Parallelogram - Definition, Formulas, Properties and Examples

Edited By Team Careers360 | Updated on Jul 02, 2025 05:13 PM IST

A quadrilateral with two opposed sides that are parallel and equal in length is referred to as a parallelogram. Hence, a parallelogram has four equal-sized angles, making it a parallelogram.

With a parallelogram, an angle's measurement can be calculated using the following formula: (4 - 2)/2 = 1. According to this equation, the total interior angle of a parallelogram is (4 – 2) 180 degrees, or 360 degrees. We can determine the size of each angle by dividing the total number of degrees by the number of angles in a parallelogram, which is 4. In this instance, each angle of a parallelogram is 90 degrees in length.

This Story also Contains

Parallelogram Definition
Angles of a parallelogram
Properties of angles of a parallelogram
Theorems Concerning a Parallelogram's Angles
Formulas of a parallelogram
Facts

It's significant to remember that the length of a parallelogram's sides has no bearing on the measure of an angle within it. This is due to the fact that the amount of rotation necessary to transfer one side of an angle to the other determines the angle's measurement.

In a parallelogram, an angle is defined as being 90 degrees in length. This idea is crucial to comprehend in geometry and has numerous practical uses in fields like building and design.

Parallelogram Definition

A quadrilateral with two sets of parallel sides is referred to as a parallelogram. In a parallelogram, the opposing sides are of equal length, and the opposing angles are of equal size. Additionally, the interior angles are additional to the transversal on the same side. The Sum of all the inside angles equals 360 degrees.

A parallelepiped is a three-dimensional shape with parallelogram-shaped faces. The is determined by its base (one of its parallel sides) and height (the distance between its top and bottom). A parallelogram's perimeter is determined by the lengths of its four sides.

The shapes of a square and rectangles share the qualities of a parallelogram.

Angles of a parallelogram

is a two-dimensional flat shape with four angles. The internal angles on either side are equal. The transversal angles on the same side are supplementary, which means they sum to 180 degrees. As a result, a parallelogram's internal angles add up to 360 degrees.

Properties of angles of a parallelogram

A quadrilateral with equal and parallel opposite sides is referred to as a parallelogram. A parallelogram stands apart from other quadrilaterals due to a few unique characteristics. Look at the parallelogram below and compare it to the attributes listed below:

the parallelogram's angles

A parallelogram's opposing angles are congruent (equal). In this case, A Equals C, and D = B.

A parallelogram has angles that sum to 360°. Here, A + B + C + D = 360 degrees.

Each consecutive angle is additional. In this case, A + B = 180°, B + C = 180°, C + D = 180°, and A + D = 180°.

Theorems Concerning a Parallelogram's Angles

Theorems of a parallelogram's angles can be used to help address parallelogram-related issues. The following two significant theorems are listed:

The opposite angles of a parallelogram are equal.
Consecutive angles of a parallelogram are supplementary.

Formulas of a parallelogram

Area= Length * breadth

Perimeter= 2 (length + breadth)

Facts

Number of sides = 4
Number of vertices = 4
Mutually Parallel sides = 2 (in pair)
Area = Base x Height
Perimeter = 2 (Sum of adjacent sides length)
Type of polygon = Quadrilateral
The opposite sides are parallel and equal
The opposite angles are equal
The consecutive or adjacent angles are supplementary
If any one of the angles is a right angle, then all the other angles will be at right angle
The two diagonals bisect each other
Each diagonal bisects the parallelogram into two congruent triangles
The Sum of the squares of all the sides of a parallelogram is equal to the sum of the squares of its diagonals. It is also called parallelogram law

Frequently Asked Questions (FAQs)

1. Are the 4 angles of a parallelogram equal?

The total of all the angles of a parallelogram, like all other quadrilaterals, is 360°. A parallelogram has neighboring or contiguous angles that sum to 180 degrees. As a result, A + B, B + C, C + D, and D + A all equal 180 degrees.

2. How to find an angle of a parallelogram?

S = (n 2) 180° is a formula that can be used to compute this, where n is the number of sides in the polygon. Here, 'n' Equals 4. The interior angle sum of a parallelogram is therefore S = (4 2) 180° = (4 2) 180° = 2 180° = 360°.

3. Are the angles of a parallelogram 90?

A quadrilateral is referred to as a parallelogram if two of its sides are parallel to one another and all four of its vertices have angles other than 90 degrees or right angles.

4. What are the angles of all 4 angles of a parallelogram?

180, A quadrilateral is a polygon that has 4 vertices and 4 sides enclosing 4 angles of each 180 degree,s and the sum of all the angles is 360°

5. Do the angles of a parallelogram add up to 360?

The flat, two-dimensional shape of a parallelogram contains four angles. Angles inside the opposite corners are equal. As they sum up to 180 degrees, the angles on the same side of the transversal are supplementary. The total internal angle of a parallelogram is therefore 360 degrees.

6. How does the concept of a parallelogram relate to the idea of shear stress in physics?

In physics, shear stress is often represented using a parallelogram. The deformation of a material under shear stress can be visualized as a rectangle transforming into a parallelogram. The angle of this parallelogram from its original rectangular shape represents the shear strain.

7. How can the concept of a parallelogram be used to understand the Parallelogram of Forces in physics?

The Parallelogram of Forces is a method for adding two forces vectorially. The two forces are represented as adjacent sides of a parallelogram, and the resultant force is represented by the diagonal from the point of application. This method visually demonstrates vector addition and is crucial in understanding force composition and resolution.

8. How is a parallelogram different from a rectangle?

While both are quadrilaterals with parallel opposite sides, a rectangle has four right angles (90 degrees each), whereas a parallelogram doesn't necessarily have right angles. In a parallelogram, opposite angles are equal, but they can be any measure.

9. What are the key properties of a parallelogram?

The key properties of a parallelogram are:

10. Why do opposite sides of a parallelogram have to be equal?

Opposite sides of a parallelogram are equal because they are parallel. When you have two parallel lines cut by two transversals (the other pair of sides), corresponding parts of the parallel lines are congruent, including the segments between the transversals.

11. What's the relationship between the diagonals of a parallelogram?

In a parallelogram, the diagonals bisect each other. This means they intersect at their midpoints, dividing each diagonal into two equal segments.

12. Can a parallelogram have right angles?

Yes, a parallelogram can have right angles. When all four angles in a parallelogram are right angles (90°), it becomes a special type of parallelogram called a rectangle.

13. What is the formula for the area of a parallelogram?

The area of a parallelogram is calculated by multiplying the base (b) by the height (h): Area = b × h. The height must be perpendicular to the base, not the slanted side.

14. Why isn't the slanted side used as the height in the area formula?

The slanted side isn't used as the height because it's not perpendicular to the base. The area formula requires the perpendicular distance between parallel sides to accurately calculate the space within the parallelogram.

15. How do you find the perimeter of a parallelogram?

The perimeter of a parallelogram is the sum of all its sides. Since opposite sides are equal, you can use the formula: Perimeter = 2(a + b), where a and b are the lengths of two adjacent sides.

16. How does changing the height of a parallelogram affect its area?

Changing the height of a parallelogram directly affects its area. If you double the height while keeping the base constant, the area doubles. If you halve the height, the area halves. This is because the area formula is A = b × h, where h is the height.

17. How can you construct a parallelogram given two sides and the included angle?

To construct a parallelogram with two sides and the included angle:

18. What is a parallelogram?

A parallelogram is a quadrilateral (four-sided shape) with two pairs of parallel sides. This means that opposite sides are equal in length and remain the same distance apart along their entire length.

19. What's the difference between a parallelogram and a rhombus?

While both are quadrilaterals with parallel opposite sides, a rhombus is a special type of parallelogram where all four sides are equal in length. In a general parallelogram, only opposite sides are equal.

20. Why does the midsegment of a trapezoid form a parallelogram?

The midsegment of a trapezoid (a line segment connecting the midpoints of the non-parallel sides) forms a parallelogram because it's parallel to both bases of the trapezoid and half the length of their sum. This creates two pairs of parallel sides, defining a parallelogram.

21. What's the relationship between the area of a parallelogram and the area of a rectangle with the same base and height?

A parallelogram and a rectangle with the same base and height will have equal areas. This is because you can "cut off" the triangle from one end of the parallelogram and "paste" it onto the other end to form a rectangle without changing the area.

22. What's special about the diagonals of a rectangle compared to those of a general parallelogram?

While the diagonals of all parallelograms bisect each other, the diagonals of a rectangle have an additional property: they are equal in length. This is not necessarily true for other types of parallelograms.

23. Can a parallelogram have an obtuse angle?

Yes, a parallelogram can have obtuse angles. In fact, if one angle in a parallelogram is obtuse (greater than 90°), the adjacent angle must be acute (less than 90°), and the two obtuse angles will be opposite each other.

24. How does shearing a rectangle relate to creating a parallelogram?

Shearing a rectangle is a transformation that creates a parallelogram. It involves keeping one side fixed while sliding the opposite side parallel to itself. This process maintains the area and the length of parallel sides but changes the angles, turning right angles into pairs of acute and obtuse angles.

25. What's the relationship between the angles of a parallelogram and circular motion?

The angles in a parallelogram relate to circular motion in that they represent phase differences in simple harmonic motion. If you plot two perpendicular simple harmonic motions with a phase difference, the resulting path is an ellipse, which is a special case of a parallelogram with curved sides.

26. How can you use coordinate geometry to prove that a quadrilateral is a parallelogram?

In coordinate geometry, you can prove a quadrilateral is a parallelogram by showing that:

27. What's the significance of the diagonals in determining whether a parallelogram is a rhombus?

In a rhombus, the diagonals are perpendicular to each other (they intersect at right angles). This property, along with the fact that diagonals bisect each other (true for all parallelograms), can be used to determine if a parallelogram is specifically a rhombus.

28. How are the angles in a parallelogram related?

In a parallelogram:

29. Can a parallelogram have two right angles?

Yes, a parallelogram can have two right angles, but if it does, it must have four right angles and therefore be a rectangle. This is because opposite angles in a parallelogram are equal, so if one angle is 90°, its opposite angle must also be 90°.

30. How can you prove that a quadrilateral is a parallelogram?

There are several ways to prove a quadrilateral is a parallelogram:

31. How does the angle between diagonals relate to the shape of a parallelogram?

The angle between the diagonals of a parallelogram provides information about its shape:

32. Why is a square considered both a rhombus and a rectangle, but not all parallelograms are squares?

A square is a special case that satisfies the definitions of both a rhombus (all sides equal) and a rectangle (all angles 90°). However, not all parallelograms have these strict conditions. Parallelograms only require opposite sides to be parallel and equal, allowing for more variation in side lengths and angles.

33. How do you find the height of a parallelogram if you know its area and base?

If you know the area (A) and base (b) of a parallelogram, you can find its height (h) by rearranging the area formula: A = b × h. Solving for h gives you: h = A ÷ b.

34. How does the area of a parallelogram change if you double both its base and height?

If you double both the base and height of a parallelogram, its area will increase by a factor of 4. This is because A = b × h, so if both b and h are doubled, the new area is (2b) × (2h) = 4bh, which is 4 times the original area.

35. Can a parallelogram have three equal sides?

No, a parallelogram cannot have exactly three equal sides. In a parallelogram, opposite sides are always equal. If three sides were equal, the fourth would also have to be equal to its opposite side, making all four sides equal. This would make it a rhombus, a special case of a parallelogram.

36. What's the relationship between a parallelogram's area and the product of its diagonals?

The area of a parallelogram is equal to half the product of its diagonals multiplied by the sine of the angle between them. This can be expressed as: A = (1/2)d₁d₂sin θ, where d₁ and d₂ are the lengths of the diagonals and θ is the angle between them.

37. How can the Pythagorean theorem be applied to a parallelogram?

The Pythagorean theorem can be applied to a parallelogram in several ways:

38. What's the relationship between a parallelogram's area and the area of the triangle formed by two adjacent sides?

The area of a parallelogram is always twice the area of the triangle formed by two adjacent sides. This is because the parallelogram can be divided into two congruent triangles by either of its diagonals, each having the same base and height as the parallelogram.

39. How does the concept of vectors relate to parallelograms?

In vector mathematics, a parallelogram is formed when two vectors are added. The sides of the parallelogram represent the magnitudes and directions of the vectors, and the diagonal from the origin to the opposite corner represents their sum. This is known as the parallelogram law of vector addition.

40. How does the area of a parallelogram change if you increase one side by 10% and decrease the corresponding height by 10%?

If you increase one side (base) by 10% and decrease the corresponding height by 10%, the area of the parallelogram remains unchanged. This is because:

41. What's the relationship between the area of a parallelogram and the area of a triangle with the same base and height?

The area of a parallelogram is always twice the area of a triangle with the same base and height. This is because a diagonal of the parallelogram divides it into two congruent triangles, each having the same base and height as the parallelogram.

42. How does the concept of symmetry apply to parallelograms?

Parallelograms have rotational symmetry of order 2, meaning they can be rotated 180° and appear the same. They also have two lines of symmetry through the midpoints of opposite sides, but only if they are rectangles or rhombi. General parallelograms do not have line symmetry.

43. Can the diagonals of a parallelogram ever be perpendicular without the parallelogram being a rhombus?

No, if the diagonals of a parallelogram are perpendicular, it must be a rhombus. Perpendicular diagonals in a parallelogram imply that all four sides are equal, which is the definition of a rhombus. The converse is also true: all rhombi have perpendicular diagonals.

44. How does the concept of parallelograms extend to three-dimensional space?

In three-dimensional space, the concept of a parallelogram extends to a parallelepiped. This is a six-faced polyhedron where each face is a parallelogram and opposite faces are parallel. Just as parallelograms have parallel sides, parallelepipeds have parallel faces.

45. What's the relationship between the area of a parallelogram and the cross product of two adjacent sides treated as vectors?

The area of a parallelogram can be calculated using the cross product of two adjacent sides treated as vectors. If a and b are vectors representing two adjacent sides, then the area of the parallelogram is equal to the magnitude of their cross product: Area = |a × b|.

46. How can the concept of a parallelogram be used to understand the distributive property in algebra?

The distributive property in algebra (a(b+c) = ab + ac) can be visualized using a parallelogram. If you draw a parallelogram with sides of length a and (b+c), its area can be calculated either as a(b+c) or as the sum of two smaller parallelograms with areas ab and ac.

47. How does the concept of a parallelogram relate to the idea of linear independence in linear algebra?

In linear algebra, two vectors are linearly independent if they don't lie on the same line. These vectors form a parallelogram, and the area of this parallelogram is non-zero only if the vectors are linearly independent. If the vectors are linearly dependent, the parallelogram collapses to a line with zero area.

48. Can a parallelogram have an angle larger than 180°?

No, a parallelogram cannot have an angle larger than 180°. The sum of the measures of the four angles in a parallelogram is always 360°, and opposite angles are equal. If one angle were larger than 180°, the sum would exceed 360°, which is impossible in a parallelogram.

49. What's the relationship between the area of a parallelogram and the determinant of a 2x2 matrix?

The area of a parallelogram formed by two vectors (a,b) and (c,d) is equal to the absolute value of the determinant of the 2x2 matrix formed by these vectors:

50. What's the relationship between the diagonals of a parallelogram and its area?

The area of a parallelogram can be calculated using its diagonals. If d₁ and d₂ are the lengths of the diagonals and θ is the angle between them, the area is given by:

51. How does the concept of a parallelogram relate to the idea of linear transformations in linear algebra?

In linear algebra, a 2x2 matrix represents a linear transformation of the plane. When this transformation is applied to a unit square, it generally results in a parallelogram. The area of this parallelogram is equal to the absolute value of the determinant of the transformation matrix, illust