How to Solve Rhomboid Problems: Step-by-Step Guide
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 What is a Rhomboid? Definition, Properties & Formulas
1. What is a rhomboid in simple terms?
A rhomboid is a type of quadrilateral, specifically a parallelogram, where adjacent sides have unequal lengths and its angles are not right angles (90°). Think of it as a 'slanted rectangle' that has been pushed over, with two pairs of equal-length sides.
2. What are the main properties of a rhomboid?
As a type of parallelogram, a rhomboid has several key properties that are important to remember for solving geometry problems:
Opposite sides are equal in length and parallel to each other.
Opposite angles are equal in measure.
Adjacent angles are supplementary, meaning they add up to 180°.
The two diagonals bisect each other, meaning they intersect at their midpoints.
The sum of all its interior angles is always 360°.
3. How is a rhomboid different from a rhombus?
The primary difference between a rhomboid and a rhombus lies in their side lengths. In a rhombus, all four sides must be equal in length. In a rhomboid, only the opposite sides are equal, while adjacent sides have different lengths. Therefore, every rhombus is a parallelogram, but a rhomboid is a more general parallelogram that isn't a rhombus.
4. What is the formula to calculate the area of a rhomboid?
The area of a rhomboid is calculated using the same formula as any parallelogram: Area = base × height. It's crucial to use the perpendicular height from the base to the opposite side, not the length of the slanted adjacent side.
5. How do you find the perimeter of a rhomboid?
The perimeter of a rhomboid is the total length of its boundary. Since opposite sides are equal, if you know the lengths of two adjacent sides (let's call them 'a' and 'b'), the formula is: Perimeter = 2 × (a + b).
6. If a rhomboid is a type of parallelogram, why does it need a specific name?
While all rhomboids are parallelograms, not all parallelograms are rhomboids. The term 'rhomboid' is useful for classifying a parallelogram that is specifically neither a rhombus (all sides equal) nor a rectangle (all angles 90°). It helps distinguish this general 'tilted' quadrilateral from its more symmetrical relatives like squares, rectangles, and rhombuses.
7. Can a rhomboid have a right angle?
No, a rhomboid cannot have a right angle. By definition, a rhomboid is a parallelogram with oblique (non-right) angles. If a parallelogram were to have just one right angle, its geometric properties would force all four of its angles to be right angles, making it a rectangle, not a rhomboid.
8. How are the diagonals of a rhomboid different from those of a square?
The diagonals of a rhomboid and a square have very different properties, which is a key concept in quadrilateral geometry.
In a rhomboid, the diagonals are of unequal length and they bisect each other, but not at a 90° angle.
In a square, the diagonals are of equal length, they bisect each other, and they are perpendicular, meaning they intersect at a perfect 90° angle.
