Geometry - How to find the length of the side of a rhombus

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

$\sqrt5$

$\sqrt6$

$\sqrt2$

Cannot be determined

Explanation

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.

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

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

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

$1^2+2^2=\text{side length}^2$

$\text{side length}^2=1+4=5$

$\text{side length}=\sqrt5$

Given that a rhombus has a perimeter of , find the length of one side of the rhombus.

Explanation

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

Since , we can set up the following equation and solve for .

$124=4\times (S)$

$S=\frac{124}{4}=31$

Find the length of a side of a rhombus that has diagonal lengths of and .

Explanation

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

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

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

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{6^2+7^2}=\sqrt{85}=9.22$

Make sure to round to places after the decimal.

A rhombus has an area of square units and a height of . Find the length of one side of the rhombus.

Explanation

To find the length of a side of the rhombus, work backwards using the formula:

$A=(base\times height)$

Since we are given the area and the height we plug these values in and solve for the base.

$360=(base\times 8)$

$base=\frac{360}{8}=45$

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

$\sqrt{29}$

$\sqrt{31}$

$\sqrt{35}$

$3\sqrt3$

Explanation

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.

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

$\frac{\text{Diagonal 2}}{2}=\frac{10}{2}=5$

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

$2^2+5^2=\text{side length}^2$

$\text{side length}^2=4+25=29$

$\text{side length}=\sqrt{29}$

A garden is shaped like a rhombus. If the diagonals of the garden are feet and feet in length, in feet, what is the length of feet for a side of the garden?

Explanation

Recall that in a rhombus, the diagonals are not only perpendicular to each other, but also bisect one another.

Thus, we can find the lengths of half of each diagonal and use that in the Pythagorean Theorem to find the length of the side of the rhombus.

First, find the lengths of half of each diagonal.

$\text{Half Diagonal 1}=\frac{10}{2}=5$

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

Now, use these half diagonals as the legs of a right triangle that has the side of the rhombus as its hypotenuse.

$\text{Side of Rhombus}=\sqrt{(\text{Half Diagonal 1})^2+(\text{Half Diagonal 2})^2}$

Plug in the lengths of the half diagonals to find the length of the rhombus.

$\text{Side of Rhombus}=\sqrt{5^2+7^2}=\sqrt{74}=8.60$

Make sure to round to places after the decimal.

A rhombus has an area of square units, and a height of . Find the length of one side of the rhombus.

Explanation

To find the length of a side of the rhombus, work backwards using the area formula:

$Area=(base\times height)$

Since we are given the area and the height we plug these values in and solve for the base.

$69=(base\times 6)$

$base=\frac{69}{6}=11.5$

Find the length of a side of a rhombus if it has diagonals possessing the following lengths: and .

$2\sqrt{34}$

$3\sqrt{15}$

$12\sqrt2$

$6\sqrt{10}$

Explanation

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.

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