What is a Parallelogram
By the end of this lesson, you will be able to
- define parallelogram.
- calculate the perimeter of parallelogram.
- evaluate the area of parallelogram.
- derive the formula for diagonal of parallelogram.
- interpret properties of parallelogram.
- demonstrate examples of parallelogram.
Parallelogram
A convex quadrilateral with two pairs of parallel sides is a parallelogram.
A parallelogram is a type of quadrilateral.
Two parallel opposite sides for a parallelogram are of equal length.
The opposite angles are equal to each other.
The sum of two adjacent angles of a parallelogram adds up to 180°. Two diagonals of a parallelogram are different in length.
Apps for calculating area and perimeter
Perimeter:
Area:
Select button and then drag the pink area of bottom right in figure and see the change.
Parallelogram formula
The parallelogram formulas that are used generally given below:
Perimeter of Parallelogram
The perimeter is the sum of lengths of all sides of a parallelogram.
Perimeter of a parallelogram can be evaluated depending on a pair of adjacent sides.
Perimeter of Parallelogram Formula
Let a and b be the adjacent of a parallelogram.
The opposite sides of a parallelogram are of equal lengths.
If the perimeter of parallelogram is P,
P = (a + b + a + b) unit
or, P = (2a+2b) unit
∴ P = 2(a+b) unit
Area of Parallelogram
Base and height of parallelogram
There are severals ways to finding out the area of a parallelogram depending upon the given conditions.
The area of a parallelogram is the base times the height.
Area of Parallelogram Formula
Let b and h be the base and height respectively.
Area of parallelogram = (base × height) square unit i.e.
Area of parallelogram = (b × h) square unit
∴ Area of Parallelogram = bh
Base and height of para are shown in figure.
Now, One can have a question,why the area of a parallelogram is the base times height; and how? Let us derive the formula.
Base, height and diagonal of parallelogram is in figure.
Step 1: Draw the parallelogram ABCD.
Step 2: Draw a perpendicular AP from vertex A to the base BC i.e. AP⊥BC where AP = h and BC = b.
Step 3: Draw the diagonal AC.
Step 4: Diagonal of a parallelogram divides it two congruent triangles.
∴ ꕔABC ≅ ꕔACD.
Step 5: ▱ABCD = △ABC + △ACD
or, ▱ABCD = △ABC + △ABC [∵ △ABC ≅ △ACD]
or, ▱ABCD = 2.△ABC
or, ▱ABCD = 2× 12 BC.AP
or, ▱ABCD = BC.AP
∴ ▱ABCD = bh
∴ Area of parallelogram = (base × height) square unit (proved).
Parallelogram Example
Diagonal of Parallelogram Formula
Diagonal of a parallelogram means the length of a line segment connecting to opposite vertices.
Let a and b be the two adjacent sides and their inclined angle be θ.
Hence, the diagonal, d = √(a² + b²- 2ab cosθ).
Properties of Parallelogram
- The four interior angles of a parallelogram add up to 3600.
- The adjacent angles are supplementary.
- Two diagonals of a parallelogram bisect to each other.
- One diagonal of a parallelogram divide it into two triangles of equal area.
- Both diagonals lie inside the parallelogram.
- The opposite sides are parallel.
- The opposite angles of parallelogram are equal to each other.
- The two triangles created by each of its diagonals are congruent.
- The opposite sides of a parallelogram are of equal lenghts.
- The sum of area of two squares formed by its two diagonals is equal to the sum of area of four squares formed by its four sides.
