✓ Area Of A Pentagon Formula
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- Area of Pentagon
- What is the Area of Pentagon?
- Area of Pentagon Formula
- Area Formula for a Pentagon
- How to Find the Area of a Regular Pentagon
- Finding the Area from the Side Length and Apothem
- Finding the Area from the Side Length
- Types of Pentagon
- Concave Pentagon
- Convex Pentagon
- Regular Pentagon
- Irregular Pentagon
- Interior and Exterior Angle of a Regular Pentagon
- Perimeter of a Pentagon
- Apothem
- Area of Regular Pentagon with Apothem
- Area of a Regular Pentagon without Apothem
- Area of Regular Pentagon with Radius
- Calculation of Area of Pentagon
- Solved Examples on Area of Pentagon
- Example Question #1 : How To Find The Area Of A Pentagon
- Example Question #2 : How To Find The Area Of A Pentagon
- Example Question #3 : How To Find The Area Of A Pentagon
- Example Question #4 : How To Find The Area Of A Pentagon
- Example Question #5 : How To Find The Area Of A Pentagon
- Example Question #6 : How To Find The Area Of A Pentagon
- Example Question #7 : How To Find The Area Of A Pentagon
- Example Question #8 : How To Find The Area Of A Pentagon
- Example Question #9 : How To Find The Area Of A Pentagon
- Example Question #10 : How To Find The Area Of A Pentagon
- Example Question #11 : How To Find The Area Of A Pentagon
- Example Question #12 : How To Find The Area Of A Pentagon
- Example Question #13 : How To Find The Area Of A Pentagon
- Example Question #14 : How To Find The Area Of A Pentagon
- Example Question #15 : How To Find The Area Of A Pentagon
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- Example Question #19 : How To Find The Area Of A Pentagon
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- Example Question #28 : How To Find The Area Of A Pentagon
- FAQs on Area of Pentagon
- What is the Area of a Pentagon?
- What is the Unit of Area of Pentagon?
- How to find the Area of Pentagon using Apothem?
- How to find the Area of Pentagon using Side Length?
- What is the Area of Pentagon Formula?
- How to Find the Area of Pentagon?
- How To Find the Area of Regular Pentagon?
- What Happens to the Area of Pentagon if the Length of the Side is Tripled?
- What Happens to the Area of Pentagon If the Length of Apothem is Halved?
Area of Pentagon
The area of a pentagon is the region that is enclosed by all the five sides of the pentagon. A pentagon is a five-sided polygon and a two-dimensional geometrical figure. Its name is derived from the Greek words “Penta” which means “five” and “gon” which means “angles”.
What is the Area of Pentagon?
The area of a pentagon is the region that is covered within the sides of the pentagon. It is expressed in square units like m2, cm2, in2, or ft2, etc. A pentagon has 5 sides and the sum of its interior angles is equal to 540°, while the sum of its exterior angles equals 360°. The area of a pentagon can be calculated by various methods depending on the dimensions that are known. The different methods are given below:
- Area of Pentagon using only the side length: In order to find the area of pentagon when only the side length is given, we use the formula:
- Area of Pentagon using the apothem: The formula that is used to calculate the area of pentagon using the apothem is, Area = 5/2 × s × a; where ‘a’ is the apothem length, and ‘s’ is the side of the pentagon.
- Area of Pentagon using the area of isosceles triangles: In this method, the pentagon can be divided into 5 equal isosceles triangles, the area of all the 5 isosceles triangles can be calculated and added to find the area of the pentagon.
Area of Pentagon Formula
The formula that is used to find the area of a pentagon using the apothem and side length is, Area = 5/2 × s × a. Here, “s” is the side length and “a” is the apothem of the pentagon. Observe the following pentagon to see the apothem ‘a’ and the side length ‘s’.
The area of a regular pentagon can also be calculated if only the side length “s” is known. The formula used to find the area of a regular pentagon,
where “s” is the length of one side of the regular pentagon.
Area Formula for a Pentagon
The Area of a Pentagon Formula is,
| A = (5 ⁄ 2) × s × a |
Where,
- “s” is the side of the Pentagon
- “a” is the apothem length
How to Find the Area of a Regular Pentagon
A pentagon is a polygon with five straight sides. Almost all problems you’ll find in math class will cover regular pentagons, with five equal sides. There are two common ways to find the area, depending on how much information you have.
Finding the Area from the Side Length and Apothem
Start with the side length and apothem. This method works for regular pentagons, with five equal sides. Besides the side length, you’ll need the “apothem” of the pentagon. The apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle.
- Don’t confuse the apothem with the radius, which touches a corner (vertex) instead of a midpoint. If you only know the side length and radius, skip down to the next method instead.
- We’ll use an example pentagon with side length 3 units and apothem 2 units.
Divide the pentagon into five triangles. Draw five lines from the center of the pentagon, leading to each vertex (corner). You now have five triangles.
Calculate the area of a triangle. Each triangle has a base equal to the side of the pentagon. It also has a height equal to the pentagon’s apothem. (Remember, the height of a triangle runs from a vertex to the opposite side, at a right angle.) To find the area of any triangle, just calculate ½ x base x height.
- In our example, area of triangle = ½ x 3 x 2 = 3 square units.
Multiply by five to find the total area. We’ve divided the pentagon into five equal triangles. To find the total area, just multiply the area of one triangle by five.
- In our example, A(total pentagon) = 5 x A(triangle) = 5 x 3 = 15 square units.
Finding the Area from the Side Length
Start with just the side length. This method only works for regular pentagons, which have five sides of equal length.
- In this example, we’ll use a pentagon with side length 7 units.
Divide the pentagon into five triangles. Draw a line from the center of the pentagon to any vertex. Repeat this for every vertex. You now have five triangles, each the same size.
Divide a triangle in half. Draw a line from the center of the pentagon to the base of one triangle. This line should hit the base at a 90º right angle, dividing the triangle into two equal, smaller triangles.
Label one of the smaller triangles. We can already label one sides and one angle of the smaller triangle:
- The base of the triangle is ½ the side of the pentagon. In our example, this is ½ x 7 = 3.5 units.
- The angle at the pentagon’s center is always 36º. (Starting with a full 360º center, you could divide it into 10 of these smaller triangles. 360 ÷ 10 = 36, so the angle at one triangle is 36º.)
Calculate the height of the triangle. The height of this triangle is the side at right angles to the pentagon’s edge, leading to the center. We can use beginning trigonometry to find the length of this side:
- In a right-angle triangle, the tangent of an angle equals the length of the opposite side, divided by the length of the adjacent side.
- The side opposite the 36º angle is the base of the triangle (half the pentagon’s side). The side adjacent to the 36º angle is the height of the triangle.
- tan(36º) = opposite / adjacent
- In our example, tan(36º) = 3.5 / height
- height x tan(36º) = 3.5
- height = 3.5 / tan(36º)
- height = (about) 4.8 units.
Find the area of the triangle. A triangle’s area equals ½ the base x the height. (A = ½bh.) Now that you know the height, plug in these values to find the area of your small triangle.
- In our example, Area of small triangle = ½bh = ½(3.5)(4.8) = 8.4 square units.
Multiply to find the area of the pentagon. One of these smaller triangles covers 1/10 of the pentagon’s area. To find the total area, multiply the area of the smaller triangle by 10.
- In our example, the area of the whole pentagon = 8.4 x 10 = 84 square units.
Using a Formula
Use the perimeter and apothem. The apothem is a line from the center of a pentagon, that hits a side at a right angle. If you are given its length, you can use this easy formula
- Area of a regular pentagon = pa/2, where p = the perimeter and a = the apothem.
- If you don’t know the perimeter, calculate it from the side length: p = 5s, where s is the side length.
Use the side length. If you only know the side length, use the following formula:
- Area of a regular pentagon = (5s2) / (4tan(36º)), where s = side length.
- tan(36º) = √(5-2√5). So if your calculator doesn’t have a “tan” function, use the formula Area = (5s2) / (4√(5-2√5)).
Choose a formula that uses radius only. You can even find the area if you only know the radius. Use this formula:
- Area of a regular pentagon = (5/2)r2sin(72º), where r is the radius.
Types of Pentagon
Any polygon has four different types: concave polygon, convex polygon, regular polygon, and irregular polygon. Similarly, the pentagon has four types. They are,
- Concave Pentagon
- Convex Pentagon
- Regular Pentagon
- Irregular Pentagon
Concave Pentagon
A pentagon in which at least one angle is more than 180∘ is called a concave pentagon. In other words, if the vertices point inwards or pointing inside a pentagon, it is known as a concave pentagon.
Here, in the pentagon ABCDE, we can see that the interior angle ∠ABC is more than 180∘. Hence, this is a concave pentagon.
Convex Pentagon
A pentagon in which each angle is less than 180∘. is called a convex pentagon. In other words, if the vertices point outwards in a pentagon or pointing outside is known as a convex pentagon.
Regular Pentagon
A polygon is said to be regular if all the sides and angles of it are equal. A pentagon with all sides equal and all the angles equal is called a regular pentagon.
The pentagon ABCDE is a regular pentagon if AB=BC=CD=DE=EA and ∠A=∠B=∠C=∠D=∠E.
Irregular Pentagon
Pentagons that aren’t regular are called irregular pentagons. In other words, a pentagon in which the sides and angles have different measures is known as an irregular pentagon.
Interior and Exterior Angle of a Regular Pentagon
We know that, for a regular polygon of nn sides, we haveSum of exterior angles equal to 360∘.
Perimeter of a Pentagon
The computation of the length of the boundary of any closed figure is known as its perimeter. The perimeter of a regular or irregular pentagon will be the sum of the lengths of its sides.
If a+b+c+d+e.
That is, perimeter =5a.
That is, perimeter
Where s= length of the side of the pentagonr= radiusss and length of apothem 5/2 times the length of the apothem and the side of the pentagon.The formula to calculate the area of the pentagon is a.a.The area of a regular pentagon with side length a is given by
Also, we can use another formula to find the area of the pentagon when only the side length is given. It is given by
Area of Regular Pentagon with Radius
Let 4cm.Ans: We know that the area of a regular pentagon with side measure a units is given by
Q.2. Find the area of the given regular pentagon whose side measure is aa units is given by
Therefore, the area of a regular pentagon with a side of 3cm is 15.484cm2.
Q.3. Find the area and perimeter of a regular pentagon whose side is 6cm, and apothem length is 5cm.Ans: Given: side measure of regular pentagon s=6cmThe measure of apothem a=5cm
⇒A=75 cm2
Now, we know that the perimeter of a regular pentagon with side length 5a.Therefore, the perimeter of the given pentagon =30cmTherefore, the area of the given pentagon is 30cm.
Example Question #1 : How To Find The Area Of A Pentagon
Find the area of the pentagon shown above.
Possible Answers:340 square units
390 square units
380 square units
320 square units
300 square units
Correct answer:380 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:
A=width×length
A=20×15=300
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=300+40+40=380
Example Question #2 : How To Find The Area Of A Pentagon
Find the area of the pentagon shown above.
Possible Answers:672 square units
502 square units
416 square units
572 square units
Correct answer:572 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:
A=width×length
A=26×16=416
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=416+78+78=572
Example Question #3 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 9 inches and an apothem length of 6.2 inches. Find the area of the pentagon.
Possible Answers:36 inches2
139.5 inches2
118 inches2
99 inches2
Correct answer:139.5 inches2
Explanation:By definition a regular pentagon must have 5 equal sides and 5 equivalent interior angles. Since we are told that this pentagon has a side length of 9 inches, all of the sides must have a length of 9 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 5 equivalent interior triangles. Each triangle will have a base of 9 and a height of 6.2.
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:
27.9×5=139.5
Example Question #4 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 14 and an apothem length of 9.6. Find the area of the pentagon.
Possible Answers:67.2 square units
84 square units
82 square units
336 square units
200 square units
Correct answer:336 square units
Explanation:By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 14 and a height of 9.6.
The area of this pentagon can be found by applying the area of a triangle formula:
Note: 67.2 is only the measurement for one of the five interior triangles. Thus, the final solution is:
67.2×5=336
Example Question #5 : How To Find The Area Of A Pentagon
A regular pentagon has a perimeter of 60 yards and an apothem length of 8.3 yards. Find the area of the pentagon.
Possible Answers:48 yards2
249 yards2
96 yards2
200 yards2
248 yards2
Correct answer:249 yards2
Explanation:To solve this problem, first work backwards using the perimeter formula for a regular pentagon:
p=5(s)
60=5(s)
s=60/5=12
Now you have enough information to find the area of this regular triangle. Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 12 and a height of 8.3.
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=49.8×5=249
Example Question #6 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 15 and an apothem length of 10.3. Find the area of the pentagon.
Possible Answers:383.5 square units
306.25 square units
400.5 square units
270 square units
386.25 square units
Correct answer:386.25 square units
Explanation:By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 15 and a height of 10.3.
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=77.25×5=386.25
Example Question #7 : How To Find The Area Of A Pentagon
A regular pentagon has a perimeter of 50 and an apothem length of 7. Find the area of the pentagon.
Possible Answers:70 square units
135 square units
170 square units
175 square units
Correct answer:175 square units
Explanation:To solve this problem, first work backwards using the perimeter formula for a regular pentagon:
p=5(s)
50=5(s)
s=50/5=10
Now you have enough information to find the area of this regular triangle. Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 10 and a height of 7.
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:
35×5=175
Example Question #8 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 25 and an apothem length of 17.2. Find the area of the pentagon.
Possible Answers:900 square units
215 square units
1,350 square units
725 square units
1,075 square units
Correct answer:1,075 square units
Explanation:By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 25 and a height of 17.2.
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:
215×5=1,075
Example Question #9 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 7 and an apothem length of 4.8. Find the area of the pentagon.
Possible Answers:68 sq. units
133.6 sq. units
84 sq. units
16.8 sq. units
33.6 sq. units
Correct answer:84 sq. units
Explanation:By definition a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 7 and a height of 4.8.
The area of this pentagon can be found by applying the area of a triangle formula:
Note: 16.8 is only the measurement for one of the five interior triangles. Thus, the solution is:
16.8×5=84
Example Question #10 : How To Find The Area Of A Pentagon
A regular pentagon has a perimeter of 65 and an apothem length of 8.9. Find the area of the pentagon.
Possible Answers:289.25
275.85
280.50
57.85
Correct answer:289.25
Explanation:To solve this problem, first work backwards using the perimeter formula for a regular pentagon:
p=5(s)
65=5(s)
s=655=13
Now you have enough information to find the area of this regular triangle. Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 13 and a height of 8.9.
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=57.85×5=289.25
Example Question #11 : How To Find The Area Of A Pentagon
Find the area of the pentagon shown above.
Possible Answers:189 sq. units
198 sq. units
180 sq. units
54 sq. units
184 sq. units
Correct answer:198 sq. 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:
A=width×length
A=18×8=144
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=144+27+27=198
Example Question #12 : How To Find The Area Of A Pentagon
A regular pentagon has a perimeter of 145 and an apothem length of 20. Find the area of the pentagon.
Possible Answers:1,580 sq. units
1,450 sq. units
1,500 sq. units
580 sq. units
450 sq. units
Correct answer:1,450 sq. units
Explanation:To solve this problem, first work backwards using the perimeter formula for a regular pentagon:
p=5(s)
145=5(s)
s=145/5=29
Now you have enough information to find the area of this regular triangle. Note: a regular pentagon must have 5 equal sides and 5 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 5 equivalent interior triangles. Each triangle will have a base of 29 and a height of 20.
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=290×5=1,450
Example Question #13 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
15
None of these
12
18
Correct answer:15
Explanation:Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(3)
Perimeter=15
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(2)(15)
Area of Pentagon=15
Example Question #14 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
400
380
390
420
Correct answer:420
Explanation:
Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(14)
Perimeter=70
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(12)(70)
Area of Pentagon=420
Example Question #15 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
65
55
50
60
Correct answer:60
Explanation:
Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(8)
Perimeter=40
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(5)(40)
Area of Pentagon=60
Example Question #16 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
88
62
78
70
Correct answer:70
Explanation:
Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(7)
Perimeter=35
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(4)(35)
Area of Pentagon=70
Example Question #17 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
420
630
700
560
Correct answer:560
Explanation:
Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(16)
Perimeter=80
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(14)(80)
Area of Pentagon=560
Example Question #18 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
325
355
340
370
Correct answer:325
Explanation: Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(13)
Perimeter=65
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(10)(65)
Area of Pentagon=325
Example Question #19 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
1220
1260
1180
1300
Correct answer:1260
Explanation: Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=1/2(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(24)
Perimeter=120
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(21)(120)
Area of Pentagon=1260
Example Question #20 : How To Find The Area Of A Pentagon
Find the area of the regular pentagon.
1300
1600
1900
1000
Correct answer:1600
Explanation: Recall that the area of regular polygons can be found using the following formula:
Area of Regular Polygon=12(apothem)(perimeter)
First, find the perimeter of the pentagon. Since it is a regular pentagon, we can use the following formula to find its perimeter:
Perimeter=5(side length)
Substitute in the length of the given side of the pentagon in order to find the perimeter.
Perimeter=5(20)
Perimeter=100
Next, substitute in the given and calculated information to find the area of the pentagon.
Area of Pentagon=1/2(16)(100)
Area of Pentagon=1600
Example Question #25 : How To Find The Area Of A Pentagon
Each side of this pentagon has a length of 5cm.
Solve for the area of the pentagon.
43cm2
25cm2
32.7cm2
21.5cm2
35.5cm2
Correct answer:43cm2
Explanation:The formula for area of a pentagon is A=5⋅(1/2⋅s⋅h), with s representing the length of one side and h representing the apothem.
To find the apothem, we can convert our one pentagon into five triangles and solve for the height of the triangle:
Each of these triangles have angle measures of 72$∘$54$∘$54$∘$, with 72$∘$being the angle oriented around the vertex. This is because the polygon has been divided into five triangles and 360/5=72.
Example Question #26 : How To Find The Area Of A Pentagon
Each side of this regular pentagon has a length of 7.5cm. Solve for the area of the pentagon. Round to the nearest tenth.
140.6cm2
35.5cm2
48.4cm2
96.8cm2
37.5cm2
Correct answer:96.8cm2
Explanation:When given the value of one side of a regular pentagon, we can assume all sides to be of equal length and we can use this formula to calculate area:
For thi formula, s represents the length of one side while N represents the number of sides. Therefore, we would plug in the values as such:
Example Question #27 : How To Find The Area Of A Pentagon
A regular pentagon has a side length of 17cm. Find the area rounded to the nearest tenth.
497.2cm2
99.4cm2
29.2cm2
225.7cm2
383.6cm2
Correct answer:497.2cm2
Explanation:We can use the following equation to solve for the area of a regular polygon with s representing side length and N representing number of sides:
Example Question #28 : How To Find The Area Of A Pentagon
In the figure below, find the area of the pentagon if the length of AF is one-fourth the length of FD.
230
130
190
150
Correct answer:230
Explanation:
Start by figuring out the lengths of AF and FD.
Let x be the length of AF. From the question, we then write the length of FD as 4x.
From the figure, you should see that AF+FD=AD=BC.
Plug in the variables and solve for x.
x+4x=20
5x=20
x=4
So then the length of AF must be 4, and the length of FD must be 16.
Now, to find the area of the pentagon, notice that we can break the shape down to one rectangle and two right triangles.
Next, find the area of rectangle ABCD.
Area of rectangle=length×width
Area of ABCD=9×20=180
Next, find the area of the right triangles.
Area of Triangle=1/2(base)(height)
For triangle AFE,
Area=1/2(4)(5)=10
For triangle EFD,
Area=1/2(16)(5)=40
Add up the component areas to find the area of the pentagon.
Area of Pentagon=180+10+40=230
FAQs on Area of Pentagon
What is the Area of a Pentagon?
The area of a pentagon is the region that is covered by all the sides of the pentagon. It can be calculated by various methods according to the dimensions given. The basic formula that is used to find the area of a pentagon is, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.
What is the Unit of Area of Pentagon?
The unit of area of the pentagon is expressed in square units, for example, m2, cm2, in2, or ft2, etc.
How to find the Area of Pentagon using Apothem?
If the apothem of the pentagon is given, its area can be calculated using the formula: Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.
How to find the Area of Pentagon using Side Length?
In case of a regular pentagon, if the side length is known then the area of the pentagon can be calculated by the formula:
where “s” is the length of one side of the regular pentagon. However, if it is not a regular pentagon, then the area can be calculated using the formula, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.
What is the Area of Pentagon Formula?
The basic formula that is used to find the area of a pentagon is, Area = 5/2 × s × a; where ‘s’ is the length of the side of the pentagon and ‘a’ is the apothem of a pentagon.
How to Find the Area of Pentagon?
We can find the area of the pentagon using the following steps:
- Step 1: Find the length of the side of the pentagon.
- Step 2: Identify the apothem length of the pentagon.
- Step 3: Use the formula, Area = 5/2 × s × a; where ‘s’ is the side length and ‘a’ is the apothem.
- Step 4: This will give the area of the pentagon and we represent the answer in square units.
How To Find the Area of Regular Pentagon?
We can find the area of the regular pentagon using the following steps:
- Step 1: Find the length of the side of the pentagon.
- Step 2: Find the area of regular pentagon using the formula
- where “s” is the length of one side of a regular pentagon.
- Step 4: This will give the area of the regular pentagon and represent the answer in square units.
What Happens to the Area of Pentagon if the Length of the Side is Tripled?
The area of the pentagon is tripled if the length of the side is tripled as the “s” is substituted to “3s” in the formula A = 5/2 × s × a = 5/2 × (3s) × a = 3 × (5/2 × s × a) which gives three times the original area of the pentagon.
What Happens to the Area of Pentagon If the Length of Apothem is Halved?
The area of the pentagon is halved if the length of the side is tripled as the “a” is substituted to “a/2” in the formula A = 5/2 × s × a = 5/2 × s × (a/2) = (1/2) × (5/2 × s × a) which gives half the original area of the pentagon.
MATHS Related Links
Area of Regular Polygon Formula
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