How to Find the Area of Regular Pentagon?

With this pentagon calculator, you'll find essential properties of a regular pentagon: side, diagonal, height, perimeter, and area, as well as the circumcircle and incircle radius. Type any value, and the remaining parameters will be calculated on the spot. If you are not sure what a pentagon is or how many sides a pentagon has, keep scrolling, and you'll find clarifying pictures with a short explanation.

Pentagon is a 5-sided polygon. Pentagon can be simple or self-intersecting.

The sum of the internal angles in a simple pentagon is 540°, so every internal angle is equal to 108°. A regular simple pentagon has all five sides equal in length. (In this article, we use the term "regular pentagon" to describe a regular simple pentagon).

Area A of a regular pentagon can be calculated from the formula:

area = a² × √(25 + 10√5) / 4, where a is a side of a regular pentagon.

Also, you can find the area having the circumscribed circle radius:

area = 5R² × √[(5 + √5)/2] / 4, where R is an circumcircle radius.

Perimeter P of a regular pentagon is equal to side length multiplied by the number of vertices. Pentagon is a 5 sided polygon, so the perimeter is:

perimeter = 5 × a

To calculate the height and diagonal of a regular pentagon, all you need to have given is the side length a:

diagonal = a × (1 + √5) / 2
height = a × √(5 + 2√5) / 2

Pentagon has five diagonals equal in length, which form a pentagram.

Now, as we know the pentagon definition, we can have a look at this step-by-step example:

Find out what is given. For a regular pentagon, one parameter is enough to find the remaining six.
Type the value into the pentagon calculator. Let's take the most famous, almost regular pentagon as an example - the Pentagon building, the headquarters of the US Department of Defense. From the Wikipedia page, we find out that it's 1414 ft wide - it's the height of the pentagram.

The Pentagon, 1,414 feet, 431m (Light blue)

RMS Queen Mary 2, 1,132 feet, 345m (Pink)

US Navy's nuclear-powered USS Enterprise, 1,123 feet, 342m (Yellow)

Airship LZ 129 Hindenburg, 804 feet, 245m (Green)

Imperial Japanese Navy's Yamato, 863 feet, 263m (Dark blue)

Empire State Building, 1,454 feet, 443m (Grey)

Knock Nevis supertanker, 1,503 feet, 458m (Red)

Apple Park main building, 1,522 feet, 458m (Green)

The pentagon parameters appear! They are:
- Side – 918.9 ft
- Diagonal – 1486.8 ft
- Perimeter – 4594 ft (0.87 mi)
- Area – 33.35 ac
- Circumcircle radius – 781.6 ft
- Incircle radius – 632.4 ft

Did you notice how enormous it is? Have a look at the perimeter – it's almost a mile! In reality, each side of the building is ~921 feet long – looks like it's practically a regular pentagon!

If you are interested in other regular shapes, have a look at our great tools:

Triangle calculator
Square calculator
Hexagon calculator
Octagon calculator

To compute the area of a regular pentagon from side length you need to apply the formula area = a² × √(25 + 10√5) / 4. Plugging in a = 2, we obtain area = 2² × √(25 + 10√5) / 4 = √(25 + 10√5) ≈ 6.882.

To compute the apothem of a pentagon with side length a, apply the formula:

apothem = 0.5 × a / tan(π/5)

By simplifying tan, we obtain:

apothem = 0.1 × a × √(25 + 10√5 ).

To compute the internal angle of a pentagon:

Divide 360° by the number of sides: 360°/5 = 72°.
Subtract 72° from 180° to get the internal angle of a pentagon: 180° - 72° = 108°.
It follows that the sum of internal angles in a pentagon equals 5 × 108° = 540°.

Find the area of the pentagon shown above.

Possible Answers:

square units

Correct answer:

square units

Explanation:

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

The area of the two right triangles can be found using the formula:

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.Thus, the solution is:

Find the area of the pentagon shown above.

Possible Answers:

square units

Correct answer:

square units

Explanation:

To find the area of this pentagon, divide the interior of the pentagon into a four-sided rectangle and two right triangles. The area of the bottom rectangle can be found using the formula:

The area of the two right triangles can be found using the formula:

Since there are two right triangles, the sum of both will equal the area of the entire triangular top portion of the pentagon.Thus, the solution is:

A regular pentagon has a side length of

inches and an apothem length of

inches. Find the area of the pentagon.

Possible Answers:

Correct answer:

Explanation:

By definition a regular pentagon must have

equal sides and

equivalent interior angles. Since we are told that this pentagon has a side length of

inches, all of the sides must have a length of

inches. Additionally, the question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into

equivalent interior triangles. Each triangle will have a base of

and a height of

. The area of this pentagon can be found by applying the area of a triangle formula:

Note: the area shown above is only the a measurement from one of the five total interior triangles. Thus, to find the total area of the pentagon multiply:

A regular pentagon has a side length of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

square units

Correct answer:

square units

Explanation:

By definition a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon--which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

Note:

is only the measurement for one of the five interior triangles. Thus, the final solution is:

A regular pentagon has a perimeter of

yards and an apothem length of

yards. Find the area of the pentagon.

Possible Answers:

Correct answer:

Explanation:

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

This question provides the length of the apothem of the pentagon—which is the length from the center of the pentagon to the center of a side. This information will allow us to divide the pentagon into equivalent interior triangles. Each triangle will have a base of

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

Thus, the area of the entire pentagon is:

A regular pentagon has a side length of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

square units

Correct answer:

square units

Explanation:

By definition a regular pentagon must have equal sides and equivalent interior angles.

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

Keep in mind that this is the area for only one of the five total interior triangles. The total area of the pentagon is:

A regular pentagon has a perimeter of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

square units

Correct answer:

square units

Explanation:

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

To find the total area of the pentagon multiply:

A regular pentagon has a side length of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

square units

Correct answer:

square units

Explanation:

By definition a regular pentagon must have equal sides and equivalent interior angles.

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

However, the total area of the pentagon is equal to:

A regular pentagon has a side length of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

sq. units

Correct answer:

sq. units

Explanation:

By definition a regular pentagon must have equal sides and equivalent interior angles.

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

Note:

is only the measurement for one of the five interior triangles. Thus, the solution is:

A regular pentagon has a perimeter of

and an apothem length of

. Find the area of the pentagon.

Possible Answers:

Correct answer:

Explanation:

To solve this problem, first work backwards using the perimeter formula for a regular pentagon:

Now you have enough information to find the area of this regular triangle.

Note: a regular pentagon must have equal sides and equivalent interior angles.

and a height of

The area of this pentagon can be found by applying the area of a triangle formula:

Thus, the area of the entire pentagon is:

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