Rhomboid Definition Properties Formula and Solved Examples
What are Plane Figures and Their Properties?
A plane figure is a closed, two-dimensional flat figure. It has only length and breadth but no thickness at all. Since it has two measurements only, it is two-dimensional. Plain figures only have vertices and sides, and owing to their two-dimensional properties, only their perimeter and area can be measured. You cannot measure their volume, which is reserved for 3-dimensional figures.
Properties of a Rhomboid
The Opposite Sides of a Rhomboid are Parallel.
Let's take a figure, where ABCD as a rhomboid. So hence, AB is parallel to DC, and AD is parallel to DC.
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The Opposite Sides of the Rhombus are Congruent as Well.
In the figure below, quadrilateral DABC is a rhomboid. In the rhomboid, DA = CB and DC = AB. So, opposite sides of the rhomboid are equal.
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The Diagonals of the Rhomboid Divide the Figure Into Two Congruent Triangles.
This is a property true of any parallelogram. In the figure below, ABCD is a rhomboid, with AC and BD as its diagonals meeting at point O. Hence Triangle ABC is congruent to Triangle ADC by SSS congruence test.
To prove it:
AB= DC (opposite sides of a rhomboid)
AD=BC (opposite sides of a rhomboid)
AC=AC (common side)
Hence proved.
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4. The Opposite Angles of a Rhomboid are Equal.
In the figure below, quadrilateral ADCB is a rhomboid. Here ∠ADC= ∠ABC and ∠DAB= ∠DCB. This is because the opposite angles of a rhomboid are always equal.
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5. The Sum of the Angles in a Rhomboid is Equal to 360 Degrees.
Owing to the angle sum property of a quadrilateral, the sum of all the angles in a rhomboid is equal to 360 degrees. In the figure below, ADCB is a rhomboid which is a quadrilateral. Hence the sum of all the angles ∠A+∠D+∠C+∠D= 360 degrees. A quadrilateral is a closed, 2-D shape which has 4 sides.
This can be verified because the Rhomboid consists of two congruent triangles whose interior angles will add up to 180 degrees each (since all interior angles of a triangle must add up to 180 degrees). Hence when you add the sum of the interior angles of both the triangles forming the quadrilateral (180+180), you can prove that the total measure is 360 degrees.
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Important formulas of a Rhomboid
1) Area of a rhomboid
Being a 2D figure, the area of the Rhomboid stands for the size of an area, i.e., space covered by the Rhomboid. To calculate the area of a Rhomboid, we first find the length of its base side and height(perpendicular).
We know that the diagonal divides the Rhomboid into two congruent triangles.
Hence the area of a rhomboid would be = 2 x [½ x base x height] (area of a triangle is: ½ x base x height)
I.e
Area of Rhomboid= base x height.
The area is expressed in unit square, eg, m2.
Solved Examples
1. Calculate the area of a rhomboid ABCD when the base is 7 cm, and the perpendicular height is 5 cm.
Solution:
It is given that:
Base, b=7 cm
Height, h= 5 cm
Area of Rhomboid= base x height.
Hence,
A = b x h
= 7 x 5 cm2
= 35 cm2
Thus, the area of rhomboid ABCD is 35 cm2
2. Find the perpendicular height of a rhomboid ABCD, where the area is 50 cm2, and the measure of the base side is 10 cm.
Solution:
It is given that
Area, A= 50 cm2
Base, b= 10 cm
We know that
A = b x h
To find h
A= b x h
50= 10 x h
50/10 = h
h= 5 cm
Hence, the perpendicular height of the rhomboid ABCD is 5 cm.
2) The perimeter of a Rhomboid
Perimeter is basically the measure of the boundary of a figure. Thus the perimeter of the Rhomboid will be the addition of all the sides.
Let's assume a rhomboid to be ABCD
Hence the perimeter of the Rhomboid would be:
P= AB+BC+CD+DA
P= 2(AB+BC) ............ ( since opposite sides of a rhomboid are congruent)
Solved Examples
1. Calculate the perimeter of a rhomboid ABCD where:
AB= 5 cm
BC=4 cm
CD=5 cm
DA= 4 cm
Solution:
It is given that:
AB = CD = 5 cm
Hence, 2AB= 5 x 2 = 10 cm
BC = DA = 4cm
Hence, 2BC=4 x 2 = 8 cm
Perimeter, P = 2AB + 2BC
= 2(AB+BC)
= 2(5+4) cm
= 2(9) cm
= 18 cm.
Thus, the perimeter of the rhomboid ABCD is 18 cm.
2. If the perimeter of the Rhomboid is 60 cm, and the measure of one side is 20 cm, find the other side of the Rhomboid.
Solution:
Let ABCD be a rhomboid.
It is given that
Perimeter, P = 60 cm
Let, the one side be AB.
Hence, AB= 20 cm
We know that the opposite sides of a rhomboid are congruent.
Hence, AB = CD = 20 cm
To find BC and DA.
P= 2AB + 2BC
60= 2(20) + 2 BC
60= 40 + 2BC
20 = 2 BC
Hence, BC = 10 cm
Since BC= DA
DA will also be 10 cm.
So finally the measurements of the sides of the rhomboid ABCD are:
AB= CD = 20 cm
BC = DA = 10 cm
FAQs on Rhomboid in Geometry Complete Guide with Key Concepts
1. What is a rhomboid in geometry?
A rhomboid is a type of parallelogram in which opposite sides are equal and parallel, but the angles are not right angles and not all sides are equal. It has the following properties:
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum to 180°).
- Diagonals bisect each other but are not equal.
2. What is the difference between a rhombus and a rhomboid?
The main difference is that a rhombus has all sides equal, while a rhomboid has only opposite sides equal. Key differences include:
- Rhombus: All four sides equal; diagonals are perpendicular.
- Rhomboid: Only opposite sides equal; diagonals are not perpendicular.
- Both are types of parallelograms.
3. What is the formula for the area of a rhomboid?
The area of a rhomboid is given by the formula Area = base × height. Here:
- Base (b) is one side of the rhomboid.
- Height (h) is the perpendicular distance between the two parallel sides.
4. How do you find the perimeter of a rhomboid?
The perimeter of a rhomboid is calculated using the formula Perimeter = 2(a + b), where a and b are adjacent sides. Since opposite sides are equal:
- Add the lengths of two adjacent sides.
- Multiply the sum by 2.
5. Are the diagonals of a rhomboid equal?
The diagonals of a rhomboid are not equal, but they bisect each other. This means:
- Each diagonal cuts the other into two equal parts.
- The diagonals are generally of different lengths.
- They are not perpendicular (unless the rhomboid is a special case like a rhombus).
6. What are the angles of a rhomboid?
In a rhomboid, opposite angles are equal and adjacent angles add up to 180°. Specifically:
- If one angle is θ, the opposite angle is also θ.
- The adjacent angles are 180° − θ.
7. Is a rhomboid a parallelogram?
Yes, a rhomboid is a type of parallelogram because it has two pairs of parallel and equal opposite sides. It satisfies all parallelogram properties:
- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Diagonals bisect each other.
8. How do you calculate the height of a rhomboid?
The height of a rhomboid can be calculated using the formula Height = Area ÷ Base. Steps:
- Find the area of the rhomboid.
- Divide the area by the chosen base.
9. Can you give an example of a rhomboid with numbers?
An example of a rhomboid is one with sides 5 cm and 9 cm, and a height of 4 cm. Using formulas:
- Area = base × height = 9 × 4 = 36 cm².
- Perimeter = 2(a + b) = 2(5 + 9) = 28 cm.
10. What is the difference between a rectangle and a rhomboid?
The key difference is that a rectangle has four right angles, while a rhomboid does not. Comparison:
- Rectangle: All angles are 90°; diagonals are equal.
- Rhomboid: Angles are not 90°; diagonals are unequal.
- Both have opposite sides equal and parallel.
