Formula For Area Of Heptagon

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 nearly the two principal formulas used to calculate the area of a heptagon. In improver, we will apply these ii formulas to find the solution to some problems.

GEOMETRY

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Learning about the area of a heptagon with examples.

Encounter examples

GEOMETRY

Relevant for…

Learning well-nigh the surface area of a heptagon with examples.

Come across examples

Formula to find the area of a heptagon

We can use two main formulas depending on the blazon of information that nosotros accept 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 1 of the sides.

Using apothem and sides

Think that the apothem is the length of the middle of the heptagon that is perpendicular to i of its sides. If we divide a heptagon into seven triangles, we accept the following figure:

Nosotros see that the apothem is equal to the height of i of the small triangles. Too, we know that the area of any triangle is equal to ane-half of the base of operations times the height of the triangle.

In this case, the area of each triangle is $latex A=\frac{1}{two}sa$, where,a is the apothem ands is the length of one of the sides. Since we have seven triangles, the expanse of the heptagon is:

$latex A=\frac{7}{2}sa$

Using just the length of the sides

If we merely know the length of 1 side of the heptagon, we can utilise the following formula to summate the area:

$latex A=\frac{7}{iv}{{s}^ii}\cot(\frac{180^{\circ} }{7})$

This formula can be simplified by computing the value of the cotangent and multiplying by the value of the fraction:

$latex A=3.634{{s}^2}$

Expanse of a heptagon – Examples with answers

The following examples utilise the two heptagon area formulas seen above depending on the information available. Endeavour to solve the exercises yourself earlier looking at the solution.

Case 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, $latex a=iv.15$ grand
Sides, $latex due south=iv$ k

Using the starting time formula with these values, nosotros accept:

$latex A=\frac{vii}{2}sa$

$latex A=\frac{seven}{ii}(iv)(four.15)$

$latex A=58.i$

The surface area of the heptagon is 58.ane m².

Case ii

What is the area of a heptagon that has an apothem of length 6.23 1000 and sides of length 6 yard?

Solution

We have the following values:

Apothem, $latex a=6.23$ grand
Sides, $latex l=6$ m

Nosotros can employ the first formula with these values:

$latex A=\frac{7}{2}sa$

$latex A=\frac{seven}{2}(half-dozen)(six.23)$

$latex A=130.eight$

The surface area of the heptagon is 130.viii thou².

EXAMPLE 3

A heptagon has an apothem of length seven.27 m and sides of length 7 1000. What is its surface area?

Solution

We accept the following values:

Apothem, $latex a=7.27$ 1000
Sides, $latex s =7$ thousand

Substituting these values in the first formula, we have:

$latex A=\frac{7}{two}la$

$latex A=\frac{7}{2}(7)(7.27)$

$latex A=178.1$

The surface area of the heptagon is 178.i m².

EXAMPLE iv

A heptagon has sides of length 11 m. What is its area?

Solution

In this case, nosotros only have the length of one of the sides of the heptagon, and so we take to apply the second formula. To make it easier, we can use the simplified version:

$latex A=3.636{{s}^2}$

$latex A=3.636{{(11)}^2}$

$latex A=iii.636(121)$

$latex A=440$

The area of the heptagon is 440 yard².

EXAMPLE 5

What is the area of a heptagon that has sides of length 21 yard?

Solution

Again, we are going to utilise the 2nd formula and use the value $latex due south=21$:

$latex A=3.636{{s}^2}$

$latex A=3.636{{(21)}^2}$

$latex A=three.636(441)$

$latex A=1603.v$

The area of the heptagon is 1603.5 m².

Area of a heptagon – Practice issues

Put into practice using the formulas for the surface area of heptagons to solve the following bug. If you lot need help with this, yous tin expect at the solved examples above.

What is the expanse of a heptagon that has sides of length 5m and an apothem of 5.19m?

Choose an respond

$latex A=88.two{{m}^2}$

$latex A=xc.8{{m}^two}$

$latex A=95.three{{m}^2}$

$latex A=105.7{{m}^2}$

What is the expanse of a heptagon that has sides of length 13m and an apothem of 13.5m?

Choose an answer

$latex A=601.5{{yard}^2}$

$latex A=606.3{{yard}^2}$

$latex A=614.1{{m}^2}$

$latex A=622.6{{chiliad}^two}$

What is the area of a heptagon that has sides of length 14m?

Choose an answer

$latex A=712.ii{{m}^ii}$

$latex A=721.6{{chiliad}^two}$

$latex A=727.five{{g}^two}$

$latex A=752.8{{chiliad}^two}$

What is the area of a heptagon that has sides of length 16m?

Choose an answer

$latex A=901.four{{one thousand}^2}$

$latex A=908.5{{m}^2}$

$latex A=921.9{{g}^2}$

$latex A=930.ii{{m}^2}$

Encounter also

Interested in learning more about parallelograms? Take a await at these pages:

Perimeter of a Parallelogram – Formulas and Examples
Diagonal of a parallelogram – Formulas and examples
Backdrop of a Parallelogram

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