Area of the rhombus:
Area is the surface measured inside the boundary of the rhombus.
Rhombus:
- Rhombus is a four sided figure whose opposite sides are parallel and its sides are equal in length
- Opposite angles are equal.
- Diagonals of the rhombus bisect each other at right angles
- Rhombus can also define as a parallelogram with all four sides is equal.
- Since the rhombus has four sides, perimeter of the rhombus is 4 x side units.
If the base and height of the rhobmus are given, then the area can be found using the formula
Area = (base x height) square units.
Calculation of area given the diagonals.
Consider the following rhombus ABCD. Let us calculate the area
Diagonal AC=d1
Diagonal BD=d2
Diagonals of the rhombus bisect each other at right angles.
Therefore OB=OD= $\frac{d_2}{2}$
Area of the rhombus = Area of triangle ABC+ Area of triangle ACD square units
= $\frac{1}{2}$ x d1 x $\frac{d_2}{2}$ +$\frac{1}{2}$ x d1 x $\frac{d_2}{2}$
=$\frac{d_1d_2}{4}+ \frac{d_1d_2}{4}$
=$\frac{2d_1d_2}{4}$
= $\frac{1}{2}$d1d2 square units
Problems:
- Find the area of rhombus given its diagonals is 9 cm and 15cm
Area of rhombus = $\frac{1}{2}$ x d1 x d2 Square units
= $\frac{1}{2}$ x 9 x 15
= 67.5 cm2
- Find the area of a rhombus with the given base 5 and height 6
Area of the rhombus = base x height square units
= 5 x 6 =30 square units
- The area of a rhombus is 120 square feet and one of its diagonal has a length of 40 feet. Find the other diagonal
Given: area =120 square feet, diagonal d1 =40 feet
Let the other diagonal be d2
Area of rhombus = $\frac{1}{2}$ x d1 x d2 Square units
$\frac{1}{2}$ x d1 x d2 = 120
$\frac{1}{2}$ x 40 x d2 = 120
20 x d2 =120 (since $\frac{40}{2}$ = 20 )
d2 = $\frac{120}{20}$
d2 = 6 feet
Area of Trapezium
Trapezium or Trapezoid:
- Trapezium is a four sided closed figure that has at least one pair of parallel sides.
- Every trapezium has 2 bases which are parallel
- Every trapezium has 2 legs which are non parallel
- Altitude of the trapezium is the perpendicular distance from one base to another
- If the trapezium has equal legs then it is called isosceles trapezium
Calculation of area of trapezium:
Drop a perpendicular from C and D to AB
We obtain two right angled triangles AMD, BNC and a rectangle MNCD. Combine the two triangles AMD and BNC.
Area of Trapezium= Area of triangle +Area of rectangle square units
=$\frac{1}{2}$ x (a-b) x h + (b x h)
= h ($\frac{1}{2}$ x (a-b) +b)
= h [ $\frac{1}{2}$ a - $\frac{1}{2}$ b + b]
= h [$\frac{a}{2}$ + b - $\frac{b}{2}$]
= h [$\frac{a}{2}$ + $\frac{2b}{2}$ -$\frac{b}{2}$]
= h [$\frac{a}{2}$+ $\frac{b}{2}$]
= $\frac{h}{2}$(a + b)
= $\frac{1}{2}$ x h x (a + b)
Area of the trapezium is half of the product of altitude and sum of the non parallel sides
Problems:
- Given a trapezium, calculate the area.
Solution:
From the diagram a = 10cm, b = 8cm, h = 5cm
Area of Trapezium= $\frac{1}{2}$ x h x (a + b) square units
= $\frac{1}{2}$ x 5 x (10 + 8)
= $\frac{1}{2}$ x 5 x 18
= 45 cm2
- Calculate the area of the trapezium
Given a = 6, b = 8, h = 10
Area of Trapezium= $\frac{1}{2}$ x h x (a + b) square units
= $\frac{1}{2}$ x 10 x (6 + 8)
= $\frac{1}{2}$ x 10 x 14
= 70 square units
- Find the unknown value of a trapezium given.
Area = 200 m2, height = 10 m, one base = 20 m, other base = x
Area of Trapezium= $\frac{1}{2}$ x h x (a + b) square units
$\frac{1}{2}$ x h x (a + b) = 200
$\frac{1}{2}$ x 10 x (20 + b) = 200
5 x (20 + b) = 200
(20 + b) = 100 ( divide both sides with 5)
20 -20 + b = 100 - 20
b = 80 cm
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