Area of Decagon

The area of a decagon is defined as the number of unit squares that can be fit within it. Decagons as shapes are around us in the form of coins, watches, designs, and patterns. A decagon is a 2-dimensional, ten-sided polygon. The word is made up of "deca" and "gon" where "deca" means ten and "gon" means sides. In this lesson, we will discuss the concept of the area of a decagon and learn to determine the area of a decagon using examples. Stay tuned to learn more!!!

The area of the decagon is the amount of region it covers. A decagon is a plane figure with 10 sides. It has 10 interior angles. A regular decagon has all its 8 sides and 8 interior angles equal. So, for a regular decagon with 10 sides, we can draw 35 diagonals and it has 10 vertices. Since the sum of all the interior angles of a decagon is 1440°, the value of each interior angle for a regular decagon is 144°. The sum of all the exterior angles of a regular decagon is 360°. The unit of area of decagon can be given in terms of m², cm², in² or ft².

A regular decagon is divided into 10 congruent isosceles triangles when all its diagonals are drawn. Therefore, the area of a decagon is given as, area of a decagon = area of 10 congruent isosceles triangles so formed
⇒ Area of a decagon = 10 × Area of each congruent isosceles triangle

We can find the area of a decagon using the following steps:

• Step 1: Find the area of each congruent isosceles triangle.
• Step 2: Multiply the value of the area of each congruent isosceles triangle by 10.
• Step 3: Write the unit in the end, once the value is obtained.

Let's use the fact that the area of a decagon is equal to the area of the 10 isosceles triangles formed in a decagon when diagonals are drawn. Therefore,
⇒ Area of a decagon = 10 × Area of each congruent isosceles triangle

Let's first find the area of each isosceles tringle first:

Area of each isosceles triangle = 1/2 × Base × Height

⇒ Area of each isosceles triangle = 1/2 × a × h

Height of the isosceles triangle, h = a/2 × tan 72° = a/2 × \({ \ \sqrt{10+2\sqrt{5}} \over \sqrt{5}-1}\)

⇒  Area of each isosceles triangle = 1/2 × a × a/2 × \({ \ \sqrt{10+2\sqrt{5}} \over \sqrt{5}-1}\)

⇒  Area of each isosceles triangle = a²/4 × \({ \ \sqrt{10+2\sqrt{5}} \over \sqrt{5}-1}\)

Substituting the value, we get:

Area of a decagon = 10 × a²/4 × \({ \ \sqrt{10+2\sqrt{5}} \over \sqrt{5}-1}\)

⇒ Area of a decagon = 5a²/2 × \({\sqrt {20 + 8\sqrt{5} \over 4}}\)

= 5a²/2 × \({\sqrt {5 + 2\sqrt{5}}}\)

Therefore, the formula for the area of a decagon is 5a²/2 × \({\sqrt {5 + 2\sqrt{5}}}\). So if we know the value of the side length of a regular octagon we can easily find its area.

FAQs on Area of a Decagon

How Many Sides Are There in a Decagon?

The word decagon is made of two words "deca" and "gon", where the word "deca" refers to ten and "gon" refers to sides. Thus, there are ten sides in a decagon.

How Can We Find the Area of a Decagon?

We can find the area of the decagon using the following steps:

• Step 1: Detemine the area of each congruent isosceles triangle.
• Step 2: Now, multiply the value of the area of each congruent isosceles triangle by 10.
• Step 3: Once the value of the area of the decagon is obtained, write the unit in the end.

What is the Formula to Find the Area of a Regular Decagon?

The formula to determine the value of the area of a decagon is 5a²/2 × \({\sqrt {5 + 2\sqrt{5}}}\), where 'a' is the side length of the decagon. 

What Are the Units Used for the Area of a Decagon?

The unit of 'area' is "square units". For example, it can be expressed as m², cm², in², etc depending upon the given units.

How to Find the Area of Decagon If the Area of an Isosceles Triangle Formed By Its Diagonal is Known?

We can determine the area of decagon if the area of an isosceles formed by its diagonal is known using the following steps:

• Step 1: Write the dimension of the area of an isosceles triangle formed.
• Step 2: Determine the area of the decagon using the formula area of a decagon = 10 × area of each congruent isosceles triangle.
• Step 3: Once the value of the area of the decagon is obtained, write the unit in the end.

What Happens to the Area of Regular Decagon If the Length of the Side of Decagon is Doubled?

The area of a regular decagon quadruples if the length of the side of the decagon is doubled as "a" in the formula gets substituted by "2a". Thus, A =  5(2a)²/2 × \({\sqrt {5 + 2\sqrt{5}}}\) = 4 (5a²/2 × \({\sqrt {5 + 2\sqrt{5}}}\)) which gives four times the original value of the area. 

What Happens to the Area of Regular Decagon If the Length of the Side of Decagon is Halved?

The area of a regular decagon becomes one-fourth if the length of the side of the decagon is halved as "a" in the formula gets substituted by "a/2". Thus, A =  5(a/2)²/2 × \({\sqrt {5 + 2\sqrt{5}}}\) = (1/4) × (5a²/2 × \({\sqrt {5 + 2\sqrt{5}}}\)) which gives one-fourth the original value of the area.