Chapter 11: Perimeter and Area

Chapter -11

Perimeter and Area

Square

Introduction to square

A square is a flat shape with 4 equal sides and every angle is a right angle i.e. 90°.                                          

Special case of square

In the above figure, ABCD is a square.
Here AB = BC = CD = DA
Each interior angle is 90°.
The Perimeter of square is 4 times the side length. Let a be the length of each side of square.
Now, perimeter of square = 4 * a = 4a
Area of square = side length squared
                        = a * a   
                        = a²

Square area, perimeter & diagonal

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

Solution

Area of square = (side)²
= (5)²
= 25 cm²
Perimeter of square = 4 × side
= 4 × 5
= 20 cm

Rectangle

Introduction to rectangle

A rectangle is a 4-sided flat shape with straight sides where all interior angles are right angles (90°).

Also opposite sides are parallel and of equal length.

In the above figure, ABCD is a rectangle where

AB = CD and BC = AD

Each interior angle is 90°.

Now, perimeter of a rectangle = 2(length + breadth)

Area of rectangle = length * breadth

Area,perimeter and diagonals

The rectangle is a quadrilateral, with equal opposite sides.

Area = Length × Breadth

Perimeter = 2(Length + Breadth)

Triangle

Introduction to triangle Area

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.

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.

The area of each triangle = 1/4 (Area of the square)

perimeter of triangle

To calculate the perimeter of a triangle, add the length of its sides. For example, if a triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c.

Types of triangle

Perimeter of an Isosceles, Equilateral and Scalene Triangle

Combination problems on square, rectangle and triangle

Introduction to square, rectangle

Square is one of the most basic shapes. A square is a quadrilateral with four equal sides & four equal angles.
A rectangle is a quadrilateral with four right angles. In a rectangle, all the angles are equal (360°/4=90°). Furthermore, the opposite sides are parallel and equal, and diagonals bisect each other.

Special case of square

A convex quadrilateral is a square in special conditions as follows:

• A rectangle with two adjacent equal sides.
• A rhombus with a right angle, all angles equal, equal diagonals.
• A quadrilateral with four equal sides and four right angles makes a square.
• A parallelogram with one right angle and two adjacent equal sides makes a square.
• An isosceles trapezoid with equal diagonals, base angle equals.
• A Kite with two disjoint pairs of adjacent sides is equal.

  Area, perimeter & diagonal of square, rectangle, triangle

Diagonals of a square: Diagonals of a square are equal in length, they bisect the angles, and they are the perpendicular bisectors of each other.
Length of the diagonal d=√(a²+a²)=√(2a²2)=a√2 units.
Where d denotes a diagonal of a square is equal to side length times square root of 2.

Diagonals of rectangle: A rectangle has two diagonals they are equal in length and intersect in the middle. The diagonal is the square root of (width squared + height squared).

Diagonal(d)= l²+b²−−−−−−√
Where l is the length of the rectangle.

Where b is the breadth of the rectangle.
Perimeter is the actual distance around a closed figure.
2. Perimeter of a regular polygon = Number of sides x Length of one side
3. Perimeter of a square = 4 x side

 

Perimeter of a triangle = AB + BC + CA (Sum of all sides of triangle)

Perimeter of a rectangle = 2 [length + breadth]

= 2(l+ b)

Types of triangle
Right angle triangle: A triangle where one of its interior angles is a right angle 90°.

Area:

Area(A)=1/2(b×h)

Thus, the height of the triangle h=Area×2/b

And, the base of triangle b=Area×2/h

Where h is denoted as height.

Where b is denoted as base.

The perimeter:

a²+b²=c²

a, b  are the lengths of the other two sides.

Where c is the length of the hypotenuse.

 Sides: The two sides that are not the hypotenuse makes the right angle.

Hypotenuse:  The side opposite the right angle is called the hypotenuse. It will always be the longest side of a right triangle. 

Isosceles triangle: A triangle which has two of its sides equal in length.

Area:

Area(A)=1/2(b×h)

Thus, the height of the triangle h=Area×2/b

And, the base of the triangle b=Area×2/h

Where h is denoted as height.

Where b is denoted as base.

Altitude h=√(a²−b²)/4

The perimeter:

P=2a+b

Where a is the lengths of the two equal sides.

Where b is the lengths of the other sides.

Equilateral triangle: A triangle which has all three of its sides equal in length.

Area:

Area(A)=√3/4s².

Where s² denotes sides of the triangle.

 The perimeter:

Perimeter(P)=a+b+c or P=s+s+s.

a, b, c are the lengths of the three equal sides.

or

S is the lengths of the three equal sides.

Parallelogram

Introduction to quadrilateral

A quadrilateral is a plane figure that has four sides or edges, so having four corners or vertices. Quadrilaterals will typically be of standard shapes with four sides like rectangle, square, trapezoid, and kite or irregular and uncharacterized. 

Quadrilateral: trapezium
Properties: two sides are parallel
Diagram:

Introduction to parallelogram

A parallelogram is a special kind of quadrilateral in which both pairs of opposite sides are parallel.

Definition and properties of a parallelogram

• Quadrilateral: parallelogram
  Properties: two sides are parallel and opposite angles are equal
  Diagram:

 Area of a parallelogram = base x height

The four basic properties of a parallelogram are:

1. Opposite sides of a parallelogram are equal
2. Opposite angles of a parallelogram are equal
3. Diagonals divide the parallelogram into two congruent triangles
4. Diagonals bisect each other 

Area of a circle

A circle 

A closed line consisting of all points on the plane that are equidistant from a given point on the plane is called a circle.
Area of circle = πr², where r is the radius of the circle.
Diameter of Circle:

The distance across the circle. The length of any chord passing through the center. It is twice the radius.
Diameter = 2⋅Radius = 2r

Radius of circle:
The radius is the distance from the centre to any point on the circle.
Radius is also known as half  of the diameter.
Radius=Diameter/2

Area of the Circle:
The area of a circle is the number of square units inside that circle.
Area of the Circle = πr2 sq.units
Here r is the radius of the circle.
The value of the π [pi] can be either 22/7 or 3.14.

Circumference of a circle

Circle radius,diameter and chord

The segment joining the center to a freely chosen point on the circumference is called the radius. We usually denote radius by the lowercase r or the uppercase R.
The  segment that joins the two points of the circle is called a chord.
The chord passing through the center of the circle is called the diameter.
AO=BO=EO=FO as circular radii.
The radii EO and FO form the diameter EF.
BC is a chord.

Parts of the circle
Diameter of Circle:

The distance across the circle. The length of any chord passing through the center. It is twice the radius.
Diameter = 2⋅Radius = 2r
Radius of circle:
The radius is the distance from the centre to any point on the circle.
Radius is also known as half  of the diameter.
Radius=Diameter/2
Chord:
The segment that joins the two points of the circle is called a chord.

A circle

Compass can look different, but all whips have two legs - a sharp-pointed pin and a pencil. 
Compass can be used to draw a circle.
If you mark a small mark or dot on a piece of paper and place the sharp crank leg at that point, you can draw a circle. The dot is called the center of the circle, and we usually denote by the capital letter O.
The segment joining center O of a circle to a point on the circle (point A in the drawing) is called the radius.

Circle circumference

Circumference of a circle is the actual distance around it.
Ratio of the circumference and the diameter of a circle is a constant
The numerical value of π is taken as (frac{22}{7}) or 3.14. (approximate)
Circumference of a circle = 2πr, where r is the radius of the circle
All circles are similar to one another. So, the ratio of the circumference to that of diameter is a constant,
That is CircumferenceDiameter=constant(piπ)    
Therefore Cd=π (Where the approximate value of π is 3.14)  
C=  d⋅π  or C= d⋅3.14
The diameter is twice the radius (2r), so the above equation can be
written as C=2r⋅π Note: d=2r
Therefore, the circumference of a circle, C=2πr units.   

Position of two circles in a common point

1. The circles have no common points, and they are inside each other. 
2. The circles have one common point, touching them externally. 
3. The circles have one common point (B), and they touch internally.
4. circles have no common points, and they are off each other

Circular pathways

Area pathways

The fundamental idea about the area of pathways

• We should observe the circular shapes around us where we need to find the area of the pathway.
• The area of the pathway is the difference between the area of the outer circle and inner circle.
• Let ‘R’ be the radius of the outer circle, and ‘r’ be the radius of the inner circle.

Therefore, the area of the circular pathway,
= πR² − πr² = π(R²−r²)sq.units
The circle is a round plane figure whose boundary (the circumference) consists of points equidistant from the fixed point (the centre). 
Area of the circle is the region enclosed by the circle.
Distance around the circular region is called the circumference or perimeter of the circle.
Area of the circle is πr² 
Here r is the radius of the circle.
Circumference of the circleis 2πr 
Here r is the radius of the circle. 

Rectangle pathways

The area of the rectangular pathway = Area of the outer rectangle – Area of the inner rectangle
The area of the rectangular pathway is (LB−lb) sq.units
The uniform path, including the park, is also a rectangle. If we consider the path as the outer rectangle, then the park will be the inner rectangle.
Let land b be the length and breadth of the park.
Area of the park (inner rectangle) = l × b  sq. units.
Let w be the width of the path. If L, B are the length and breadth of the outer rectangle, then
L =L + 2w and b=B + 2w.
Similarly for inner rectangle l=L−2wand b=B−2w