summary of Area maths chapter
Answers
Area
It refers to the surface of the enclosed figure.
Perimeter and Area
Area and Perimeter of Square
Square is a quadrilateral, with four equal sides.
Area = Side × Side
Perimeter = 4 × Side
Example
Find the area and perimeter of a square-shaped cardboard whose length is 5 cm.
Perimeter
Solution
Area of square = (side)2
= (5)2
= 25 cm2
Perimeter of square = 4 × side
= 4 × 5
= 20 cm
Area and Perimeter of Rectangle
The rectangle is a quadrilateral, with equal opposite sides.
Area = Length × Breadth
Perimeter = 2(Length + Breadth)
Example
What is the length of a rectangular field if its width is 20 ft and Area is 500 ft2?
Rectangular field
Solution
Area of rectangular field = length × width
500 = l × 20
l = 500/20
l = 25 ft
Note: Perimeter of a regular polygon = Number of sides × length of one side
Triangles as Parts of Rectangles
If we draw a diagonal of a rectangle then we get two equal sizes of triangles. So the area of these triangles will be half of the area of a rectangle.
Rectangles
The area of each triangle = 1/2 (Area of the rectangle)
Likewise, if we draw two diagonals of a square then we get four equal sizes of triangles .so the area of each triangle will be one-fourth of the area of the square.
Area of the rectangle
The area of each triangle = 1/4 (Area of the square)
Example
What will be the area of each triangle if we draw two diagonals of a square with side 7 cm?
Solution
Area of square = 7 × 7
= 49 cm2
The area of each triangle = 1/4 (Area of the square)
= 1/4 × 49
= 12.25 cm2
Congruent Parts of Rectangles
Two parts of a rectangle are congruent to each other if the area of the first part is equal to the area of the second part.
Example
Congruent Parts of Rectangles
The area of each congruent part = 1/2 (Area of the rectangle)
= 1/2 (l × b) cm2
=1/2 (4 × 3) cm2
= 1/2 (12) cm2
= 6 cm2
Parallelogram
It is a simple quadrilateral with two pairs of parallel sides.
Also denoted as ∥ gm
Area of parallelogram = base × height
Or b × h (bh)
We can take any of the sides as the base of the parallelogram. And the perpendicular drawn on that side from the opposite vertex is the height of the parallelogram.
Example
Find the area of the figure given below:
Parallelogram
Solution
Base of ∥ gm = 8 cm
Height of ∥ gm = 6 cm
Area of ∥ gm = b × h
= 8 × 6
= 48 cm
Area of Triangle
Triangle is a three-sided closed polygon.
If we join two congruent triangles together then we get a parallelogram. So the area of the triangle will be half of the area of the parallelogram.
Area of Triangle = 1/2 (Area of ∥ gm)
= 1/2 (base × height)
Example
Find the area of the figure given below:
Triangle
Solution
Area of triangle = 1/2 (base × height)
= 1/2 (12 × 5)
= 1/2 × 60
= 30 cm2
Note: All the congruent triangles are equal in area but the triangles equal in the area need not be congruent.
Circles
It is a round, closed shape.
The circumference of a Circle
The circumference of a circle refers to the distance around the circle.
Radius: A straight line from the Circumference till the centre of the circle.
Diameter: It refers to the line from one point of the Circumference to the other point of the Circumference.
π (pi): It refers to the ratio of a circle's circumference to its diameter.
Circumference(c) = π × diameter
C = πd
= π × 2r
Circumference
Note: diameter (d) = twice the radius (r)
d = 2r
Example
What is the circumference of a circle of diameter 12 cm (Take π = 3.14)?
Solution
C = πd
C = 3.14 × 12
= 37.68 cm
Area of Circle
Area of the circle = (Half of the circumference) × radius
= πr2
Area of Circle
Example
Find the area of a circle of radius 23 cm (use π = 3.14).
Solution
R = 23 cm
π = 3.14
Area of circle = 3.14 × 232
= 1,661 cm2
Conversion of Units
Sometimes we need to convert the unit of the given measurements to make it similar to the other given units.
Unit Conversion
1 cm 10 mm
1 m 100 cm
1 km 1000 m
1 hectare(ha) 100 × 100 m
Unit Conversion
1 cm2 100 mm2
1 m2 10000 cm2
1 km2 1000000 m2 (1e + 6)
1 ha 10000 m2
Example: 1
Convert 70 cm2 in mm2
Solution:
1 cm = 10 mm
1 cm2 = 10 × 10
1 cm2 = 100 mm2
70 cm2 = 700 mm2
Example: 2
Convert 3.5 ha in m2
Solution:
1 ha = 10000 m2
3.5 ha = 10000 × 3.5
ha = 35000 m2
Applications
We can use these concepts of area and perimeter of plane figures in our day to day life.
If we have a rectangular field and want to calculate that how long will be the length of the fence required to cover that field, then we will use the perimeter.
If a child has to decorate a circular card with the lace then he can calculate the length of the lace required by calculating the circumference of the card, etc.
Example:
A rectangular park is 35 m long and 20 m wide. A path 1.5 m wide is constructed outside the park. Find the area of the path.
Rectangular Park
Solution
Area of rectangle ABCD – Area of rectangle STUV
AB = 35 + 2.5 + 2.5
= 40 m
AD = 20 + 2.5 + 2.5
= 25 m
Area of ABCD = 40 × 25
= 1000 m2
Area of STUV = 35 × 20
= 700 m2
Area of path = Area of rectangle ABCD – Area of rectangle STUV
= 1000 – 700
= 300 m2
Answer:
area =2(l+b) - for a rectangle
= 4 × sides - for square
= ½ × b × h - for triangle
hope this helped you