Important Facts and Formulae:
Results On Triangle
1.Sum of the angles of a triangle is 180 degrees.
2.The sum of any two sides of a triangle is greater than third side.
3.Pythagoras Theorem:
In a right angled triangle (Hypotenuse)2 = (Base)2 +(Height)2
4.The line joining the mid point of a side of a triangle to the opposite vertex is called the MEDIAN.
5.The point where the three medians of a triangle meet, is called CENTROID. The centroid divides each of the medians in the ratio 2:1
6.In an isosceles triangle, the altitude from the vertex bisects the base
7.The median of a triangle divides it into two triangles of the same area.
8.The area of the triangle formed by joining the mid points of the sides of a given triangle is one-fourth of the area of the given triangle.
Results On Quadrilaterals
1.The diagonals of a Parallelogram bisect each other.
2.Each diagonal of a Parallelogram divides it into two triangles of the same area.
3.The diagonals of a Rectangle are equal and bisect each other
4.The diagonals of a Square are equal and bisect each other at right angles.
5.The diagonals of a Rhombus are unequal and bisect each other at right angles.
6.A Parallelogram and a Rectangle on the same base and between the same parallels are equal in area.
7.Of all he parallelogram of given sides the parallelogram which is a rectangle has the greatest area.
Formulae
1.Area of a RECTANGLE = length * breadth
Length = (Area/Breadth) and Breadth = (Area/Length)
2.Perimeter of a RECTANGLE = 2(Length + Breadth)
3.Area of a SQUARE = (side)2 = ½ ( Diagonal)2
4.Area of four walls of a room = 2(length + breadth) * height
5.Area of a TRIANGLE = ½ * base * height
6.Area of a TRIANGLE = √[s * (s-a) * (s-b) * (s-c)],where a,b,c are the sides of the triangle and s = 1/2(a+b+c)
7.Area of EQUILATERAL TRIANGLE = √(3/4)* (side)2
8.Radius of in circle of an EQUILATERAL TRIANGLE ofside a = r / 2√3
9.Radius of circumcircle of an EQUILATERAL TRIANGLE of side a = r / √3
10.Radius of incircle of a triangle of area ∆ and semi perimeter S = ∆ / s
11.Area of a PARALLELOGRAM = (base * height)
12.Area of RHOMBUS = 1/2 (product of diagonals)
13.Area of TRAPEZIUM =
=1/2 * (sum of parallel sides)* (distance between them)
14.Area of a CIRCLE =  r2 where r is the radius
15.Circumference of a CIRCLE = 2r
16.Length of an arc = 2 rø / 360, where ø is central angle
17.Area of a SECTOR = ½ (arc * r) = r2ø / 360
18.Area of a SEMICIRCLE = r2 / 2
19.Circumference of a SEMICIRCLE = r
Simple Problems
1.One side of a rectangular field is 15m and one of its diagonal is 17m. Find the area of field?
Sol: Other side = √[(17*17) – (15*15)] = √(289-225) = 8m
Area = 15 * 8 =120 sq. m
2.A lawn is in the form of a rectangle having its sides in the ratio 2:3 The area of the lawn is 1/6 hectares. Find the length and breadth of the lawn.
Sol: let length = 2x meters and breadth = 3x mt
Now area = (1/6 * 1000)sq m = 5000/3 sq m
2x * 3x = 5000/3 =>x * x =2500 / 9
x = 50/3
length = 2x = 100/3 m and breadth = 3x = 3*(50/3) = 50m
3.Find the cost of carpeting a room 13m long and 9m broad with a carpet 75cm wide at the rate of Rs 12.40 per sq meter
Sol: Area of the carpet = Area of the room = 13* 9 =117 sq m
length of the carpet = (Area/width) = 117 * (4/3) = 156 m
Cost of carpeting = Rs (156 * 12.40) = Rs 1934.40
4.The length of a rectangle is twice its breadth if its length is decreased by 5cm and breadth is increased by 5cm, the area of the rectangle is increased by 75 sq cm. Find the length of the rectangle.
Sol: let length = 2x and breadth = x then
(2x-5) (x+5) – (2x*x)=75
5x-25 = 75 => x=20
length of the rectangle = 40 cm
5.In measuring the sides of a rectangle, one side is taken 5% in excess and the other 4% in deficit. Find the error percent in the area, calculate from the those measurements.
Sol: let x and y be the sides of the rectangle then
correct area = (105/100 * x) * (96 / 100 *y)
=(504/500 xy) – xy = 4/500 xy
Error% = 4/500 xy*(1/xy)*100 % = 4/5% = 0.8%
Results On Triangle
1.Sum of the angles of a triangle is 180 degrees.
2.The sum of any two sides of a triangle is greater than third side.
3.Pythagoras Theorem:
In a right angled triangle (Hypotenuse)2 = (Base)2 +(Height)2
4.The line joining the mid point of a side of a triangle to the opposite vertex is called the MEDIAN.
5.The point where the three medians of a triangle meet, is called CENTROID. The centroid divides each of the medians in the ratio 2:1
6.In an isosceles triangle, the altitude from the vertex bisects the base
7.The median of a triangle divides it into two triangles of the same area.
8.The area of the triangle formed by joining the mid points of the sides of a given triangle is one-fourth of the area of the given triangle.
Results On Quadrilaterals
1.The diagonals of a Parallelogram bisect each other.
2.Each diagonal of a Parallelogram divides it into two triangles of the same area.
3.The diagonals of a Rectangle are equal and bisect each other
4.The diagonals of a Square are equal and bisect each other at right angles.
5.The diagonals of a Rhombus are unequal and bisect each other at right angles.
6.A Parallelogram and a Rectangle on the same base and between the same parallels are equal in area.
7.Of all he parallelogram of given sides the parallelogram which is a rectangle has the greatest area.
Formulae
1.Area of a RECTANGLE = length * breadth
Length = (Area/Breadth) and Breadth = (Area/Length)
2.Perimeter of a RECTANGLE = 2(Length + Breadth)
3.Area of a SQUARE = (side)2 = ½ ( Diagonal)2
4.Area of four walls of a room = 2(length + breadth) * height
5.Area of a TRIANGLE = ½ * base * height
6.Area of a TRIANGLE = √[s * (s-a) * (s-b) * (s-c)],where a,b,c are the sides of the triangle and s = 1/2(a+b+c)
7.Area of EQUILATERAL TRIANGLE = √(3/4)* (side)2
8.Radius of in circle of an EQUILATERAL TRIANGLE ofside a = r / 2√3
9.Radius of circumcircle of an EQUILATERAL TRIANGLE of side a = r / √3
10.Radius of incircle of a triangle of area ∆ and semi perimeter S = ∆ / s
11.Area of a PARALLELOGRAM = (base * height)
12.Area of RHOMBUS = 1/2 (product of diagonals)
13.Area of TRAPEZIUM =
=1/2 * (sum of parallel sides)* (distance between them)
14.Area of a CIRCLE =  r2 where r is the radius
15.Circumference of a CIRCLE = 2r
16.Length of an arc = 2 rø / 360, where ø is central angle
17.Area of a SECTOR = ½ (arc * r) = r2ø / 360
18.Area of a SEMICIRCLE = r2 / 2
19.Circumference of a SEMICIRCLE = r
Simple Problems
1.One side of a rectangular field is 15m and one of its diagonal is 17m. Find the area of field?
Sol: Other side = √[(17*17) – (15*15)] = √(289-225) = 8m
Area = 15 * 8 =120 sq. m
2.A lawn is in the form of a rectangle having its sides in the ratio 2:3 The area of the lawn is 1/6 hectares. Find the length and breadth of the lawn.
Sol: let length = 2x meters and breadth = 3x mt
Now area = (1/6 * 1000)sq m = 5000/3 sq m
2x * 3x = 5000/3 =>x * x =2500 / 9
x = 50/3
length = 2x = 100/3 m and breadth = 3x = 3*(50/3) = 50m
3.Find the cost of carpeting a room 13m long and 9m broad with a carpet 75cm wide at the rate of Rs 12.40 per sq meter
Sol: Area of the carpet = Area of the room = 13* 9 =117 sq m
length of the carpet = (Area/width) = 117 * (4/3) = 156 m
Cost of carpeting = Rs (156 * 12.40) = Rs 1934.40
4.The length of a rectangle is twice its breadth if its length is decreased by 5cm and breadth is increased by 5cm, the area of the rectangle is increased by 75 sq cm. Find the length of the rectangle.
Sol: let length = 2x and breadth = x then
(2x-5) (x+5) – (2x*x)=75
5x-25 = 75 => x=20
length of the rectangle = 40 cm
5.In measuring the sides of a rectangle, one side is taken 5% in excess and the other 4% in deficit. Find the error percent in the area, calculate from the those measurements.
Sol: let x and y be the sides of the rectangle then
correct area = (105/100 * x) * (96 / 100 *y)
=(504/500 xy) – xy = 4/500 xy
Error% = 4/500 xy*(1/xy)*100 % = 4/5% = 0.8%
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