Aptitude Volume and Surface Area. Theory

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Brandon Copeland
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1 Aptitude Volume and Surface Area Theory Volume Volume is the amount of space inside a three-dimensional (length, width and height.) object, or its capacity. measured in cubic units. Surfce Area Total area of the surface of a 3D object, measured in square units. Difference between Volume & Surfce Area with example Surface area is the area of the thin sheet of aluminum required to make a soda can. Volume is the amount of soda in the can. CUBOID A cuboid is a three-dimensional shape with a length, width, and a height.

2 Volume of a Cuboid Formula Volume of a Cuboid= length breadth height Example Find the volume of a cuboid of length 20 cm, breadth 15 cm and height 10 cm. here, length - 20 cm breadth - 15 cm height - 10 cm Now calculating Volume of Cuboid Volume of cuboid = length breadth height Now applying this values to formula Volume = 20cm x 15 cm x 10 cm. = 3000 cm 3

3 Exercise Calculate the volume of a cuboid which has sides 4cm, 6cm and 10cm A. 220 cm 3 B. 340 cm 3 C. 180 cm 3 D. 240 cm 3 Answer: 240 cm 3 Explanation : Volume of a Cuboid= length breadth height = = 240 cm 3. Surface area of a Cuboid Formula Surface area of a Cuboid = 2 (lw + wh + hl) Diagonal = l 2 + w 2 + h 2 The Surface area of a cuboid is the sum of the areas of its 6 faces. (i.e) Surface Area of a Cuboid = 2lw + 2wh +2 hl = 2 (lw + wh + hl)

4 Example Find the total surface area and diagonal length of a cuboid with dimensions 8 cm by 6 cm by 5 cm here, l = 8 cm w = 6 cm h = 5 cm First calculating Surface Area Surfacearea of a Cuboid = 2 (lw + wh + hl) = 2 [(8 6)+(6 5)+(5 8)(8 6)+(6 5)+(5 8)] = 2 ( ) = 2(118) = 236 cm 2. Next Calculating diagonal length Diagonal = l 2 + w 2 + h 2 = =

5 = 125 = 5 3 = = 5 5 cm Exercise Find the surface area of a 10cm * 4cm * 3cm brick. A. 154 cm square B. 156 cm square C. 160 cm square D. 164 cm square Answer: 164 cm square Explanation : Surface area of a cuboid = 2(lb+bh+hl) cm square Surface area of a brick = 2( *10) cm square = 2(82) cm square = 164 cm square CUBE A cube has Six sides (faces). Each face is a square of side length 'a'. which are all the same size.

6 Volume of a Cube Formula Volume of a Cube = S 3 So, the formalu of volume is = length breadth height = S x S x S Volume = S 3 Example The length of a cubic shape box is 15 cm. What is the volume of the box? here, The length of cubic box = 15 cm Now calculating Volume of Cube

7 Volume of Cube = S 3 = 15x15x15 = 3375 cm 3 Exercise The side of a cube is 8cm. What is the volume of the cube? A.488 B.512 C.64 D.552 Answer: 512 Explanation : Volume of a Cube = a 3 = = 512 cm 3 Surface Area of a Cube Formula Surfacearea of a Cube = 6a 2 DiagonalofaCube = 3a

8 The area of a square is 6a 2 The total surface area is the sum of the area of each surface. Surface Area = a 2 + a 2 + a 2 + a 2 + a 2 + a 2 = 6a 2 The total surface area is the sum of the area of each surface. The main diagonal of any cube is multiplying the length of one side by the square root of 3. Example Find the total surface area and diagonal of a box whose edges are all 5.5 cm long. here, Edge (face) = 4.5 cm First, calculating Surface area Now calculating Surface area of Cube Surface area of Cube = 6a 2 = 6(5.5) 2 = = cm 2

9 Now calculating the diagonal of the cube Diagonal of a Cube = 3a = = = cm Exercise The length of a cube is 8.7 cm, find the surface area of a cube? A cm 2 B cm 2 C cm 2 D cm 2 Answer: cm 2 Explanation : Surfacearea of a Cube = 6a 2 = 6 (8.7) 2 = cm 2 CONE A solid (3-dimensional) object with a circular flat base joined to a curved side that ends in an apex point.

10 Volume of a Cone Formula Volume of a Cone = V = 1 3 Bh or V = 1 3 πr2 h Where B= πr 2 The volume V of a cone with radius r is one-third the area of the base B times the height h where π is a number that is approximately equals to 3.14 or 22/7 Example Calculate the volume of a cone if the height is 12 cm and the radius is 7 cm. here, Height of the cone = 12 cm Radius of the cone = 7 cm.

11 V = 1 3 πr2 h V = V = 616cm 3 Exercise The height and the slant height of a cone are 21 cm and 28 cm respectively. Find the volume of the cone. A cm 3 B cm 3 C cm 3 D cm 3 Answer: 7546 cm 3 Explanation : l 2 = r 2 + h 2 r = l 2 h 2 = = 7 7 volume of the cone = 1 3 πr2 h = =7546 cm 3

12 Surface Area of the Cone Formula Curved surface area of the cone = πrlπrl Total Surface area of a Cone = πrl + πr 2 or πr(l+r) slant height l = h 2 + r 2 The Curved surface area of a Cone is the surface area of the outside of the cone The Total surface area of a cone = The surface area of the outside of the cone + The surface area of the circle r - is the radius, h - is the height, l - is the slant height Example The diameter of a cone is 16 cm and its height is 6 cm. Find the Curved surface area and Total surface area of cone. here, d = 16 => r = d/2 = 16/2 = 8cm r= 8 cm h = 6 cm l = h 2 + r 2 = l = = l = = l = l = 100 = 10 cm

13 l= 10cm First calculating Curved surface area of the cone Curved surface area of the cone = πrl = = cm 2 Now calculating Total Surface area of a cone Total Surface area of a Cone = πrl + πr 2 = ( ) = = cm 2 Exercise Find the total surface area of a right cone if the radius is 6 inches and the slant height is 10 inches. A inches 2 B inches 2 C inches 2 D inches 2 Answer: inches 2

14 Explanation : Total Surface area of a Cone = πr(l + r) = (10 + 6) = inches 2 CYLINDER A cylinder is a closed solid that has two parallel (usually circular) bases connected by a curved surface. Volume of a Cylinder Formula volume of a cylinder = πr 2 h where, r: Radius of a cylinder, h: Height of a cylinder and π is equal to 22/7 or 3.14 Example Find the Volume of a cylinder whose height is 28 cm and radius 6 cm here,

15 Height of the cylinder = 28 cm Radius of the cylinder = 6 cm Now calculating Volume of cylinder volume of a cylinder = πr 2 h = 22 / = cm 3 Exercise Calculate the volume of a cylinder of height 12cm and radius 6cm. A cm 3 B cm 3 C cm 3 D cm 3 Answer: cm 3 Explanation : volume of a cylinder = πr 2 h = = cm 3

16 Surface Area of a Cylinder Formula curved surface area of a cylinder = 2πrh Total surface area of a cylinder = 2πrh + 2 πr 2 The curved surface area is defined as the area of only curved surface, leaving the circular top and base The Total surface area of the curved surface as well as the bases. The Total surface area = curved surface area + (2 area of circle) The area of a circle is πr 2, so the combined area of the two disks is twice that, or 2 πr 2 Example The diameter of the base of a cylinder is 10 cm and the height is 8 cm. Find the surface area of the solid cylinder here, Diameter (base) = 10 cm Radius = Diameter /2 = 10/2 = 5 cm r = 5m First calculate the curved surface area

17 curved surface area of a cylinder = 2πrh = 2 22/7 5 8 = cm 2 now find the total surface area of a cylinder Total surface area of a cylinder =2πrh + 2 πr 2 = = cm 2 Exercise Find the surface area of the solid cylinder. The diameter of the base of a cylinder is 30 cm and the height is 17 cm. A cm 2 B cm 2 C cm 2 D cm 2 Answer: cm 2 Explanation : Total surface area of a cylinder = 2πr(r + h) = ( ) = cm 2

18 SPHERE A 3-dimensional object shaped like a ball. Every point on the surface is the same distance from the centre. Volume of a Sphere Formula volume of a sphere V = 4 3 πr3 Where, r: Radius of a Sphere The volume V of a sphere is four-thirds times pi times the radius cubed. Example Calculate the volume of sphere with radius 4 cm. here, Radius of the sphere = 4 cm Now calculating Volume of sphere volume of a sphere V = 4 3 πr3

19 = = cm 3 Exercise Find the volume of a sphere of radius 11.2 cm. A cm 3 B cm 3 C cm 3 D cm 3 Answer: cm 3 Explanation : Required volume = 4 3 πr3 = = cm 3 Surface Area of a Sphere Formula Surface area of a sphere = 4π r 2 Where r is the radius of the sphere. The surface area of a sphere is exactly four times the area of a circle with the same radius.

20 The area of a circle = π r 2, and the surface area of a sphere = 4π r 2 Example Find the surface area of a sphere of radius 25cm. here, r = 25 cm Surface area of a sphere = 4π r 2 π = = = 7855 cm 2 Exercise Find the surface area of a sphere of radius 7 cm A. 462 cm 2 B.487 cm 2 C. 587 cm 2 D.616 cm 2 Answer: 616 cm 2

Explanation : The surface area of a sphere of radius 7 cm would be 4πr 2 = 4 22/7 7 7 cm 2 = 616 cm 2 HEMISPHERE A hemisphere is half a sphere, with one flat circular face and one

21 Explanation : The surface area of a sphere of radius 7 cm would be 4πr 2 = 4 22/7 7 7 cm 2 = 616 cm 2 HEMISPHERE A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face. Volume of a Hemisphere Formula Volume of a sphere V = 2 3 πr3 The volume of a hemisphere is equal to two thrids of the cube of the radius by pi. Where, r: Radius of a Sphere

22 Example Find the volume of the hemisphere, whose radius is 10 cm. here, Radius of the sphere = 10 cm Now calculating Volume of sphere volume of a hemisphere V = 2 3 πr3 = = 23 x 3.14 x 1000 = 23 x 3140 = cm 3 Exercise A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain? A cm 3 B cm 3 C cm 3 D cm 3

23 Answer: 89.8 cm 3 Explanation : The volume of water the bowl can contain = 2 3 πr3 = = 89.8 cm 3 Surface Area of Hemisphere Formula Curved surface area of Hemisphere = 2πr 2 Total surface area of Hemisphere = 3πr 2 The hemisphere is to include the base then the surface area is 2 π r 2 + π r 2 = 3 r 2 Example Find the curved surface area and total surface area of a hemisphere having the radius of 7 cm? here, r = 7 cm First calculate the curved surface area Curved surface area of Hemisphere = 2πr 2

24 = = 308 cm 2 Now calculate the total surface area Total surface area of Hemisphere = 3πr 2 = = 462 cm 2 Exercise Find the curved surface area of a hemisphere of radius 21 cm. A cm 2 B cm 2 C cm 2 D cm 2 Answer: 2772 cm 2 Explanation : The curved surface area of a hemisphere of radius 21 cm would be = 2πr 2 = 2 22/ cm 2 = 2772cm 2 More Questions at::

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth.

Volume of Cylinders. Volume of Cones. Example Find the volume of the cylinder. Round to the nearest tenth. Volume of Cylinders As with prisms, the area of the base of a cylinder tells the number of cubic units in one layer. The height tells how many layers there are in the cylinder. The volume V of a cylinder