=2(1+\sqrt{2}) \times s^2 \\\\ & A=2(1+\sqrt{2}) \times(14.7)^2 \\\\ & A=2 \times(1+\sqrt{2}) \times(14.7)^2 \\\\ & A=1043.37482 \\\\ & A=1043.37 \text { meters } \end{aligned}

6. Find the area of the given octagon and round to two decimal places.

3,401.04 \, feet

3,519.92 \, feet

3,019.22 \, feet

2,456.04 \, feet

Substitute the length of the side into the area of an octagon formula, then solve.

\begin{aligned}A & =2(1+\sqrt{2}) \times s^2 \\\\ A & =2(1+\sqrt{2}) \times(27)^2 \\\\ A & =2 \times(1+\sqrt{2}) \times(27)^2 \\\\ A & =3519.92337 \\\\ A & =3519.92 \text { feet } \end{aligned}

## Octagon shape FAQs

What does the word octagon mean?

The word octagon comes from the Greek word oktágōnon, which means eight angles. This is why a shape with 8 sides and angles is called an octagon.

What is the angle of an octagon?

A regular octagon has interior angles that are all equal to 135^{\circ} ( the sum of the angles should equal 1080^{\circ}). This means that all the exterior angles are equal to 45^{\circ}. The side of a regular octagon can be any length but will be equal in length.

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