how do you find the area of a rhombus

What is a Rhombus?
Surface area of a Rhombus Formula
How to Discover the Area of a Rhomb

Formula Using the Distance and Side
Formula Using the Side and Angle
Formula Using the Diagonals

What is a Rhombus?

A rhombus is a airplane effigy, and so information technology is two-dimensional. It is a closed figure with straight (linear) sides, one of the many quadrilaterals (iv-sided shapes). It is a special case of a parallelogram. All four sides are of equal length, and both pairs of opposite sides are parallel. Opposite angles are equal, too. That's it!

A rhombus tin also be called a rhomb, lozenge, or diamond. A foursquare is a rhombus with 4 equal (correct) angles.

Sometimes you see a rhombus with two sides horizontal, as if a square has been run into by a jitney and tilted over (that is a handy mnemonic to remember its proper name: run, bus; rhomb). In this presentation, the altitude (height) of the rhombus can be very easy to see.

$rhombus properties$

Sometimes the rhomb is fatigued then 1 of its two diagonals (lines connecting opposite vertices) is horizontal, making the diamond shape more obvious.

One unusual quality of a rhombus is that its diagonals are always perpendicular to each other, no matter the angles of the rhombus's four vertices.

These diagonals likewise bifurcate each other, which ways they divide the rhombus upwardly into 4 correct-bending triangles. The squares of the lengths of the two diagonals are always four times the square of a side.

For such a unproblematic shape, a rhomb has many parts and measurements. Knowing how to utilise these measurements tin can help yous find surface area, perimeter and other facts almost the rhomb.

Area of a Rhombus Formula

There are three different formulas for finding the surface area of a rhombus. In that location is 1 using the altitude and side, another using the side and angle, and i for the diagonals. The three formulas to find area depend on information you know almost the rhomb.

If you know Altitude (height) and side s the formula is:
$a r e a = h east i g h t \times s$
If you know the length of one side s and the measure of ane angle the formula is:
$a r due east a = s^{2} \sin (∠ A) = {south}^{ii} \sin (∠ B)$
If you know the lengths of the diagonals the formula is:
$a r eastward a = \frac{(d_{1} \times d_{2})}{2}$

How to Observe the Area of a Rhomb

Let'due south construct a rhombus with sides $due south$ and 4 vertices with interior angles $A$ , $B$ , $C$ and $D$ . We tin connect reverse angles with diagonals $d_{1}$ and $d_{2}$ . Connecting one side to the other with a perpendicular line gives the altitude or elevation. Our rhombus has:

$area of rhombus$

Four equal-length sides: $A B$ , $B C$ , $C D$ and $D A$
Four interior angles with opposite angles equal: $∠ a = ∠ c$ and $∠ b = ∠ d$
Two diagonals: $d_{i}$ and $d_{two}$ ; merely in a square will these be equal-length diagonals
Altitude, or superlative -- When the rhombus sits with ii sides horizontal (apartment), the elevation $h$ is the distance from one side to the opposite side; the line segment perpendicular to one side, connecting to the opposite side

Since the iv sides are equal, if you know the length of whatever side $due south$ , yous know the length of all four sides.

Since contrary angles are equal, and the four angles add to $360 °$ , if you lot know i angle, you tin find all angles. Since reverse sides are parallel, next angles in a rhombus add together to $180 °$ .

For finding area, you need to know the height, or altitude, $h$ , of the rhomb.

Remember that the superlative is not the same as the length of a side.

Formula Using the Distance and Side

If y'all accept a mental moving picture of a rhombus being a tilted foursquare, this first method will make a lot of sense.

If the rhombus were a square, its area in square units would be side ten side, right? Well, when the rhombus is tilted, you could imagine cut off the triangular part on one side of the rhomb and sliding information technology over to the matching other side, restoring the shape to its foursquare-ness.

You cannot actually cut up every rhombus you meet, though, so consider what that ane synthetic, perpendicular side really is: the tiptop, or distance, of the rhomb.

So, the showtime and mayhap easiest style to find expanse of a rhombus is to find the length of one side and the rhombus's elevation. Multiply these and you have the expanse, in square units:

Instance:

So, if y'all had a rhombus whose altitude is 3 inches with sides that are 5 inches, then the area of this rhomb is:

$3 i n c h e s \times 5 i north c h e s = fifteen i n c h e s^{ii}$

$area of rhombus$

Another example is where side southward is 15 feet and the height of 11 anxiety. This area of this rhombus is:

$15 \times eleven = 165 south q u a r e f e eastward t$

Formula Using the Side and Angle

The second way to find the area of a rhomb is by knowing the length of a side s and the measure of one bending $(∠ A o r ∠ B)$ . Here you need to detect the sine of the angle, but the formula is withal straightforward:

$a r eastward a = s^{2} \sin (∠ A)$

$a r due east a = {southward}^{2} \sin (∠ B)$

Yous can see that these 2 formulas give the same result, so

$a r due east a = {due south}^{2} \sin (∠ A) = {southward}^{2} \sin (∠ B)$

Example:

$area of rhombus example$

In a rhombus with sides of 10 yards and interior adjacent angles of $60 °$ and $120 °$ then to observe the expanse of this rhombus, nosotros would plug this into our formula for area using the side and an bending.

$a r due east a = 10^{2} \sin (60 °)$

which is also the same as

$a r east a = 10^{2} \sin (120 °)$

And then, we multiply these two numbers together:

$a r e a = 100 \times 0.866$

Then, we get our reply:

$a r e a = 86.half dozen southward q u a r east y a r d s$

Remember that for this method the angle you choose does not affair. The formula is the same for both angles; you just have to pick i.

The sine of $60 °$ and $120 °$ is the aforementioned, $0.866$

Formula Using the Diagonals

Recollect that the diagonals of a rhomb always intersect each other at right angles, and bifurcate each other. That ways the two diagonals form two sides of a foursquare that is twice the size of the rhombus.

You can find the area in foursquare units of the rhombus by multiplying the lengths of the two diagonals ( $d_{i}$ and $d_{2}$ ) and dividing by two.

If our rhombus just has the measurements for the diagonals, this is the formula we would employ.

Case:

If our rhomb had diagonals measuring 24 and 18 meters in length, then to find the area of this rhomb, we would plug the numbers into our formula.

$a r e a = \frac{(24 \times 18)}{2}$

$area of rhombus diagonal formula$

We multiply the two diagonals:

$a r e a = \frac{(432)}{2}$

And then we get our answer:

$a r e a = 216 due south q u a r eastward m e t due east r s$

Lesson Summary

You lot have reviewed what a rhombus is, how information technology fits into the family of quadrilaterals, what its diverse parts are, and how to find its area.

$area of rhombus formulas$