Important Geometry Formulas-Volume & Surface area

1. CUBE

  • Volume =  a^{3}
  • Surface area of cube =  6a^{2}
  • Diagonal of the cube = a\sqrt{3}

2. CUBOID

  • Volume = lbh
  • Surface area = 2(lb + bh +lh)
  • Diagonal of cuboid = \sqrt{ l^{2}+ b^{2}+ h^{2}}

3. Room

(Rectangular room has 4 walls and opposite walls have equal areas)

  • Total area of walls = 2(l+b)\times\,h
  • Total volume of the room = lbh
  • Area of floor or roof = l\times\,b

4. BOX

  • Surface area of an open box = 2(length+Breadth)\times\,h+lb = 2(l+b)\times\,h+lb
  • Capacity of box = (l-2t)(b-2t)(h-2t)
  • Volume of the material of the box = External volume –Internal volume(or capacity)

lbh-(l-2t)(b-2t)(h-2t)

5. CYLINDER

  • Volume = Area of base × height =  \pi\,r^{2}h
  • Curved Surface Area =Perimeter of base × height= 2\pi\,rh
  • Total Surface Area =C.S.A + 2 × Area of one circular end=  2\pi\,r(r+h)

6. HOLLOW CYLINDER

  • Volume = \pi h(R^{2}- r^{2})
  • Curved Surface Area = 2\pi h(R+r)
  • Total Surface Area of Hollow Cylinder = 2\pi h(R+r)+2\pi (R^{2}-r^{2})

7. CONE

  • Volume = \frac{1}{3}\pi r^{2}h
  • Slant Height (l) = \sqrt{ r^{2}+ h^{2}}
  • Curved Surface Area = \pi\,rl=  \pi\,r\sqrt{ r^{2}+ h^{2}}
  • Total Surface Area= \pi\,r(l+r)
  • If cone is formed by sector of a circle then.
    (a) Slant height = radius of circle
    (b) circumference of base of cone = length of arc of sector
  • Radius of maximum size sphere in a cone = \frac{h\times\,r}{l +r}

where , r is radius of cone; l is slant height of cone and h is height of cone.

  • If cone is cut parallel to its base and ratio of heights, radius or slant height of both parts is given as x : y. Then ratio of there volume =  x^{3}\,:\,y^{3}

8. FRUSTUM OF CONE

Frustum

  • Slant Height (l)= \sqrt{ h^{2}+(R-r)^{2}}
  • Curved Surface Area = \pi(R+r)l
  • Total Surface Area = \pi\left \{ (r+R)l+r^{2}+R^{2} \right \}
  • Volume = \frac{\pi\,h}{3}(r^{2}+R^{2}+rR)

9. SPHERE

Sphere

  • Volume = \frac{4}{3}\pi\,r^{3}
  • Total Surface Area = 4\pi\,r^{2}
  • If a sphere is cut into n parts, then T.S.A of n parts = 4\pi\,r^{2} + n\,\pi\, r^{2}

10.  HOLLOW SPHERE OR SPHERICAL

Hollow Sphere

  • Volume = \frac{4}{3} \pi(R^{3}- r^{3})
  • Internal Surface Area = 4\pi r^{2}
  • External Surface Area = 4\pi R^{2}

11. HEMISPHERE

Hemisphere

  • Volume = \frac{2}{3} \pi\,r^{3}
  • Total Surface Area= 3 \pi\,r^{2}
  • Curved Surface Area= 2 \pi\,r^{2}

12. RIGHT PRISM

  • Volume = Area of Base \times Height
  • Lateral Surface Area = Perimeter of base \times  Height
  • Total Surface Area = Lateral  Surface Area + 2 \times  Area of Base

13. PYRAMID

For Triangular Pyramid

Triangular pyramid

  • Volume = \frac{1}{3}\times Area of Base \times Height
  • Total Surface Area = Area of Base \times  Area of Slant Surface
  • Base =abc
  • Slant Surface =abd+bcd+acd

For Square Pyramid

Square Pyramid

  • Volume =  \frac{1}{2}\times Area of abcd \times Height
  • Total Surface Area = Area of abcd + Area of (apd+dpc+pcb+apb)
  • Base =abcd
  • Slant Surface =apd+dpc+pcb+apb

14. TETRAHEDRON

Tetrahedron

Height = \sqrt{\frac{2}{3}}a

Volume = \frac{\sqrt{2}}{12} a ^{3}

Lateral Surface Area= \frac{3\sqrt{3}}{4} a ^{2}

Total Surface Area = \sqrt{3} a ^{2}

Slant Height = \frac{\sqrt{3}}{2}a

Slant Edge = a

IMPORTANT POINTS TO REMEMBER

1. If length, breadth and height of a cuboid are increased by x%, Y% and z% then its volume increased by

\left [ X+Y+Z+\frac{XY+YZ+ZX}{100}+\frac{XYZ }{(100)^{2}} \right ]%

2. If side of a cube is increased by x% then its volume increases by

\left [ \left ( 1+\frac{X}{100} \right )^{3} \right ]\times100%

3. If the radius (or diameter) of a sphere is changed by x% then its volume changed by

\left [ \left ( 1+\frac{X}{100} \right )^{3} \right\,-1 ]\times100%

4. If height of a cylinder is changed by x% and radius remains the same then the volume changes by x%.

5. If radius of a cylinder is changed by  x% and height remains unchanged , then volume changes by \left ( 2x+\frac{ x^{2}}{100} \right )%

6. If radius of a cylinder or cone is changed by x% and height is changed by y% , then volume changes by

\left [ 2x+y+\frac{x ^{2}+2xy}{100}+\frac{x ^{2}y}{100^{2}} \right ]%

7.  If in a cylinder or cone , height & radius both change by x% , then volume changes by   \left [ \left \ \left ( 1+\frac{x}{100}\right )^{3} \right \,-1 \right ]\times100%

8. If radius a share or edge of a cube is changed by x%, then the change in surface area is given by \left [ 2x+\frac{x ^{2}}{100} \right ]%

 

 

 

 

 

 

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