Rhombus
- Definition of Rhombus
- Relation between a Rhombus and a Square
- Relation between a Rhombus and a Parallelogram
- What are the Properties of a Rhombus?
- Area of a Rhombus When its Base and Altitude are Known
- Area of a Rhombus When its Diagonals are Known
- The Perimeter of a Rhombus
- Practice Problems
- Frequently Asked Questions
Definition of Rhombus
A rhombus is a unique type of quadrilateral that stands out for its distinctive properties. Other than rhombus various other commonly seen quadrilaterals are:
- Square
- Rectangle
- Parallelogram
- Trapezoid
- Kite
To understand a rhombus, let’s first look into the characteristics of a parallelogram. A parallelogram is a four-sided figure with opposite sides that are parallel and equal in length. To visualize what does a rhombus look like, picture a parallelogram where not only the opposite sides are equal, but all four sides are also identical. This special case is precisely what defines a rhombus.
In simpler terms, a rhombus is a special kind of parallelogram having all its sides equal. This makes it unique and earns it the nickname of a "tilted square." Picture a square slightly tilted, and you have a mental image of a rhombus. It's like a square that decided to lean to one side while still maintaining its distinctive symmetry. On playing cards, the diamond suit symbol exemplifies the rhombus shape.
With all its sides being equal, the rhombus qualifies as a regular polygon. Other examples of regular polygons include equilateral triangles, regular pentagons, regular hexagons, regular octagons, etc.
Other names of a rhombus are:
- Diamond
- Lozenge
Relation between a Rhombus and a Square
How are a Rhombus and a Square Similar?
- Both a square and a rhombus have four sides of equal length.
- The diagonals of both a square and a rhombus bisect each other at right angles.
- In both shapes, opposite angles are equal.
- Both a square and a rhombus have their opposite sides parallel.
How is a Rhombus Different from a Square?
- In a square, all angles are right angles, while the opposite angles of a rhombus may have varying measurements.
- The diagonals of a square are equal in length, however the diagonals of a rhombus may not be equal in length.
Facts about Rhombus and a Square
- A rhombus with right angles is equivalent to a square.
- Thus, all squares are rhombuses, but not every rhombus is a square.
Relation between a Rhombus and a Parallelogram
How are a Rhombus and a Parallelogram Similar?
- Both a rhombus and a parallelogram have opposite sides that are equal in length.
- The diagonals of a rhombus bisect each other at right angles, similar to a parallelogram.
- In both a rhombus and a parallelogram, opposite angles are equal.
- Both shapes are considered parallelograms, meaning their opposite sides are parallel.
How is a Rhombus Different from a Parallelogram?
- A rhombus has four equal sides. A parallelogram may have unequal side lengths.
- The diagonals of a rhombus bisect each other at right angles. In a parallelogram, the bisecting diagonals do not form right angles.
Facts about Rhombus and a Parallelogram
- A rhombus is a type of parallelogram with all four sides equal in length.
- Thus, all rhombuses are parallelograms, but not every parallelogram is a rhombus.
What are the Properties of a Rhombus?
After comparing to parallelogram and square, we can summarize the properties of a rhombus as follows:
Equal Side Lengths: A rhombus has all four sides of equal length, creating a uniform and symmetrical shape.
Opposite Angles Equality: Opposite angles in a rhombus are equal.
Diagonal Bisecting: The diagonals of a rhombus bisect each other at right angles, splitting it into four congruent right-angled triangles.
Opposite Sides Parallel: Opposite sides of a rhombus are parallel.
Supplementary Adjacent Angles: Adjacent angles in a rhombus are supplementary.
Area of a Rhombus When its Base and Altitude are Known
To find the area of a rhombus when you know the length of its base and the corresponding altitude (height), you simply multiply these two values. In mathematical terms, the area `(A)` is equal to the product of the base `(b)` and height `(h)`:
`\color{#38761d}A = \color{#6F2DBD}b \times \color{#fb8500}h`
This formula expresses the area in square units, making it easy to visualize the space within the rhombus.
Example `1`: A rhombus has a base length of `10` units and an altitude (height) of `12` units. Determine the area of the rhombus.
Solution:
Area of a rhombus `=` base `×` height
Plugging in the values, where the base `(b)` is `10` units and the height `(h)` is `12` units:
` \color{#38761d}A = \color{#6F2DBD}(10) \ \color{#6F2DBD}\text{units} \times \color{#fb8500}(12) \ \color{#fb8500}\text{units} `
\( \color{#38761d}A = 120 \, \text{square units} \)
Therefore, the area of the rhombus is \( 120 \, \text{square units} \).
Area of a Rhombus When its Diagonals are Known
Alternatively if the lengths of the diagonals of a rhombus are known, you can determine the area using a different formula. The area `(A)` in this case is found by taking half of the product of the two diagonals `(d_1` and `d_2)`:
`\color{#38761d}A = 1/2 × (\color{#6F2DBD}(d_1) × \color{#6F2DBD}(d_2))`
Understanding these two methods allows for flexibility in calculating the area of a rhombus, depending on the available information. Whether provided with the base and altitude or the lengths of diagonals, these formulas provide a way to determine the area of this geometric shape.
Example `1`: A rhombus has diagonals measuring `24` units and `16` units. Determine the area of the rhombus.
Solution:
`\color{#38761d}A = 1/2 × (\color{#6F2DBD}(d_1) × \color{#6F2DBD}(d_2))`
Plugging in the values, where `d_1` is `\color{#6F2DBD}(24)` units and `d_2` is `\color{#6F2DBD}(16)` units:
`\color{#38761d}A = \frac{1}{2} \times \color{#6F2DBD}(24) \ \color{#6F2DBD}\text{units} \times \color{#6F2DBD}(16) \ \color{#6F2DBD}\text{units}`
\(\color{#38761d}A = 192 \, \text{square units} \)
Therefore, the area of the rhombus is \( 192 \, \text{square units} \).
The Perimeter of a Rhombus
Calculating the perimeter of a rhombus involves adding the lengths of all four sides.
Let's denote the length of one side as `s`.
Since all sides are equal, the perimeter `(P)` can be expressed as the sum of these four sides:
`\color{#38761d}P = \color{#fb8500}s + \color{#fb8500}s + \color{#fb8500}s + \color{#fb8500}s = 4\color{#fb8500}s`
In simpler terms, you can find the perimeter by multiplying the length of one side by `4`.
Example `1`: The sides of a rhombus measure `15` meters each. Determine the perimeter of the rhombus.
Solution:
Perimeter of the rhombus `\color{#38761d}P = 4\color{#fb8500}s`, \( s \) being the length of each side.
Plugging in `\color{#fb8500}(15)` for `\color{#fb8500}(s)`:
`\color{#38761d}P = 4 \times \color{#fb8500}(15)`
\( \color{#38761d}P = 60 \)
Therefore, the perimeter of the rhombus is `60` meters.
Practice Problems
Q`1`. A rhombus has a base of `10` centimeters, and its altitude measures `8` centimeters. Calculate the area of the rhombus.
- `40\ \text{cm}^2`
- `60\ \text{cm}^2`
- `80\ \text{cm}^2`
- `100\ \text{cm}^2`
Answer: c
Q`2`. Lisa has a rhombus-shaped garden with a base measuring `18` feet, and the altitude is `14` feet. What is the area of her garden?
- `126` sq. feet
- `252` sq. feet
- `504` sq. feet
- `756` sq. feet
Answer: b
Q`3`. The diagonals of a rhombus measure `10` cm and `24` cm. Calculate the area of the rhombus.
- `120\ \text{cm}^2`
- `240\ \text{cm}^2`
- `480\ \text{cm}^2`
- `600\ \text{cm}^2`
Answer: a
Q`4`. A garden is designed in the shape of a rhombus with its diagonals measuring `16` meters and `30` meters. The owner wants to know the area of the garden. What is the approximate area of the garden in square meters?
- `240\ \text{m}^2`
- `360\ \text{m}^2`
- `480\ \text{m}^2`
- `720\ \text{m}^2`
Answer: a
Q`5`. If one side of a rhombus measures `14` meters, what is the perimeter of the rhombus?
- `28` meters
- `42` meters
- `56` meters
- `70` meters
Answer: c
Frequently Asked Questions
Q`1`. What is a rhombus?
Answer: A rhombus is a type of parallelogram that has all four sides of equal length.
Q`2`. How is the perimeter of a rhombus calculated?
Answer: The perimeter of a rhombus is found by adding up the lengths of all four sides, as expressed by the formula \( P = 4s \), where \( s \) represents the length of one side.
Q`3`. Can a rhombus have right angles?
Answer: Yes, it is possible for a rhombus to form right angles, making it a special case known as a square.
Q`4`. What is the relationship between the diagonals of a rhombus?
Answer: The diagonals of a rhombus are perpendicular bisectors of each other. This means they intersect at a right angle, and each diagonal divides the other into two equal parts.
Q`5`. How do you find the area of a rhombus when the diagonals are known?
Answer: The area of a rhombus can be determined using the formula \( A = \frac{d_1 \times d_2}{2} \), where \( d_1 \) and \( d_2 \) represent the lengths of the two diagonals.