MENSURATION WORKSHOP CALCULTION AND SCIENCE 1ST YEAR PRESENTATION

WORKSHOPCALCULATION AND SCIENCE FIRST YEAR

1. MENSURATION WORKSHOP CALCULATION AND SCIENCE

2. In Engineering field, an Engineer has to estimate the material, manpower, machinery, etc. required to prepare the geometrical objects. Hence we must be very conversant with all relevant formulae connected with geometrical objects. Length = l Breadth or width = b Diagonal = d Diameter = d or ꝋ Radius = r unit Semiperimeter = S unit Perimeter = P unit Circumference = C unit Area = A unit2 Total surface area = T.S.A unit2 Lateral surface area = L.S.A unit2 Volume = V unit3

3. SQUARE This is also a four sided figure, opposite sides are parallel. All the four sides are equal. All the sides are inclined at 90º Area = axa (unit 2 ) Perimeter = 4a (unit) Diagonal = √2 x a (unit)

4. Question: The perimeter of one square is 748 cm and that of another is 336 cm. Find the perimeter of a square which is equal in area of the sum of the two. =

5. Perimeter1 = 4a1 a = perimeter/4 a1 = 748/4 = 187 cm, area1 = a1 2 = 187 2 = 34969 cm 2 Perimeter2 = 4a2 a = perimeter/4 a2 = 336/4 = 84 cm, area2 = a2 2 = 84 2 = 7056 cm 2 A3 2 = a1 2 + a2 2 = 34969 + 7056 = 42025 cm 2 A3 = √42025 = 205 cm Perimeter3 = 4a3 = 4 x 205 = 820 cm

6. RECTANGLE This is a four sided figure. Opposite sides are parallel. Angles between adjacent sides are 90º Area = l x b (unit 2 ) Perimeter = l + b + l + b = 2l + 2b = 2( l+b ) (unit) Diagonal = (unit)

7. The perimeter of a rectangle is 320 metre . Its sides are in the ratio of 5:3. Find the area of the rectangle Ratio = l : b = 5:3 length l = 5x, breadth b = 3x Perimeter 2(l + b) = 2(5x + 3x) = 320 2(8x) = 320 x = 20 l = 5x = 5 x 20 = 100 m b = 3x = 3 x 20 = 60 m Area = l x b = 100 x 60 = 6000 m2

8. PARELLELOGRAM Perimeter = a + b + a + b = 2( a+b )

9. TRIANGLES Tri means three. Hence tri- angle means three angled figure. For construction of three angled figure, there should be three sides. Hence triangle means three sided figure. Sum of the three angles of any triangle = 180º.

10. Area of any triangle = ½ x Base x Height unit 2

11. ISOSCELES TRIANGLE Area of any triangle = ½ x base x height HERE base = 2b, height = h S 2 = h 2 + b 2 EQUILATERAL TRIANGLE Since all sides are equal,

12. SCALENE TRIANGLE

13. PYTHAGORAS THEOREM

14. Circle It is the path of a point which is always equal from its centre is called a circle .

15. CIRCUMFERENCE OF A CIRCLE: Draw a circle of dia = 7cm Measure the circumference using scale = 22 cm Circumference / Diameter = 22/7 = ∏ Hence Circumference C = ∏ x Diameter = ∏ (2r) = 2 ∏r AREA OF CIRCLE: Convert the shape into a rectangle by dividing as sectors as many as possible Now area of rectangle is length x height Here height = r, length = half of circumference length Area of Circle = r x ∏r = ∏r 2 7CM DIA

16. AREA OF SEMI CIRCLE = ∏r 2 /2 LENGTH OF THE ARC = 2 ∏r/2 = ∏r SEMI CIRCLE CIRCULAR RING Radius of outer circle = R1, Radius of inner circle = R2 AREA OF CIRCULAR RING = ∏(R1 2 – R2 2 )

17. SECTOR OF CIRCLE

18. ELLIPSE ∏

19. HEXAGON Circumference of hexagon = 6 a Area of hexagon = area of 6 equal triangle a

20. SOLIDS CUBE All sides of cube are same i.e length, breadth and height have same value. It is bounded by six equal square faces. Volume of cube = side x side x side = a 3 unit 3 Lateral surface area = 4a 2 unit 2 Total surface area = 6 x side x side = 6a 2 unit 2

21. Rectangular solid (or) cuboid Rectangular soild is bounded by six rectangular surfaces and opposite surfaces are equal and parallel to each other. Volume of rectangular solid = Length x breadth x height = l . b . h unit 3 Lateral surface area = 2h( l+b ) unit 2 Total surface area = 2lb + 2bh + 2hl = 2( lb+bh+hl ) unit 2

22. CYLINDER This is a prism whose top and bottom surfaces are equal and circular. Curved area of cylinder = 2 ∏ rh unit 2 Total surface area of cylinder = 2 ∏rh + ∏r 2 + ∏r 2 = 2 ∏ r( h+r ) unit 2 r = Radius of base d = Diameter of base h = Height of cylinder

23. HOLLOW CYLINDER Hollow means empty space. In hollow cylinder there is an empty place. Water pipe is an example of hollow cylinder. Volume of hollow cylinder = Area of circular ring x height = π (R 2 – r 2 ) h Total surface area of hollow cylinder = Inner + outer curved area + area of top and bottom circular part = 2 π rh + 2 π Rh + π (R 2 – r 2 ) + π (R 2 – r 2 ) R = outer radius r = inner radius D = outer diameter d = inner diameter h = height of cylinder t = thickness

24. Mean dia = (D − d)/2 If thickness given then: Volume of hollow cylinder = length x breadth x height (Here length = mean circumference of the ring = π (D-d)/2) Volume = π x mean dia x thickness x height

25. Cone Cone is a pyramid with a circular base Curved area = ½ x l x r= ½ x (2 x π x r) x s = π rs

26. Frustum of a cone When a cone is cut by a plane parallel to the base, and upper part is removed, the formation appears, is termed as frustum of a cone. Buckets, oil cans etc. are such frustums in shape. TSA = π l (R + r) + A1 + A2 unit2

27. SPHERE Sphere is a solid circular body Volume of sphere = Volume of two Cones of same radius & height = π r 2 h + π r 2 h = π r 2 (2r)   = +

28. HEXAGONAL BAR Volume of Hexagonal bar = Area of hexagonal x height = a 2 h Lateral surface area of hexagonal bar = 6 x height of the bar x side of hexagon = 6 h a Total surface area of hexagonal bar = lateral surface area + (2 x area of hexagon) = 6ah + 2 x a 2