In this post we will discuss area of parallelogram questions. All the questions are solved step by step so that you can understand the concept in full details.
Let us understand the properties of parallelogram so that you can solve the questions easily
Properties of Parallelogram
Parallelogram is a quadrilateral with four sides where the opposite sides are parallel and equal.
1. In parallelogram opposite sides are equal
From the above figure we can say that AB = CD and AD = CB
2. Opposite angles are also equal
Angle A = Angle C
Angle D = Angle B
3. Consecutive angles of parallelogram are 180
Angle A + Angle B = 180
Angle B + Angle C = 180
Angle C + Angle D = 180
Angle D + Angle A = 180
4. The diagonals of Parallelogram bisect each other into equal length
These are some of the important properties of parallelogram. Let us study some formulas which are helpful to solve aptitude questions of parallelogram
Area of Parallelogram
Area of Parallelogram when base and height is given
When in the question, the base and height of the parallelogram is given, then you can easily calculate its area with the help of below formula
Area of Parallelogram = 1/2 * Base * Height
Area of Parallelogram with the side length and angle is given
If the question provides you with the data of the length of two sides of parallelogram and the angle between the sides, then you can calculate the formula using following formula
Area of Parallelogram = a * b sin (z)
Where a & b are the sides and z is angle between the sides
When the Length of diagonals and the bisecting angle is given
If you are provided with the length of the sides of the diagonals and the angle between them, then you can easily calculate the area of parallelogram using below formula
Area of Parallelogram = 1/2 * d1 * d2 sin(z)
Where d1 and d2 are the length of diagonals
and z is the angle between the diagonals
Perimeter of Parallelogram
Perimeter is nothing but just the sum of all sides
So formula for perimeter of parallelogram can be written as:
Perimeter = 2 ( a + b)
where a & b are the sides of parallelogram
Calculate Area of Parallelogram
Area of parallelogram = Base * height
Here base = 18 cm
And height = 8 cm
Using the above formula, we get
Area of Parallelogram = 18 * 8 = 144 sq cm
Hence 144 sq cm is the required area of parallelogram
ABCD is the parallelogram.
You can observe from the above figure that:
Area of Parallelogram= 2 * Area of Triangle
Lets take triangle ABD
Now calculate the area of triangle ABD using Herons Formula= \sqrt { S(S-a)(S-b)(S-c) }
Let us first calculate the value of S
we know that S =( a + b + c)/2
Here, a =30 cm, b = 14 cm , c= 40 cm
so, S => (30 + 14 + 40)/ 2
S => 84/2 => 42 cm
Now putting all the values in in Herons Formula
Area of Scalene Triangle ABD =
\sqrt { 42(42-30)(42-14)(42-40) } \\\ \\ \sqrt { 42\quad *\quad 12\quad *\quad 28\quad *\quad 2 } \\\ \\ Splitting\quad into\quad Multiples\quad to\quad get\quad fast\quad solution\\ \quad \\ \sqrt { (7*6)\quad *\quad (6*2)\quad *\quad (7*4)\quad *\quad 2 } \\After solving the above equation we will get
Area of Triangle ABD ==> 7 * 6 * 4
Area of Triangle ABD ==> 168 sq cm
Now the area of parallelogram = 2* area of triangle ==> 2 * 168 -=> 336 sq cm
Hence the area of parallelogram of 336 sq cm
(03) One diagonal of parallelogram is 70 cm and the perpendicular distance of this diagonal from either of the outlying vertices is 27 cm. The area of parallelogram is
Area of Parallelogram ABCD =Area of triangle ADB + Area of Triangle BDC
==> (1/2 * b * h) + (1/2 * b * h)
==> (1/2 * DB * AP) + (1/2 * DB * CQ)
==> (1/2 * 70 * 27) + (1/2 * 70 * 27) sq cm
==> ( 35 * 27) + (35 * 27)
==> 945 + 945
==> 1890 sq cm
Hence the required area of parallelogram is 1890 sq cm
Let the common base of triangle and parallelogram is b m
Altitude of Triangle = h1 m
Altitude of parallelogram = h2 m
Now, Area of Triangle = 1/2 * b * h1
Area of Parallelogram = b * h2
Its given that
Area of Triangle = Area of parallelogram
==> 1/2 * b * h1 = b * h2
==> h1 = 2 * h2
we know from question that h2 = 100
==> h1 = 2 * 100
==> h1 = 200
hence the altitude of triangle is 100 meter