How to find the perimeter of a pentagon - Geometry

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Question

An apothem is a line drawn from the center of a regular shape to the center of one of its edges. The line drawn is perpendicular to the edge. The apothem of a regular pentagon is . What is the perimeter of this pentagon?

Answer

From the given information, we can imagine the following image

In order to solve for the perimeter, we need to solve for the length of one of the sides. This can be accomplished by creating a right triangle from the apothem and the top angle that's marked.

This angle measure can be calculated through:

$\frac{360^{\circ}}{5} = 72^{\circ}$ , where the sum of all the angles around the center of the pentagon sum up to $360^{\circ}$ and the is for the number of sides. We can do this because all of the interior angles of the pentagon will be equal because it is regular. Then, this answer needs to be divided by because the pictured right triangle is half of a larger isosceles triangle.

$\frac{72^{\circ}}{2}=36^{\circ}$

Therefore, the marked angle is $36^{\circ}$ .

Now that one side and one angle are known, we can use SOH CAH TOA because this is a right triangle. In this case, the tangent function will be used because the mystery side of interest is opposite of the known angle, and we are already given the adjacent side (the apothem). In solving for the unknown side, we will label it as .

$\tan(\theta )=\frac{opp}{adj}$

$\tan (36^{\circ})=\frac{x}{3}$

$3 \cdot \tan (36^{\circ})=x$

For a more precise answer, keep the entire number in your calculator. This will prevent rounding errors.

Keep in mind that the base of the triangle is actually only half the length of the side. This means that the value for must be multiplied by .

$2.17963...\cdot 2 = 4.3596...$

In order to solve for the perimeter, the length of the side needs to be multiplied by the number of sides; in this case, there are five sides.

$4.3596... \cdot 5 = 21.7963$

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is .

Compare your answer with the correct one above

Question

If a side length of a pentagon is , what is the perimeter of the pentagon?

Answer

Write the formula to find the perimeter of a pentagon.

Substitute the side length into the equation and simplify.

Compare your answer with the correct one above

Question

If the side of a pentagon is $5\sqrt5$ , what is the perimeter?

Answer

Write the perimeter formula for pentagons and substitute the side length.

$P=5(5\sqrt5)=25\sqrt5$

Compare your answer with the correct one above

Question

Determine the perimeter of a pentagon if the side length is $\frac{\sqrt2}{10}$ .

Answer

Write the formula to find the perimeter for a pentagon.

Substitute the side into the perimeter formula.

$P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

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Question

In the figure below, if the length of is one-third the length of , what is the perimeter of the pentagon?

Answer

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of and .

Let be the length of . Then, it follows that is the length of .

From the figure, we know that . Since we are working with a rectangle, .

Now, solve for .

So then, $AF=6 \text{ and }FD=18$ .

Now that we know , we can use the Pythagorean theorem to solve for the length of .

$AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of by using the Pythagorean theorem.

$DE=\sqrt{(FD)^2+(EF)^2}$

$DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

In pentagon below, the length of is one-third that of . Find the perimeter of the pentagon.

Answer

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 38=\frac{9}{AF}$

$AF=\frac{9}{\tan38}=11.52$

Now, we know that is three times the length of .

Since we have rectangle , .

Thus,

Now, we can use cosine to find the length of .

$\sin38=\frac{9}{AE}$

$AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

Compare your answer with the correct one above

Question

In pentagon below, the length of is one-third the length of . Find the perimeter of the pentagon.

Answer

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 26=\frac{13}{AF}$

$AF=\frac{13}{\tan 26}=26.65$

Now, we know that is three times the length of .

Since we have rectangle , .

Thus,

Now, we can use sine to find the length of .

$\sin 26=\frac{13}{AE}$

$AE=\frac{13}{\sin 26}=29.66$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{13^2+79.95^2}=81.00$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=28+106.60+28+81.00+29.66=273.26$

Compare your answer with the correct one above

Question

An apothem is a line drawn from the center of a regular shape to the center of one of its edges. The line drawn is perpendicular to the edge. The apothem of a regular pentagon is . What is the perimeter of this pentagon?

Answer

From the given information, we can imagine the following image

In order to solve for the perimeter, we need to solve for the length of one of the sides. This can be accomplished by creating a right triangle from the apothem and the top angle that's marked.

This angle measure can be calculated through:

$\frac{360^{\circ}}{5} = 72^{\circ}$ , where the sum of all the angles around the center of the pentagon sum up to $360^{\circ}$ and the is for the number of sides. We can do this because all of the interior angles of the pentagon will be equal because it is regular. Then, this answer needs to be divided by because the pictured right triangle is half of a larger isosceles triangle.

$\frac{72^{\circ}}{2}=36^{\circ}$

Therefore, the marked angle is $36^{\circ}$ .

Now that one side and one angle are known, we can use SOH CAH TOA because this is a right triangle. In this case, the tangent function will be used because the mystery side of interest is opposite of the known angle, and we are already given the adjacent side (the apothem). In solving for the unknown side, we will label it as .

$\tan(\theta )=\frac{opp}{adj}$

$\tan (36^{\circ})=\frac{x}{3}$

$3 \cdot \tan (36^{\circ})=x$

For a more precise answer, keep the entire number in your calculator. This will prevent rounding errors.

Keep in mind that the base of the triangle is actually only half the length of the side. This means that the value for must be multiplied by .

$2.17963...\cdot 2 = 4.3596...$

In order to solve for the perimeter, the length of the side needs to be multiplied by the number of sides; in this case, there are five sides.

$4.3596... \cdot 5 = 21.7963$

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is .

Compare your answer with the correct one above

Question

If a side length of a pentagon is , what is the perimeter of the pentagon?

Answer

Write the formula to find the perimeter of a pentagon.

Substitute the side length into the equation and simplify.

Compare your answer with the correct one above

Question

If the side of a pentagon is $5\sqrt5$ , what is the perimeter?

Answer

Write the perimeter formula for pentagons and substitute the side length.

$P=5(5\sqrt5)=25\sqrt5$

Compare your answer with the correct one above

Question

Determine the perimeter of a pentagon if the side length is $\frac{\sqrt2}{10}$ .

Answer

Write the formula to find the perimeter for a pentagon.

Substitute the side into the perimeter formula.

$P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

Compare your answer with the correct one above

Question

In the figure below, if the length of is one-third the length of , what is the perimeter of the pentagon?

Answer

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of and .

Let be the length of . Then, it follows that is the length of .

From the figure, we know that . Since we are working with a rectangle, .

Now, solve for .

So then, $AF=6 \text{ and }FD=18$ .

Now that we know , we can use the Pythagorean theorem to solve for the length of .

$AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of by using the Pythagorean theorem.

$DE=\sqrt{(FD)^2+(EF)^2}$

$DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

In pentagon below, the length of is one-third that of . Find the perimeter of the pentagon.

Answer

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 38=\frac{9}{AF}$

$AF=\frac{9}{\tan38}=11.52$

Now, we know that is three times the length of .

Since we have rectangle , .

Thus,

Now, we can use cosine to find the length of .

$\sin38=\frac{9}{AE}$

$AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

Compare your answer with the correct one above

Question

In pentagon below, the length of is one-third the length of . Find the perimeter of the pentagon.

Answer

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 26=\frac{13}{AF}$

$AF=\frac{13}{\tan 26}=26.65$

Now, we know that is three times the length of .

Since we have rectangle , .

Thus,

Now, we can use sine to find the length of .

$\sin 26=\frac{13}{AE}$

$AE=\frac{13}{\sin 26}=29.66$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{13^2+79.95^2}=81.00$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=28+106.60+28+81.00+29.66=273.26$

Compare your answer with the correct one above

Question

An apothem is a line drawn from the center of a regular shape to the center of one of its edges. The line drawn is perpendicular to the edge. The apothem of a regular pentagon is . What is the perimeter of this pentagon?

Answer

From the given information, we can imagine the following image

In order to solve for the perimeter, we need to solve for the length of one of the sides. This can be accomplished by creating a right triangle from the apothem and the top angle that's marked.

This angle measure can be calculated through:

$\frac{360^{\circ}}{5} = 72^{\circ}$ , where the sum of all the angles around the center of the pentagon sum up to $360^{\circ}$ and the is for the number of sides. We can do this because all of the interior angles of the pentagon will be equal because it is regular. Then, this answer needs to be divided by because the pictured right triangle is half of a larger isosceles triangle.

$\frac{72^{\circ}}{2}=36^{\circ}$

Therefore, the marked angle is $36^{\circ}$ .

Now that one side and one angle are known, we can use SOH CAH TOA because this is a right triangle. In this case, the tangent function will be used because the mystery side of interest is opposite of the known angle, and we are already given the adjacent side (the apothem). In solving for the unknown side, we will label it as .

$\tan(\theta )=\frac{opp}{adj}$

$\tan (36^{\circ})=\frac{x}{3}$

$3 \cdot \tan (36^{\circ})=x$

For a more precise answer, keep the entire number in your calculator. This will prevent rounding errors.

Keep in mind that the base of the triangle is actually only half the length of the side. This means that the value for must be multiplied by .

$2.17963...\cdot 2 = 4.3596...$

In order to solve for the perimeter, the length of the side needs to be multiplied by the number of sides; in this case, there are five sides.

$4.3596... \cdot 5 = 21.7963$

This is the final answer, so the entire decimal is no longer needed. Rounded, the perimeter is .

Compare your answer with the correct one above

Question

If a side length of a pentagon is , what is the perimeter of the pentagon?

Answer

Write the formula to find the perimeter of a pentagon.

Substitute the side length into the equation and simplify.

Compare your answer with the correct one above

Question

If the side of a pentagon is $5\sqrt5$ , what is the perimeter?

Answer

Write the perimeter formula for pentagons and substitute the side length.

$P=5(5\sqrt5)=25\sqrt5$

Compare your answer with the correct one above

Question

Determine the perimeter of a pentagon if the side length is $\frac{\sqrt2}{10}$ .

Answer

Write the formula to find the perimeter for a pentagon.

Substitute the side into the perimeter formula.

$P=5(\frac{\sqrt2}{10})=\frac{\sqrt2}{2}$

Compare your answer with the correct one above

Question

In the figure below, if the length of is one-third the length of , what is the perimeter of the pentagon?

Answer

Notice that the pentagon is made up of one rectangle and two right triangles. Start by finding the lengths of and .

Let be the length of . Then, it follows that is the length of .

From the figure, we know that . Since we are working with a rectangle, .

Now, solve for .

So then, $AF=6 \text{ and }FD=18$ .

Now that we know , we can use the Pythagorean theorem to solve for the length of .

$AE=\sqrt{(AF)^2+(FE)^2}$

Plug in the lengths of the legs to find the length of the hypotenuse.

$AE=\sqrt{6^2+4^2}=\sqrt{52}$

Next, find the length of by using the Pythagorean theorem.

$DE=\sqrt{(FD)^2+(EF)^2}$

$DE=\sqrt{18^2+4^2}=\sqrt{340}$

Finally, add up all the side lengths of the pentagon to find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=8+24+8+\sqrt{340}+\sqrt{52}=65.65$

Make sure to round to places after the decimal.

Compare your answer with the correct one above

Question

In pentagon below, the length of is one-third that of . Find the perimeter of the pentagon.

Answer

Notice that this pentagon is made up of two right triangles and one rectangle.

Start by finding the length of . Since we are given an angle measurement and a length of a leg of the triangle, we can use tangent to find the length of .

$\tan 38=\frac{9}{AF}$

$AF=\frac{9}{\tan38}=11.52$

Now, we know that is three times the length of .

Since we have rectangle , .

Thus,

Now, we can use cosine to find the length of .

$\sin38=\frac{9}{AE}$

$AE=\frac{9}{\sin38}=14.62$

Next, use the Pythagorean Theorem to find the length of .

$ED=\sqrt{(EF)^2+(FD)^2}$

$ED=\sqrt{9^2+34.56^2}=35.71$

Now that we have all the side lengths of the pentagon, we can find its perimeter.

$\text{Perimeter}=AB+BC+CD+DE+EA$

$\text{Perimeter}=25+46.08+25+35.71+14.62=146.41$

Compare your answer with the correct one above

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