Rhombus

Here we will learn about a rhombus, including what it is and how to identify it.

Students will first learn about a rhombus as part of geometry in 4 th grade.

What is a rhombus?

A rhombus is a geometric shape with the following properties:

• Four congruent sides
• Four vertices
• Two pairs of opposite parallel sides
• Two pairs of congruent opposite angles

For example,

Let’s look at each of the properties of a rhombus.

• Four congruent sides:

Since a rhombus is a type of quadrilateral, it has four sides. All the sides are equal lengths (congruent). This can be shown with a tick mark on each side or with given measurements. Since all the sides of a rhombus are equal, it is an equilateral quadrilateral.

• Four vertices:

The sides of a rhombus come together at points called vertices. Since there are four sides, there are four vertices.

• Two pairs of opposite parallel sides:

Since a rhombus is a type of parallelogram, the opposite sides will be the same distance apart and never intersect.

• Two pairs of congruent opposite angles:

Since the pairs of opposite sides are parallel, the opposite angles formed are congruent. The curved lines within the angles are used to show equivalence.

Since a four sided polygon will always have four vertices, these properties can be combined under the property of four equal sides when classifying.

Since the two pairs of opposite parallel sides create two congruent opposite angles, these properties can be combined under the property of two pairs of opposite parallel sides when classifying.

Finally, a square is a special type of rhombus. The square shape fits all the properties of a rhombus.

Note on vocabulary: More than one rhombus can be called ‘rhombuses’ or ‘rhombi.’

What is a rhombus?

Common Core State Standards

How does this relate to 3 rd grade math to 6 th grade math?

• Grade 3 – Measurement and Data (3.MD.8)
  Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas, or with the same area and different perimeters.

• Grade 4 – Geometry (4.G.A.2)
  Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

• Grade 5 – Geometry (5.G.B.3)
  Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.

• Grade 5 – Geometry (5.G.B.4)
  Classify two-dimensional figures in a hierarchy based on properties.

• Grade 6 – Geometry (6.G.1)
  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

How to identify a rhombus

In order to identify a rhombus:

1. Look for a shape with four equal sides.
2. Decide if the shape has two pairs of opposite parallel sides.

Rhombus examples

Example 1: identifying a shape

Is the shape a rhombus?

1. Look for a shape with four equal sides.

The shape has four equal sides.

2Decide if the shape has two pairs of opposite parallel sides.

The opposite sides are the same distance apart and will never intersect.

The shape is a rhombus.

Example 2: identifying a shape

Is the shape a rhombus?

Look for a shape with four equal sides.

The shape has four equal sides.

Decide if the shape has two pairs of opposite parallel sides.

The opposite sides are the same distance apart and will never intersect.

The shape is a rhombus. It is also classified as a square.

Example 3: identifying a shape

Is the shape a rhombus?

Example 4: identifying a shape

The sides of a polygon are 7 \, cm, 7 \, cm, 7 \, cm, and 7 \, cm , and the opposite sides are parallel. Is the shape a rhombus?

Decide if the shape has two pairs of opposite parallel sides.

Yes, the statement says “opposite sides are parallel.”

The shape is a rhombus.

Note: We don’t know anything about the angles, so the overall form of the rhombus is not clear. It could be…

Or any other angle measurements between 0^{\circ} and 180^{\circ}.

Example 5: finding rhombuses in geometric shapes

How many rhombuses are formed by the geometric shape?

Look for a shape with four equal sides.

The shapes highlighted have four equal sides.

Decide if the shape has two pairs of opposite parallel sides.

The opposite sides are the same distance apart and will never intersect.

There are 4 rhombuses in the geometric shape.

Example 6: finding rhombuses in the real world

How many rhombuses are in the image below?

Look for a shape with four equal sides.

The shape of the road sign highlighted has four equal sides.

Decide if the shape has two pairs of opposite parallel sides.

The opposite sides are the same distance apart and will never intersect.

There is one rhombus in the image.

How to calculate the perimeter and area of a rhombus

In order to calculate the perimeter and area of a rhombus:

1. Identify the side lengths and the height.
2. Calculate.

Example 7: calculating perimeter of a rhombus

A rhombus has a side length of 6.7 \, cm. What is the perimeter?

Example 8: calculating area of a rhombus

What is the area of the rhombus?

Teaching tips for rhombus

• Printable worksheets that include real world drawings or images are a good way for students to practice identifying rhombuses. They can also use a tool to measure each side, to create justifications for whether or not a shape is a rhombus.

• There are many real life items that are in the shape of a rhombus. Encourage students to look for examples of rhombuses everywhere they go.

Easy mistakes to make

• Mistaking a rectangle for a rhombus
  A rectangle has parallel opposite sides, but all the sides are not equal length, so it is NOT a rhombus.

• Forgetting the interior angles can be any measure between \bf{0}^{\circ} and \bf{180}^{\circ}
  Some commonly shown rhombuses are…
  Angles of 45^{\circ} and 135^{\circ} and Angles of 90^{\circ}
  
  

  

  
  However, the interior angles of a rhombus can vary from >0^{\circ} to <180^{\circ} .
  For example,
  From angles close to 0^{\circ} and 180^{\circ} to angles close to 90^{\circ}.
  
  

  

• Thinking adjacent angles in a rhombus are always equal
  Adjacent sides are equal, but not necessarily the angles. The only time the adjacent angles are equal is when the rhombus can also be classified as a square.
  For example,
  
  

  

Related quadrilateral lessons

• Quadrilateral
• Types of quadrilaterals
• Parallelogram
• Rectangle
• Square shape
• Trapezoid

The next lessons are

• Perimeter
• Symmetry
• Angles in polygons

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