Ever noticed how some shapes in math also pop up in real life—like diamonds on a playing card or the shape of a kite? That’s a rhombus for you! In geometry, a rhombus is a special type of parallelogram where all four sides are exactly the same length. Because of this, it’s also called an equilateral quadrilateral (which is just a fancy way of saying all sides are equal).

Understanding rhombuses isn’t just cool for spotting shapes around you—it’s super useful for exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, and MCAT. In this article, we’ll break down everything you need to know about the definition, properties, area, perimeter, and walk you through some solved examples. Ready to ace those geometry questions? Let’s get started!

What is a Rhombus?

A rhombus is a type of quadrilateral whose opposite sides are parallel and all four sides are equal in length. Moreover, it is also known as a diamond. It is a special case of parallelogram and kite in terms of sides and angles. If all the angles of the rhombus-shaped Geometric image are right angles then it is known as a square.

Check out the below image to understand the same.

Here you can check that, AB, BC, CD and AD are sides of the rhombus and the line joining AC represents the diagonal \(d_1\) and the line joining BD denotes the diagonal \(d_2\) of the rhombus.

Rhombus Formulas

As per the definition, we can say that every square is a rhombus but not all rhombuses are squares. Let us now learn the different formulas related to the topic.

Area of a Rhombus

We can define the area of a rhombus in maths as the amount of space covered by all four sides of a rhombus in a two-dimensional space. The area of the rhombus formula taking the diagonals, height and base as well as using trigonometric ratios are as follows.

The area of the rhombus using the diagonals:

If \(d_1\) and \(d_2\), are the two diagonals then the area of the rhombus formula is equal to half of the product of the length of the diagonal. Mathematically this is written as: \(\frac{1}{2}\times d_1\times d_2=\frac{d_1\times d_2}{2}\).

The area of the rhombus using height and base:

If we are given with the length of base and height (perpendicular distance from base to the opposite side of rhombus), then the area of rhombus = \(b\times h\).

The area of the rhombus using side and trigonometric function:

In terms of a trigonometric function, we can say that the product of the square of the length of one side of the rhombus with the sine of any interior angle is equal to the area.

As per the property, ∠A = ∠C and ∠D = ∠B

The area of rhombus = \(a^2\sin A\text{ or}\ a^2\sin B\).

For the above formula, a is the length of the side and A and B are the interior angle.

Perimeter of a Rhombus

Perimeter is the total length of boundaries or the total distance around the edges of a rhombus.

The perimeter of rhombus formula = 4a

As all the side of rhombus are equal i.e. AD = DC = CB = BA = a.

Properties of Rhombus

At this point of the article, we know the definition and different formula links with the area and perimeter. Let us now learn the important properties:

• All the sides are of equal length.
• Opposite sides are parallel to one another and the opposite angles are equal.
• The summation of two adjacent angles is equivalent to 180 degrees.
• The two diagonals bisect each other at right angles.
• You will get a rectangle if you connect all 4 midpoints of the sides of the rhombus and the area of the rectangle will be half of the area of the rhombus.
• If we connect the diagonals of the rhombus then they split the rhombus into four congruent triangles.
• We can’t inscribe a rhombus in a circle. Similarly, we can’t circumscribe the circle around the rhombus.
• A rhombus has 4 interior angles.
• Adjacent angles of the rhombus are supplementary pairs.

Similarities between Squares and Rhombus

Throughout our discussion, we read the connection between squares and rhombuses. Let us understand the various similarities between these two shapes.

• Both the square and rhombus are quadrilateral.
• The diagonals of both square and rhombus bisect each other at right angles.
• The opposite sides are parallel in both rhombus and square.
• All the 4 sides of a rhombus as well as a square are of equal length.
• The summation of all the interior angles of a square as well as a rhombus is equal to 360°.

Solved Examples of Rhombus

We have read all the details regarding the definition, formula and properties related to rhombus shaped objects. Let us now practise some solved examples.

Solved Example 1: Madhuri has a kite where the lengths of the two diagonals are 6 cm and 12 cm, respectively. Now she wants to colour the kite, then determine the area of the kite.

Solution:
Given:

Diagonal of the kites, \(d_1\) = 12 cm, and \(d_2\) = 6 cm

\(A=\frac{d_1\times d_2}{2}\)

\(A=\frac{6\times12}{2}\)

A = 36 \(cm^2\)

The area of the rhombus shaped kite = 36 \(cm^2\).

Solved Example 2: Determine the measure of the diagonal for a rhombus diagram, if the area is 169 \(cm^{2}\) and the longest diagonal is 26 cm long.

Solution: 

Given:

Area of rhombus diagram = 169 \(cm^{2}\) and let the longest diagram i.e. \(d_1\) = 26 cm and the other diameter be denoted by \(d_2\).

Using area formula;

\(A=\frac{d_1\times d_2}{2}\)

\(169=\frac{\left(26\times d_2\right)}{2}\)

\(d_2\)=13

Thus, the length of another diagonal is 13 cm.

Solved Example 3: What is the perimeter of a rhombus shape tile, when the measure of all sides is equivalent to 4.6 cm?

Solution:

The side of rhombus shape tile = 4.6 cm

According to the formula of;

Perimeter = 4 x side

P = 4 x 4.6

P = 18.4 cm

Conclusion

So, now you’ve got a solid grip on what a rhombus is all about! From its equal sides and special diagonals to formulas for area and perimeter—you’re all set to tackle those tricky geometry questions on exams like the SAT, ACT, PSAT/NMSQT, and more. Keep practicing, and spotting rhombuses (or diamonds!) in real life will feel just as easy as solving them on paper!