How to find the perimeter of a rhombus - Intermediate Geometry

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Question

Given a rhombus with diagonal lengths of 12cm and 16cm, find the perimeter.

Answer

A rhombus is a parallelogram with all of its sides being equal. A square is a rhombus with all of the angles being equal as well as all of the sides. Both squares and rhombuses have perpendicular diagonal bisectors that split each diagonal into 2 equal pieces, and also splitting the quadrilateral into 4 equal right triangles.

With this being said, we know the Pythagorean Theorem would work great in this situation, using half of each diagonal as the two legs of the right triangle.

$\left ( 8 cm \right )^{2}+\left ( 6cm \right )^{2}=h^{2}$ where is the hypotenuse.

$h^{2}=100$ $cm^{2}$ -->

We find the hypotenuse to be 10cm. Since the hypotenuse of each triangle is a side of the rhombus, we have found what we need to find the perimeter.

Each side is the same, so we add all 4 sides to find the perimeter.

$10+10+10+10=40\ cm$

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Question

The diagonals of a rhombus have lengths and units. What is its perimeter?

Answer

We should begin with a picture.

We should recall several things. First, all four sides of a rhombus are congruent, meaning that if we find one side, we can simply multiply by four to find the perimeter. Second, the diagonals of a rhombus are perpendicular bisectors of each other, thus giving us four right triangles and splitting each diagonal in half. We therefore have four congruent right triangles. Using Pythagorean Theorem on any one of them will give us the length of our sides.

With a side length of 17, our perimeter is easy to obtain.

$\small P=4\cdot17=68$

Our perimeter is 68 units.

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Question

A rhombus has a side length of $\frac{5}{6}$ foot, what is the length of the perimeter (in inches).

Answer

To find the perimeter, first convert $\frac{5}{6}$ foot into the equivalent amount of inches. Since, $\frac{5}{6}=\frac{10}{12}$ and $\frac{12}{12}=1foot$ , $\frac{5}{6}$ is equal to inches.

Then apply the formula , where is equal to the length of one side of the rhombus.

Since,

The solution is:

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Question

Find the perimeter of a rhombus that has a side length of $6\frac{5}{8}$ .

Answer

In order to find the perimeter of this rhombus, first convert $6\frac{5}{8}$ from a mixed number to an improper fraction:

$6\frac{5}{8}=\frac{48+5}{8}=\frac{53}{8}$

Then apply the formula: , where is equal to one side of the rhombus.

Since, $S=\frac{53}{8}$ the solution is:

$p=4(\frac{53}{8})$

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Question

A rhombus has an area of square units and an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$
$42=(base \times 7)$
$base=\frac{42}{7}=6$

Then apply the perimeter formula:

, where a side of the rhombus.

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Question

A rhombus has an area of square units, an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$108=(base\times 12)$

$base=108\div12=9$

Then apply the perimeter formula:

, where is equal to the length of one side of the rhombus.

The solution is:

$P=4\times9=36$

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Question

A rhombus has an area of square units and an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$56=(base\times 7)$

$base=\frac{56}{7}=8$

Then apply the perimeter formula:

, where the length of one side of the rhombus.

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Question

Given that a rhombus has an area of square units and an altitude of , find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$240=(base\times 16)$

$base=\frac{240}{16}=15$

Then apply the formula: , where

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Question

A rhombus has an area of square units and an altitude of , find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$9=(base\times 1.8)$

$base=\frac{9}{1.8}=5$
Then apply the formula: , where

$P=4\times5=20$

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Question

A rhombus has an area of square units, and an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$15=(base\times 2)$

$base=\frac{15}{2}=7.5$

Since, perimeter , where is equal to

Perimeter= $4\times7.5=30$

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Question

A rhombus has a side length of $9\frac{1}{2}$ . Find the perimeter of the rhombus.

Answer

To find the perimeter, apply the formula: , where

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Question

A rhombus has an area of square units, and an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$120=(base\times 7.5)$

$base=120\div7.5=16$

, where

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Question

A rhombus has an area of square units, and an altitude of . Find the perimeter of the rhombus.

Answer

In order to solve this problem, use the given information to work backwards to find a side length of the rhombus:

$A=(base\times altitude)$

$66=(base\times 6)$

$base=\frac{66}{6}=11$

, where

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Question

Given that a rhombus has a side length of $4\frac{3}{4}$ , find the perimeter of the rhombus.

Answer

To find the perimeter of this rhombus, apply the formula: , where $S=4\frac{3}{4}=\frac{19}{4}$

$P=4(\frac{19}{4})=19$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{14}{2}=7$

$\frac{\text{Diagonal 2}}{2}=\frac{16}{2}=8$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$7^2+8^2=\text{side length}^2$

$\text{side length}^2=49+64=113$

$\text{side length}=\sqrt{113}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4\sqrt{113}$

Solve and round to two decimal places.

$\text{Perimeter}=42.52$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{8}{2}=4$

$\frac{\text{Diagonal 2}}{2}=\frac{12}{2}=6$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$4^2+6^2=\text{side length}^2$

$\text{side length}^2=16+36=52$

$\text{side length}=2\sqrt{13}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(2\sqrt{13})$

Solve and round to two decimal places.

$\text{Perimeter}=28.84$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{14}{2}=7$

$\frac{\text{Diagonal 2}}{2}=\frac{18}{2}=9$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$7^2+9^2=\text{side length}^2$

$\text{side length}^2=49+81=130$

$\text{side length}=\sqrt{130}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(\sqrt{130})$

Solve and round to two decimal places.

$\text{Perimeter}=45.61$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{12}{2}=6$

$\frac{\text{Diagonal 2}}{2}=\frac{16}{2}=8$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$6^2+8^2=\text{side length}^2$

$\text{side length}^2=36+64=100$

$\text{side length}=10$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(10)$

Solve.

$\text{Perimeter}=40$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{24}{2}=12$

$\frac{\text{Diagonal 2}}{2}=\frac{26}{2}=13$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$12^2+13^2=\text{side length}^2$

$\text{side length}^2=144+169=313$

$\text{side length}=\sqrt{313}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(\sqrt{313})$

Solve and round to two decimal places.

$\text{Perimeter}=70.77$

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Question

Find the perimeter of a rhombus if it has diagonals of the following lengths: and .

Answer

Recall that the diagonals of a rhombus bisect each other and are perpendicular to one another.

In order to find the length of the side, we can use Pythagorean's theorem. The lengths of half of the diagonals are the legs, and the length of the side of the rhombus is the hypotenuse.

First, find the lengths of half of the diagonals.

$\frac{\text{Diagonal 1}}{2}=\frac{28}{2}=14$

$\frac{\text{Diagonal 2}}{2}=\frac{36}{2}=18$

Next, substitute these half diagonals as the legs in a right triangle using the Pythagorean theorem.

$14^2+18^2=\text{side length}^2$

$\text{side length}^2=196+324=520$

$\text{side length}=2\sqrt{130}$

Since the sides of a rhombus all have the same length, multiply the side length by in order to find the perimeter.

$\text{Perimeter}=4(\text{side length})$

$\text{Perimeter}=4(2\sqrt{130})$

Solve and round to two decimal places.

$\text{Perimeter}=91.21$

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