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ASPIRE M.Sc., CLASS HANDOUT
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Formula Derivation And Numerical Practice, On, , ‘Mensuration’, Delivered by: Toufique Shahnowaz
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Area
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©Aspire_Zoology@Toufique, , I. Rectangle, a. Perimeter of a rectangle= 2(Length + Breadth) unit., , A, , b. Diagonal of a rectangle= Length𝟐 + Breadth𝟐 unit., , a, , b, , D, a, , c. Area of a rectangle= (Length × Breadth) square unit., B, , b, , C
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©Aspire_Zoology@Toufique, , II. Square, a. Perimeter of a square= (4 × Side) unit., b. Diagonal of a square=, , 𝟐 × 𝑺𝒊𝒅𝒆 unit., , c. Area of a square= (Side)² square unit., = ½ × (Diagonal)² square unit., , A, , a, , D, , a, , a, , B, , C, , a
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III. Triangle, , ©Aspire_Zoology@Toufique, , a. Perimeter of a triangle= (AB+BC+CA) unit., b. Area of a rectangle= ½ ×(Base×Height) square unit., , = 𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄) square unit. [where a,b,c are the sides of, the triangle and s= ½ (a+b+c)], A, , a, , c, , B, b, , C
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Equilateral Triangle:, , ©Aspire_Zoology@Toufique, , a. Perimeter of a equilateral triangle= (3× Side) unit., b. Area of a equilateral rectangle=, , 𝟑, 𝟒, , ×(Side)² square unit.
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Equilateral Triangle:, a. Radius of circumcircle of an equilateral triangle of side a= a/ 𝟑 unit., b. Radius of incircle of an equilateral triangle of side a= a/2 𝟑 unit., , ©Aspire_Zoology@Toufique
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IV. Area of a Parallelogram= (Base×Height) square unit., , ©Aspire_Zoology@Toufique, , V. Area of a Rhombus= ½ ×(Product of diagonal) square unit., , VI. Area of Trapezium= ½ ×(sum of parallel sides × distance between them) square unit.
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VIII. Semi-circle, a. Perimeter of a semi-circle= r(π+2) unit., b. Area of a circle= πr²/2 square unit., , ©Aspire_Zoology@Toufique
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Volume And Surface Area
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I. Cuboid & II. Cube, In case of cuboid,, Let,, Length= l, Breadth= b, Height= h, I.a. Whole surface area of cuboid= 2(lb+bh+lh) square units., , ©Aspire_Zoology@Toufique
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In case of cube, let each edge of a cube be of length a. So,, Length= a, Breadth= a, Height= a, II.a. Whole surface area of cube= 6a² square units., , ©Aspire_Zoology@Toufique
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II.b. In case of cube total surface area of 4 wall (without roof and floor)= 4a² square units., , ©Aspire_Zoology@Toufique, , I.b. In case of cuboid total surface area of 4 wall (without roof and floor)= (Perimeter of floor, × height) square units.
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I.d. Volume of cuboid= (l×b×h) cubic units., , II.d. Volume of cube= a³ cubic units., , ©Aspire_Zoology@Toufique
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Lets practice, , ©Aspire_Zoology@Toufique, , Q2. Find the length of the longest pole that can be placed in a room 12 m long, 3 m broad and 9, m high?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q3. The volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is 12.8, cubic meters. Find the breadth of the wall?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q4. Water flows into a tank 200 m × 150 m through a rectangular pipe 1.5 m × 1.25 m @ 20, kmph. In what time (in minutes) will the water rise by 2 meters?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q5. The dimensions of an open box are 50 cm, 40 cm and 23 cm. its thickness is 3 cm. if 1 cubic, of metal used in the box weighs 0.5 gms, find the weight of the box?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q6. A rectangular block 6 cm by 12 cm by 15 cm is cut up into an exact number of equal cubes., Find the least possible number of cubes?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q8. Three solid cubes of sides 1 cm, 6 cm and 8 cm are melted to form a new cube. Find the, surface area of the cube so formed?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q9. If each edge of a cube is increased by 50%, find the percentage increase in its surface, area?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q10. Two cubes have their volumes in the ratio 1:27. Find the ratio of their surface areas?
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III. Cylinder, Let,, Radius of base= r, Height/ length= h, a. Curved surface area= 2πrh square units., b. Total surface area= 2πr(h+r) square units., , ©Aspire_Zoology@Toufique
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Lets practice, , ©Aspire_Zoology@Toufique, , Q11. Find the volume, curved surface area and the total surface area of a cylinder with, diameter of base 7 cm and height 40 cm?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q12. If 1 cubic cm of cast iron weighs 21 gms, then find the weight of a cast iron pipe of length 1, meter with a bore of 3 cm and in which thickness of the metal is 1 cm?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q13. The radii of two cylinders are in the ratio 3:5 and their heights are in the ratio of 2:3. Find, the ratio of their curved surface areas?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q14. Find the slant height, volume, curved surface area and the whole surface area of a cone, of radius 21 cm and height 28 cm?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q15. The heights of two right circular cones are in the ratio 1:2 and the perimeter of their bases, are in the ratio 3:4. Find the ratio of their volumes?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q16. The radii of the bases of a cylinder and a cone are in the ratio of 3:4 and their heights are, in the ratio 2:3. Find the ratio of their volumes?
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V. Sphere, Let the radius of the sphere be r. Then,, a. Surface area= 4πr² square units., b. Volume= 4/3 πr³ cubic units., , ©Aspire_Zoology@Toufique
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Lets practice, Q17. Find the volume and surface area of a sphere of radius 10.5 cm?, , ©Aspire_Zoology@Toufique
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Lets practice, , ©Aspire_Zoology@Toufique, , Q18. If the radius of a sphere is increased by 50%, find the increase percent in volume and the, increase percent in the surface area?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q19. How many spherical bullets can be made out of a lead cylinder 28 cm high and with base, radius 6 cm, each bullet being 1.5 cm in diameter?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q20. A cone and a sphere have equal radii and equal volumes. Find the ratio of the diameter, of the sphere to the height of the cone?
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VI. Hemisphere, Let, the radius of a hemisphere be r. Then,, a. Curved surface area= 2πr² square units., b. Total surface area= 3πr² square units., , ©Aspire_Zoology@Toufique
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Lets practice, , ©Aspire_Zoology@Toufique, , Q21. Find the volume, curved surface area and the total surface area of a hemisphere of, radius 10.5 cm?
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Lets practice, , ©Aspire_Zoology@Toufique, , Q22. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Find, the ratio of their volumes?
