## Mensuration

Area: area of a bounded figure is the space covered by it.

Perimeter: The length of the boundary of a closed figure is called perimeter

Triangle: A plane figure bounded by three straight lines is called a triangle

Area of a triangle = 12base height = 12CDAB

s(s-a) (s-b) (s-c)

And s = a + b + c2

Where a, b, c are the lengths of the sides of triangle

Equilateral triangle if all the three sides of a triangle are equal , then there is an equilateral triangle

Area of an equilateral triangle = 34a2

Perimeter of an equilateral triangle with side a = 3a

Isosceles triangle if at least two sides of a triangle are equal, there is a isosceles triangle

Area of isosceles triangle = a44b2 – a2

And height of isosceles triangle, h = b2 – a24

Perimeter of isosceles triangle = 2b + a

Square:

Area of a square = (side)2= a2 or

d22 where d is the diagonal of a square

Perimeter of a square = 4a

Diagonal of a square = 2a

Area of a path which is outside of a square of uniform width x = 4×2+ 4ax

Area of a path which is inside of a square of uniform width x = 4ax – 4×2

Perimeter of a path which is outside of a square of uniform width x = 4(a + 2x)

Perimeter of a path which is inside of a square of uniform width x = 4(a- 2x)

Rectangle:

Area of a rectangle = lb, where l is the length of a rectangle and b is the breadth of a rectangle

Perimeter of a rectangle = 2(l + b)

Diagonal of a rectangle, d = l2 + b2

Area of a path which is outside of a rectangle of uniform width x =4×2+2lx + 2bx

Area of a path which is inside of a rectangle of uniform width x = 2lx + 2bx – 4×2

Perimeter of a path which is outside of a square of uniform width x = 2(l + b + 2x)

Perimeter of a path which is outside of a square of uniform width x = 2(l + b – 2x)

Parallelogram:

Area of parallelogram = 12baseheight = 12bh

Rhombus: it is a parallelogram whose all sides are equal

Its diagonal bisect each other at right angle.

Area of rhombus = 12 product of diagonals = 12d1d2 or

Area = 12bh

Where b is the side of rhombus, h is the height and d1,d2are the diagonals of the rhombus

Perimeter of rhombus = 4b

Side of rhombus = ( d12)2 + ( d22)2

Trapezium:

Area of trapezium = 12height (sum of parallel sides)

Median of trapezium = 12(sum of parallel sides)

Median is the line segment joining the midpoints of non parallel side

Circle:

Area of a circle= r2

Diameter (d) = 2r

Circumference = 2r

Area of a sector =360 r2

Length of arc = 360 2r

Area of the ring or circular path = (R+r) (R-r)

Cube:

Volume of a cube = a3

Diagonal of a cube = 3a

Surface area of a cube = 6a2

Area of four walls = 4a2

Cuboid:

Volume of a cuboid = lbh

Diagonal of a cuboid = l2+ b2+h2

Surface area of a cuboid = 2(lb + bh + hl)

Area of four walls = 2(lh + bh)

Cylinder:

Volume of a cylinder = r2h

Curved surface area = 2rh

Total surface area = 2r(r + h)

Cone:

Volume of a cone = 13r2h

Curved surface area = rl

Total surface area = r (l + r)

Slant height = h2+ r2

Sphere:

Volume of a sphere = 43r3

Surface area = 4r2

Hemisphere:

Volume of a hemisphere = 23r3

Curved surface area = 2r2

Total surface area = 3r2

Frustum of a cone

Volume = 13h(r2+R2+ r.R)

Surface area = (r + R)(R -r)2+h2

Total surface area = (r + R)(R -r)2+h2 + r2 + R2

Slant height, l = (R -r)2+h2

Right pyramid

Volume of a pyramid = 13(area of the base)height

Lateral surface area of a pyramid = 12(perimeter of base) slant height

Total surface area of a pyramid = lateral surface area + area of the base

Regular tetrahedron it is a polyhedron with four equilateral triangle faces

Let the edge of a regular tetrahedron is a

Volume = 212a3

Lateral surface area = 334a2

Total surface area = 3a2

Relation between edges, vertices and faces of a polyhedron

Faces + edges = vertices – 2