**Important Formulas on Quadrilateral and Circle**

**Rectangle**

A four-sided shape that is made up of two pairs of parallel lines and that has four right angles; especially: a shape in which one pair of lines is longer than the other pair.

The diagonals of a rectangle bisect each other and are equal.

Area of rectangle = length x breadth = *l* x b

**OR** Area of rectangle = if one sides (*l*) and diagonal (d) are given.

**OR** Area of rectangle = if perimeter (P) and diagonal (d) are given.

Perimeter (P) of rectangle = 2 (length + breadth) = 2 (*l *+ b).

**OR** Perimeter of rectangle = if one side (*l*) and diagonal (d) are given.

**Square**

A four-sided shape that is made up of four straight sides that are the same length and that has four right angles.

The diagonals of a square are equal and bisect each other at 90^{0}.

Area (a) of a square

Perimeter (P) of a square

= 4a, i.e. 4 x side

Length (d) of the diagonal of a square

**Circle**

A circle is a path travelled by a point which moves in such a way that its distance from a fixed point remains constant.

The fixed point is known as the centre and the fixed distance is called the radius.

(a) Circumference or perimeter of circle =

where r is radius and d is the diameter of the circle

(b) Area of circle

is radius

is circumference

circumference x radius

(c) The radius of circle =

**Sector :**

A sector is a figure enclosed by two radii and an arc lying between them.

here AOB is a sector

length of arc

Area of Sector

**Ring or Circular Path:**

R=outer radius

r=inner radius

area=π(R^{2}-r^{2})

Perimeter=2π(R+r)

**Rhombus**

Rhombus is a quadrilateral whose all sides are equal.

The diagonals of a rhombus bisect each other at 90^{0}

Area (a) of a rhombus

= a × h, i.e. base × height

Product of its diagonals

since d^{2}_{2 }

since d^{2}_{2 }

Perimeter (P) of a rhombus

= 4a, i.e. 4 x side

Where d1 and d2 are two-diagonals.

Side (a) of a rhombus

**Parallelogram**

A quadrilateral in which opposite sides are equal and parallel is called a parallelogram. The diagonals of a parallelogram bisect each other.

**Area (a) of a parallelogram** = base × altitude corresponding to the base = b × h

Area (a) of the parallelogram

where a and b are adjacent sides, d is the length of the diagonal connecting the ends of the two sides and

In a parallelogram, the sum of the squares of the diagonals = 2

(the sum of the squares of the two adjacent sides).

i.e.,

Perimeter (P) of a parallelogram

= 2 (a+b),

Where a and b are adjacent sides of the parallelogram.

**Trapezium (Trapezoid)**

A trapezoid is a 2-dimensional geometric figure with four sides, at least one set of which are parallel. The parallel sides are called the bases, while the other sides are called the legs. The term ‘trapezium,’ from which we got our word trapezoid has been in use in the English language since the 1500s and is from the Latin meaning ‘little table.’

Area (a) of a trapezium

1/2 x (sum of parallel sides) x perpendicular

Distance between the parallel sides

i.e.,

Where, l = b – a if b > a = a – b if a > b

And

Height (h) of the trapezium

**Pathways Running across the middle of a rectangle:**

X is the width of the path

Area of path= (l+b-x)x

perimeter= 2(l+b-2x)

**Outer Pathways:**

Area=(l+b+2x)2x

Perimeter=4(l+b+2x)

**Inner Pathways:**

Area=(l+b-2x)2x

Perimeter=4(l+b-2x)

**Some useful Short trick:**

- If there is a change of X% in defining dimensions of the 2-d figure then its perimeter will also change by X%
- If all the sides of a quadrilateral are changed by X% then its diagonal will also change by X%.
- The area of the largest triangle that can be inscribed in a semi-circle of radius r is r
^{2.} - The number of revolution made by a circular wheel of radius r in travelling distance d is given by

** number of revolution =d/2πr**

- If the length and breadth of the rectangle are increased by x% and y% then the area of the rectangle will be increased by.

** (x+y+xy/100)%**

- If the length and breadth of a rectangle are decreased by x% and y% respectively then the area of the rectangle will decrease by:

** (x+y-xy/100)%**

- If the length of a rectangle is increased by x%, then its breadth will have to be decreased by
**(100x/100+x)%**in order to maintain the same area of the rectangle. - If each of the defining dimensions or sides of any 2-D figure is changed by x% its area changes by

** x(2+x/100)%**

**where x=positive if increase and negative if decreases.**

**Thanks**

**Prep Smart. Stay Safe. Go BYJU'S Exam Prep.**

## Important Mensuration (3D) Formulas

**Cube**

- s = side
- Volume: V = s^3
- Lateral surface area = 4a
^{2} - Surface Area: S = 6s^2
- Diagonal (d) = s√3

**Cuboid**

- Volume of cuboid: length x breadth x width
- Total surface area = 2 ( lb + bh + hl)

**Right Circular Cylinder**

- Volume of Cylinder = π r^2 h
- Lateral Surface Area (LSA or CSA) = 2π r h
- Total Surface Area = TSA = 2 π r (r + h)

**Hollow-Cylinder**

r_{1} = outer radius

r_{2} = inner radius

*** Volume of Hollow Cylinder = ****π(pie) h(r**_{1(Square)} - r_{2(Square)})

**Right Circular Cone**

- l^2 = r^2 + h^2
- Volume of cone = 1/3 π r^2 h
- Curved surface area: CSA= π r l
- Total surface area = TSA = πr(r + l )

**Important relation between radius, height and slant height of similar cone.**

**Frustum of a Cone**

**r**= top radius**, R =**base radius,- h = height, s = slant height
**Volume**: V = π/ 3 (r^2 + rR + R^2)h**Surface Area:**S = πs(R + r) + πr^2 + πR^2

**Sphere**

- r = radius
- Volume: V = 4/3 πr^3
- Surface Area: S = 4π^2

**Hemisphere**

- Volume-Hemisphere = 2/3 π r^3
- Curved surface area(CSA) = 2 π r^2
- Total surface area = TSA = 3 π r^2

**Quarter-Sphere**

Let 'r' is the radius of given diagram. You have to imagine this diagram, this is 1/4^{th} part of Sphere.

**Prism**

- Volume = Base area x height

- Lateral Surface area = perimeter of the base x height

**Pyramid**

- Volume of a right pyramid = (1/3) × area of the base × height.
- Area of the lateral faces of a right pyramid = (1/2) × perimeter of the base x slant height.
- Area of whole surface of a right pyramid = area of the lateral faces + area of the base.

**Important:**

1.From a solid cylinder no. of maximum solid cone of same height and radius as cylinder are 3.

2.From a solid sphere, no. of maximum solid cone having height and radius equal can be made are 4.

3.From a solid hemisphere, no. of maximum solid cone having height and radius equal can be made are 2.

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