PERIMETER ,AREA &VOLUME


 

  • Perimeter of a closed plane figure is the length of its boundary.
  • Area of a closed plane figure is the measure of the region (surface) enclosed by its boundary.
  • Pythagoras theorem
    In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
  • Area of a triangle
    Area of a triangle = (1/2) × base × height
    Also, Area of a triangle = [s (s -a)(s -b)(s-c)] where a, b, c are the lengths of the sides and s = (a +b +c)/2
    Area of an equilateral triangle = 3a²/4, where a is the side.
  • Rectangle
    If l = length and b = breadth of a rectangle, then
    perimeter = 2 (l +b)
    length of diagonal =  l²+b²
    area = l×b.
  • Square
    If a is the length of side of a square, then
    perimeter = 4a
    length of diagonal = 2 a
    area = a²
  • Parallelogram
    Area of a parallelogram = base×height.
  • Trapezium
    Area of a trapezium = (1/2) (sum of parallel sides)×height
  • Circle
    If r is the radius of a circle, then
    length of a diameter = 2r
    circumference = 2r
    area = r²
    Take  = 22/7 (unless given otherwise)
  • Circular ring (track)
    If R and r are the radii of two concentric circles then
    area of the circular ring (track) =  (R²-r²).

  • Cuboid
    If l, b and h are length, breadth and height of a cuboid, then
    Volume = l×b×h
    Surface are = 2 (lb +l>h +bh)
    Lateral surface area = 2 (l +b)×h
    Length of a diagonal = (l² +b² +c²)

  • Cube
    If a be the length of an edge of a cube, then
    Volume = a³
    Surface area = 6a²
    Length of a diagonal = 3a     

  • Let h be the height and a be the side of an equilateral triangle, then h = (3/2) a.
  • Circumference and area of a circle.
    If r is the radius of a circle, then
    (i) the circumference of the circle = 2r
    (ii) the area of the circle = r²
  • Area of a circular ring.
    If R and r are the radii of the bigger and smaller (concentric) circles, then
    area of the ring = (R² -r²).
  • Circumference and area of a sector of a circle.
    If r is the radius of the circle and the arc subtends an angle of n° at the center, then
    (i) the length of the arc = (n/360).2r = nr/180
    (ii) the area of the sector = (n/360).r²
  • Circumference and area of circumscribed and inscribed circles of an equilateral triangle.
    If R and r are the radii of the circumscribed and inscribed circles of the triangle, then
    (i) R = (2/3)h and r = (1/3)h
    (ii) the circumference of the circumscribed circle = 2R = (4/3)h
    (iii) the area of the circumscribed circle = R² = (4/9)h²
    (iv) the circumference of the inscribed circle = 2  r = (2/3)h
    (v) the area of the inscribed circle = r² = (1/9)  h²
  • Circumference and area of circumscribed and inscribed circles of a regular hexagon.
    Let a be the side of a regular hexagon and R, r be the radii of the circumscribed and inscribed circles respectively of the hexagon, then
    (i) R = a and r = (3/2) a
    (ii) the circumference of the circumscribed circle = 2R = 2a
    (iii) the area of the circumscribed circle =  R² =  a²
    (iv) the circumference of the inscribed circle = 2r = 3 a
    (iv) the area of the inscribed circle = r² = (3/4)a².
  • Surface area and volume (of solids)
  • Solid Cylinder.
    • Let r be the radius and h be height of a solid cylinder, then (i) curved (lateral) surface area = 2 rh (ii) total surface area = 2 r(h +r) (iii) volume = r²h
  • Hollow cylinder.
    Let R and r be the external and internal radii, and h be the height of a hollow cylinder, then
    (i) external curved surface area = 2Rh
    (ii) internal curved surface area = 2rh
    (iii) total surface area = 2(Rh +rh +R² -r²)
    (iv) volume of material = (R² -r²)h
  • Cone.
    Let r, h and l be the radius, height and slant height respectively of a cone, then
    (i) slant height = r² +h²
    (ii) curved (lateral) surface area =  rl
    (iii) total surface area = r(l +r)
    (iv) volume = (1/3) r²h
  • Solid sphere.
    Let r be the radius of a solid sphere, then
    (i) surface area = 4r²
    (ii) volume = (4/3) r³
  • Spherical shell.
    Let R and r be the radii of the outer and inner spheres, then
    (i) thickness of the shell = R -r
    (ii) volume of material = (4/3) (R³ -r³)
  • Solid hemisphere.
    Let r be the radius of a hemisphere, then
    (i) curved (lateral) surface area = 2r²
    (ii) total surface area = 3 r²
    (iii) volume = (2/3) r³
  • Hemispherical shell.
    Let R and r be the radii of the outer and inner hemispheres, then
    (i) the thickness of the shell = R -r
    (ii) external curved surface area = 2 R²
    (iii) internal curved surface area = 2r²
    (iv) total surface area = (3R² +r²)
    (v) volume of material = (2/3)(R³ -r³    )