Rhombuses

By this point, you may have come across quadrilaterals quite a lot. If not, know that a quadrilateral is any four-sided shape. You may also remember learning about the different types of quadrilaterals. Of course, we have the square and the rectangle. But we also have those others that are less common and can be harder to remember, such as parallelograms, trapezoids, and kites. In this article, we discuss rhombuses. Let's start by talking about what we mean by a rhombus

Rhombuses Rhombuses

Create learning materials about Rhombuses with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Contents
Table of contents

    Rhombuses meaning

    • A rhombus is a quadrilateral with all sides equal in length and two pairs of parallel sides.
    • A quadrilateral is any four-sided shape.
    • A parallelogram is any quadrilateral with two pairs of parallel sides.

    A quadrilateral with 2 pairs of parallel opposite sides is called a parallelogram.

    From the definition of the rhombus and parallelogram, we see that all rhombuses are special types of parallelograms, since they have two pairs of parallel sides. It is helpful to be familiar with parallelograms for the purposes of this article.

    The following figure illustrates a rhombus, ABCD. Since it is a rhombus, all of its sides are of equal length.

    Rhombuses rhombus illustration StudySmarter

    Rhombus illustration - StudySmarter Originals

    From the figure of ABCD, we have:

    AB=BC=CD=DA

    Since a rhombus is a type of parallelogram, the opposite sides are parallel. Thus, AD is parallel to BC and AB is parallel to BC.

    Properties of rhombuses

    Now that we have discussed the basic characteristics of a rhombus, let's consider its properties in more detail.

    Rhombuses as parallelograms

    We have mentioned that rhombuses are a special type of parallelogram, so we can say that all the properties of parallelograms apply to rhombuses as well. Let's see how the properties of parallelograms in general apply specifically to rhombuses:

    • Both pairs of opposite sides of a rhombus are parallel.

    • Opposite angles of a rhombus are equal.

    • The diagonals of a rhombus bisect each other. In other words, the intersection of the two diagonals is at the mid point of each diagonal.

    • Each diagonal of a rhombus divides the rhombus into 2 congruent triangles.

    Properties unique to rhombuses

    In addition to the properties concerning parallelograms, there are also additional properties specific and unique to rhombuses. We will describe this with reference to the rhombus ABCD below:

    Rhombuses illustration with diagonals StudySmarter

    Rhombus illustration with diagonals shown - StudySmarter Originals
    • The diagonals of a rhombus are perpendicular to each other. This means they are at right angles to each other.
    • Thus, in the diagram above, AC BD.
    • From this, we could also say that AOD=COD=COB=AOB=90°
    • Each diagonal of a rhombus bisects a pair of opposite interior angles.
    • In other words, CAB = CAD =ACB=ACD and ABD=CBD=ADB=CDB
    • The four triangles created when we add in the rhombus diagonals are congruent. So, they are mathematically identical but just oriented differently.

    The congruent triangles of rhombuses

    From the properties of rhombuses, we know that its diagonals divide the shape into four triangles that are congruent. What does it mean for triangles to be congruent? Two or more triangles are congruent if they are mathematically identical. In other words, all of the sides and angles are the same, even if they are oriented differently. Also recall that the internal angles in a triangle sum to 180 degrees.

    Consider the rhombus below. Prove that for the given rhombus, AC ⊥ BD.

    The mathematical symbol means "perpendicular to."

    Rhombuses illustration with diagonals shown StudySmarter

    Rhombus example - StudySmarter Originals

    Solution:

    By the definition of a rhombus:

    AB=BC=CD=DA

    Now consider the triangles AOB and COB:

    The side OB is a side of both triangles. Now, AB=BC, since ABCD is a rhombus. We also have OA=OC, since ABCD is also a parallelogram and the diagonals of a parallelogram bisect each other. Therefore, triangle AOB is congruent to triangle COB. In other words, they are exactly the same triangles, just rotated in different positions.

    This implies that:

    AOB=COB

    Additionally, AOB and COB lie on the same straight line. Thus,

    AOB+COB=180°

    Therefore,

    AOB=COB=180÷2=90°

    Similarly we can show that:

    AOD=COD=90°

    Thus, AC and BD are perpendicular. That is, ACBD.

    Area formula for rhombuses

    We have a specific formula to find the area of a rhombus. Consider the following rhombus:

    Rhombuses illustration with diagonals shown StudySmarter

    Rhombus example - StudySmarter Originals

    Now, let's label the diagonals such that BD=d1 and AC=d2.

    The area of the rhombus is given by the formula:

    Area = 12d1d2

    Since a rhombus is a type of quadrilateral, we also have an alternative formula for finding the area.

    The area of any quadrilateral is given by the formula:

    Area = base × height

    So, depending on what lengths we have, we can use either of the above formulas to work out the area of a rhombus.

    Note that when we discuss area, we use square units. For example, if the lengths of the base and height are given both in centimeters, the units for the area are centimeters squared (cm2).

    A rhombus has diagonals of lengths 10 cm and 15 cm . What is the area of the rhombus?

    Solution:

    Using the specific formula for the area of a rhombus, we have:

    Area = 12d1d2

    Substituting in d1=10 and d2=15 , we have:

    Area = 12×10 cm×15 cm = 75 cm2

    Thus, the area of this rhombus is 75 cm2.

    Further rhombuses examples

    Now we will look at some further example problems on rhombuses.

    Consider the below rhombus. Given that ACB=35°, find DBA.

    Rhombuses illustration with diagonals shown StudySmarter

    Rhombus example - StudySmarter Originals

    Solution:

    Recall that in a rhombus, each diagonal bisects, which means that we have a pair of equal angles. We also know that opposite angles in a rhombus are equal.

    Thus,

    ACB=OAB=35°

    Now, we mentioned earlier that the diagonals in a rhombus are perpendicular to each other. Therefore,

    AOB=90°

    Since OAB forms a triangle, and angles in a triangle sum to 180°, we can work out DBA:

    DBA + AOB +OAB=180°

    So, substituting in the known angles, we have

    DBA + 90° +35°=180°

    which implies that

    DBA + 125°=180°

    Subtracting 180° from both sides, we obtain:

    DBA=180°-125°=55°

    So, we have that DBA =55°.

    Consider the following rhombus depicted below. Given that ADC=120°, find OAD.

    Rhombuses illustration with diagonals shown StudySmarter

    Rhombus example - StudySmarter Originals

    Solution:

    Recall that diagonal BD bisects ADC. Therefore, we have,

    ADO=120°÷2=60°.

    We also have that DOA=90° due to the perpendicular bisector.

    So, since angles in a triangle sum to 180°, we have:

    OAD=180°-90°-60°=30°.

    Thus, OAD=30°.

    Rhombuses - Key takeaways

    • A rhombus is a special quadrilateral with all four sides equal in length and two pairs of parallel sides.
    • A parallelogram is any quadrilateral with two pairs of parallel sides, so a rhombus is also a parallelogram.
    • Opposite angles of a rhombus are equal.
    • The diagonals of a rhombus bisect each other. In other words, the intersection of the two diagonals is at the midpoint of each diagonal.
    • The diagonals of a rhombus divide the shape into four congruent right-angled triangles.
    • The area of a rhombus is given by 12d1d2 where d1 and d2 are the diagonals.
    Frequently Asked Questions about Rhombuses

    What are rhombuses?

    A rhombus is a quadrilateral with all 4 of its sides equal.

    What shape is a rhombus?

    A rhombus is a quadrilateral with all 4 of its sides equal.

    Is a square a rhombus?

    A square is a special type of rhombus.

    How to find the area of a rhombus?

    Consider a rhombus with diagonals of length d1 and d2. The area of the rhombus is given by: A = 0.5×d1×d2.

    What are the properties of rhombuses?

    • The diagonals of a rhombus are perpendicular to each other.
    • If the diagonals of a parallelogram are perpendicular, then it implies that the given parallelogram is a rhombus.
    • Each diagonal of a rhombus bisects a pair of opposite angles.
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Rhombuses Teachers

    • 7 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App