The area of a heptagon is the region enclosed by the seven sides of the heptagon. The area can be considered as the region covered by a geometric figure in the two-dimensional (2D) plane. Depending on the information we have available, we can calculate the area of a heptagon using the length of the apothem and the length of the sides or simply using the length of the sides.
Here, we will learn about the two main formulas used to calculate the area of a heptagon. In addition, we will apply these two formulas to find the solution to some problems.
Formula to find the area of a heptagon
We can use two main formulas depending on the type of information that we have available. It is possible to use a formula to calculate the area of regular heptagons using the apothem and one of the sides or simply using the length of one of the sides.
Using apothem and sides
Recall that the apothem is the length of the center of the heptagon that is perpendicular to one of its sides. If we divide a heptagon into seven triangles, we have the following figure:
We see that the apothem is equal to the height of one of the small triangles. Also, we know that the area of any triangle is equal to one-half of the base times the height of the triangle.
In this case, the area of each triangle is $latex A=\frac{1}{2}sa$, where, a is the apothem and s is the length of one of the sides. Since we have seven triangles, the area of the heptagon is:
| $latex A=\frac{7}{2}sa$ |
Using only the length of the sides
If we only know the length of one side of the heptagon, we can use the following formula to calculate the area:
| $latex A=\frac{7}{4}{{s}^2}\cot(\frac{180^{\circ} }{7})$ |
This formula can be simplified by calculating the value of the cotangent and multiplying by the value of the fraction:
| $latex A=3.634{{s}^2}$ |
Area of a heptagon – Examples with answers
The following examples use the two heptagon area formulas seen above depending on the information available. Try to solve the exercises yourself before looking at the solution.
EXAMPLE 1
A heptagon has an apothem of length 4.15 m and sides of length 4 m. What is its area?
Solution
We have the following lengths:
- Apothem, $latex a=4.15$ m
- Sides, $latex s=4$ m
Using the first formula with these values, we have:
$latex A=\frac{7}{2}sa$
$latex A=\frac{7}{2}(4)(4.15)$
$latex A=58.1$
The area of the heptagon is 58.1 m².
EXAMPLE 2
What is the area of a heptagon that has an apothem of length 6.23 m and sides of length 6 m?
Solution
We have the following values:
- Apothem, $latex a=6.23$ m
- Sides, $latex l=6$ m
We can use the first formula with these values:
$latex A=\frac{7}{2}sa$
$latex A=\frac{7}{2}(6)(6.23)$
$latex A=130.8$
The area of the heptagon is 130.8 m².
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EXAMPLE 3
A heptagon has an apothem of length 7.27 m and sides of length 7 m. What is its area?
Solution
We have the following values:
Apothem, $latex a=7.27$ m
Sides, $latex s =7$ m
Substituting these values in the first formula, we have:
$latex A=\frac{7}{2}la$
$latex A=\frac{7}{2}(7)(7.27)$
$latex A=178.1$
The area of the heptagon is 178.1 m².
EXAMPLE 4
A heptagon has sides of length 11 m. What is its area?
Solution
In this case, we only have the length of one of the sides of the heptagon, so we have to use the second formula. To make it easier, we can use the simplified version:
$latex A=3.636{{s}^2}$
$latex A=3.636{{(11)}^2}$
$latex A=3.636(121)$
$latex A=440$
The area of the heptagon is 440 m².
EXAMPLE 5
What is the area of a heptagon that has sides of length 21 m?
Solution
Again, we are going to use the second formula and use the value $latex s=21$:
$latex A=3.636{{s}^2}$
$latex A=3.636{{(21)}^2}$
$latex A=3.636(441)$
$latex A=1603.5$
The area of the heptagon is 1603.5 m².
Area of a heptagon – Practice problems
Put into practice using the formulas for the area of heptagons to solve the following problems. If you need help with this, you can look at the solved examples above.
See also
Interested in learning more about parallelograms? Take a look at these pages:
