Mensuration: Areas & Volumes

Perimeter is the distance around a closed figure while area is the part of plane or region occupied by the closed figure.

Square

Perimeter = 4a

Diagonal = √2 × a

Area = a2

Rectangle

Perimeter = 2(l+b)

Diagonal = √(l2 + b2)

Area = l × b

Triangles

Area = bh/2

Hero's Formula

Area of a triangle with its sides as a, b and c can calculated by using Hero’s formula,

Area of triangle = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2

Results on Triangles

  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.

Quadrilateral

Area = d(h1+h2)/2

Area of a quadrilateral whose sides and one diagonal are given, can be calculated by dividing the quadrilateral into two triangles and then using the Hero’s formula.

Area of a trapezium is half of the sum of the lengths of parallel sides × perpendicular distance between them.

Parallelogram

Area = base x height

Rhombus

Area of Rhombus = (d1d2)/2

Side = (√(d12 + d22))/2

Circles

Circumference of a circle = 2πr

Area of a circle = πr2

Length of an arc = θ/360 x (Circumference)

Area of a sector = θ/360 x (Area of Circle)

Cuboid

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

Lateral Surface Area = 2(l + b)h

Volume of a cuboid = l × b × h

Longest Diagonal = √(l2 + b2 + h2)

Cube

Surface area of a cube = 6a2

Volume of a cube = a3

Longest Diagonal = √(3a)

Cylinder

Curved Surface Area = 2πrh

Total Surface Area = 2πr(r + h)

Volume = πr2h

Cone

Slant Height = √(h2 + r2)

Curved Surface Area = πrl

Total Surface Area of right circular cone = πrl + πr2 = πr (l + r)

Volume = (πr2h)/3

Sphere

Surface area of a sphere = 4πr2

Curved surface area of a hemisphere = 2πr2

Total surface area of a hemisphere = 3πr2

Volume of a sphere = (4πr3)/3

Volume of a hemisphere = (2πr3)/3

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.

Prism

A solid having two congruent and parallel faces, called bases and whose other faces, the lateral faces are parallelograms, formed by joining corresponding vertices of the bases is called a Prism. 

Right Prism

A prism in which bases are perpendicular to the lateral edges is called a Right Prism. The base of the prism can be a polygon. In a right prism

Number of lateral surfaces = Number of sides of the base of the prism

Total number of surfaces of a prism = Number of lateral surfaces + 2

Lateral surface area = Perimeter of base × Height

Total surface area = Lateral surface area + 2 (Area of base)

Volume = Area of base × Height