SURFACE AREA AND VOLUME

CLASS-IX
SURFACE AREA AND VOLUME
ALL FORMULAE
SOME DERIVATIONS

1. MATHS NCERT CHAPTER- 13 SURFACE AREA AND VOLUME CLASS- IX

2. CUBE A A A A CUBE IS A SYMMETRICAL THREE-DIMENSIONAL SHAPE, EITHER SOLID OR HOLLOW, CONTAINED BY SIX EQUAL SQUARES.

3. LOOKING AT THE CUBE TEMPLATE, IT IS EASY TO SEE THAT THE CUBE HAS SIX SIDES AND EACH SIDE IS A SQUARE THE AREA OF ONE SQUARE IS A × A = A 2 SINCE THERE ARE SIX SIDES, THE TOTAL SURFACE AREA IS- SA = A 2  + A 2  + A 2  + A 2  + A 2  + A 2 SA = 6 × A 2   SURFACE AREA

4. THE VOLUME OF A CUBE IS MEASURED BY MULTIPLYING THE LENGTH, WIDTH AND HEIGHT OF THE CUBE. THE FORMULA FOR THE VOLUME OF A CUBE = LENGTH × WIDTH × HEIGHT.  IN A CUBE, THE LENGTH, WIDTH, AND HEIGHT ARE ALL SAME. EQUATION FOR VOLUME OF A CUBE, V = S VOLUME OF A CUBE =  S CUBIC UNITS WHERE, 'S' IS THE LENGTH OF ANY EDGE OF THE CUBE. 3 3 VOLUME

5. CUBOID A CUBOID IS A BOX-SHAPED SOLID OBJECT. IT HAS SIX FLAT SIDES AND ALL ANGLES ARE RIGHT ANGLES. AND ALL OF ITS FACES ARE RECTANGLES.

6. To calculate the surface area of the cuboid we need to first calculate the area of each face and the add up all the areas to get the total surface area. Total area of top and bottom surfaces is  lw  +  lw =  2 lw Total area of front and back surfaces is  lh  +  lh =  2 lh Total area of the two side surfaces is  wh  +  wh =  2 wh Surface area of cuboid = 2 lw  + 2 lh  + 2 wh  = 2( lw  +  lh  +  wh ) SURFACE AREA

7. VOLUME   Look at this shape. There are 3 different measurements: Length,   Width,     Height The volume is found using the formula: Volume = Length × Width ×    Height Which is usually shortened to: V = l × w × h Or more simply: V = lwh

8. CYLINDER A solid object with: •two identical flat ends that are circular or elliptical •one curved side.

9. SURFACE AREA (LATERAL) SURFACE AREA OF CURVE  = 2 × π × r × h = 2 πrh WHERE, π = pi r = RADIUS OF THE BASE h = HEIGHT OF THE CYLINDER

10. SURFACE AREA (TOTAL) THE SURFACE AREA HAS THESE PARTS: SURFACE AREA OF BOTH ENDS  = 2 ×  Π  × R 2 SURFACE AREA OF SIDE  = 2 ×  Π  × R × H WHICH TOGETHER MAKE: SURFACE AREA = 2 ×  π  × r × (r+h) AND IN SHORT- 2 πr(r+h) WHERE, π = pi r = RADIUS OF THE BASE h = HEIGHT OF THE CYLINDER

11. VOLUME To calculate the volume we multiply the area of the base by the height of the cylinder: Area of the base:  π  × r 2 Height: h And we get: Volume =  π  × r 2  × h Or, Volume= π r 2  h

12. RIGHT CIRCULAR CONE 90 A CONE WHOSE SURFACE IS GENERATED BY LINES JOINING A FIXED POINT TO THEPOINTS OF A CIRCLE, THE FIXED POINT LYING ON A PERPENDICULAR THROUGH THECENTER OF THE CIRCLE.

13. SLANT HEIGHT IN A RIGHT CIRCULAR CONE, WE CAND FIND SLANT HEIGHT USING PATHAGORAS THEOREM, i.e.- a + b = c HERE, a = radius (r) b = height (h) c = slant height (l) SO, r + h = l OR, l= √(h + r ) 2 2 2 2 2 2 2 2

14. SURFACE AREA (LATERAL) THE AREA OF THE CURVED (LATERAL) SURFACE OF A CONE =  π rl

15. SURFACE AREA (TOTAL) THE SURFACE AREA OF THE CONE EQUALS THE AREA OF THE CIRCLE PLUS THE AREA OF THE CONE AND THE FINAL FORMULA IS GIVEN BY: SA = π R 2  + π RL WHERE, R  IS THE RADIUS π IS pi L  IS THE SLANT HEIGHT

16. VOLUME The volume of a cylinder is:   π  × r 2  × h The volume of a cone is:   1 / 3   π  × r 2  × h The volume formulas for cones and  cylinders  are very similar: So a cone's volume is exactly one third (  1/ 3  ) of a cylinder's volume.

17. SPHERE A ROUND SOLID FIGURE, OR ITS SURFACE, WITH EVERY POINT ON ITS SURFACE EQUIDISTANT FROM ITS CENTRE.

18. SURFACE AREA The surface area of a sphere is given by the formula:- Area=4πr Where r is the radius of the sphere. 2

19. VOLUME The volume enclosed by a sphere is given by the formula:- volume=4/3πr Where r is the radius of the sphere. 3

20. HEMISPHERE IN GEOMETRY IT IS AN EXACT HALF OF A SPHERE.

21. SURFACE AREA (LATERAL) THE SURFACE AREA OF A SPHERE OF RADIUS r IS 4πr 2 . HALF OF THIS IS 2πr 2 .  SO, LATERAL SURFACE IS 2πr 2

22. SURFACE AREA (TOTAL) THE AREA OF A CIRCLE OF RADIUS R IS Π R 2  AND THUS IF THE HEMISPHERE IS MEANT TO INCLUDE THE BASE THEN THE SURFACE AREA IS:-  2πR 2  +  πR 2  = 3πR 2 .

23. VOLUME  The volume of a sphere is 4/ 3 π  r 3 . So the volume of a hemisphere is half of that: V = (2 / 3) π r 3

24. CONCLUSION

25. THE END