Area of Parallelograms
The part of the plane enclosed by a simple closed figure is called a planar region corresponding to that figure. The magnitude or measure of this planar region is called its area.
If two figures are congruent, then they have equal areas. However, the converse of this statement is not true.
If R1 and R2 are two regions such that both are congruent to each other than their areas are also equal. This is called congruence area axiom.
If a planar region formed by a figure T is made up of two non-overlapping planer regions formed by figures P and Q, then
area(T) = area(P) + area(Q).
Height or altitude of a parallelogram is the perpendicular distance from base to the opposite side.
Two figures are said to be on the same base and between the same parallels, if they have a common base and the vertex opposite to the common base of each figure lies on a line parallel to the base.
Area of a parallelogram is the product of its base and the corresponding altitude.
Parallelograms on the same base and between the same parallels are equal in area.
A Parallelogram and a rectangle on the same base and between the same parallels are equal in area.
Parallelograms on equal bases and between the same parallels are equal in area.
Parallelograms on the same base and having equal areas lie between the same parallels.
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The adjacent sides of a parallelogram are 48 cm and 36 cm. If the distance between shorter sides is 16 cm. Then the distance between the longer sides isMarks:1
Consider AD as base and BL as corresponding height.
Area of parallelogram ABCD = base × height
= AD × BL
= 36 × 16
= 576 cm2
Area remains the same taking AB as base and DM as its corresponding height.
Area = AB × DM
576 = 48 × DM
Therefore, we have
DM = 576/48 = 12 cm
PQRS is a parallelogram, whose one side is ‘a’ and the other side is ‘b’. If ABCD is a rectangle with the same sides ‘a’ and ‘b’, thenMarks:1
area (ABCD) ≥ area (PQRS)
Area of parallelogram = h b
Area of rectangle = a b
As we see from the figure that
⟹ h b a b
Therefore, we see that the area of rectangle is greater than the area of parallelogram.
If a parallelogram with area P, a rectangle with area R and a triangle with area T, all are constructed on the same base and all have the same altitude, then the true statement isMarks:1
P = R.
Area of ABCD = R
Area of ABXY = P
Area of △ AYB = T = area of △ ABC
= 1/2 area of ABCD
= 1/2 R
Also, area ABXY = 2 area △ AYB
⟹ P = 2T = R.
In the following parallelogram ABCD, X and Y are the mid-points of BC and AD respectively. If the area of the CDYX = 36 cm2, then the area of parallelogram ABCD is equal to
It is given that X and Y are the mid-points of BC and AD.
Therefore, area of CDYX = 1/2 area of ABCD
Hence, area of ABCD = 2 area of CDYX
= 2 36
= 72 cm2
A rectangle and a parallelogram have equal areas. If the sides of the rectangle are 10 m and 12 m and the base of the parallelogram is 20 m, then the altitude of the parallelogram isMarks:1
Area of rectangle = 10 x 12 = 120 m2
If h is the altitude of the parallelogram, then the area of parallelogram = 20 x h
20 x h = 120
or h = 6 m