Algebra Tutorial-simple Shapes

Algebra Tutorial Simple shapes Cube, Rectangular Prism, Cylinder and Sphere

These shapes are most basic ones we use in various objects. let us explore two properties for each shape: Volume &Surface Area We learn to calculate these one with the other.

for each shape ; and also

compare

CUBE A cube is an object Length, width and height are

with all the three equal.

equal.

sides

Volume V = a xa x a V= a ^3 is the side of the cube. Surface area: A cube has six sides, all of equal area a xa. Therefore Surface area: SA= 6 a.a Where a

1 Find the volume and surface area of a box of cubical shape with side a = 2 feet. Volume V = 2 x 2 x2 = 8 cubic feet. Note the unit: cubic feet. What is a cubic feet? Well- it is the volume of a cube with a= 1 foot. This is a convenient measure of volume of these shapes. We will use cubic feet [or cubic meter ] for volume ume inin ann all calculations. calculations. Surface Area= 6 x 2 x 2 = 24 square feet. Note: For area, we use square feet or square meter as units. 2 John has a truck with internal ternalvolume vol ume of 10000cubi of 8000 cu.feet feet How many cubical boxes of side 2 feet he can load into his truck? Volume of each box = 2 x 2 x 2= 8 cu.feet Internal Volume of a truck = 8000 cu ft. No of cubical boxes you can fit in = 8000/8 = 1000 boxes. Rectangular Prism It is a shape in which length ,width and height are not the same. The simplest prism is a square bottom with some height. In this length and width are the same ,but hieght height isis different. different. Volume = area of the square face x height Suppose the bottom of a toy box is a square with side 8inches. The height is 6 inches. Find its volume. Volume = area of the square face x height V= 8 x 8 x 6 384 cubic inches. What is a cubic inch? It is the volume of a cube with one inch side.! Surface area : If th ebox has lid also, find its surface area. Area of bottom and top side [squares] = 2 x 8 x 8 = 128 sq in Area of the four sides = 4 x 8 x6 =

192 sq in Total

surface area =

324 sq in

General Prism The general case: Arenctangular prism in which length,width and height are all different. Volume = length x widh x height = L x W x H Surface area = 2 ( L +W) + 2 ( W x H) + 2 ( H x L) 3 A shoe box has these dimensions: length = 9 in width =6in Height = 5 in Find the volume & surface area of the box. Volume = 9 x 6 x 5 V = 270 cu in Surface area = 2(9+5) + 2(5 x 6) + 2(6+9) SA = 2x14 + 2 x 30 + 2 x 15 SA= 28 + 60 + 30 = 118 sq in Applied Problems Airconditioning and ventilation engineers use these fromulas formulastoto calculate the volume of air they have to cool or pump out. Space heating firms have to calculate the air psace space they . have to heat. Packing companies and painters have to find volumes anddsurfae surface areas areas almost daily several several times. 4 Find the air volume of a large auditorium to be air-conditioned.Inside dimensions are: length = 100 feet width = 60 feet height = 30 feet. Volume = 100 x 60 x 30 = 180000 cu.ft 5 A class room has intenal length of 20 feet, width 12 feet and height 12 feet. Linda wants to paint the four walls and the roof.The cost of painting is 60 cents per sq ft. Find the cost estimate for the paint job. Surface area : 4 walls = 4 x 20 x 12 960 sq feet Roof = 20 x 12 = 240 Total area = 1200 sq feet Cost of painting= 1200x0.6 720 $ 6 James 1x,2x,3x

wants to make a box either as a cube or rectangular box with width: height:length in the ratios The side of the cube would be 10 feet. Find the dimensions of rectangular box of the same volume . Then find out which one ---cube or rectanular would need more material or surface area to make the box. Volume of cubical box= 10x 10 x10 = 1000 cu ft Volume of rectangular box = X.2X.3X = 6 x^3 = 1000 What is X now? x^3 = 1000/6= 133 cubic feet 5.1 width X = 5.1 feet height 2x= 10.2 ft length 3x= 15.3 SA of cube = 6 *10*10 = 600 sq feet SA of rect box SA = 2 (5.1 X 10.2) + 2 ( 10.2 X15.3) + 2(15.3 x 5.1)

572.22

Cylinder A cylinder has a circular base or cross section. The dimensions of a cylinder are: diameter (or raidus of the base) & its height. Volume = base area x height Volume = pi x r x r x h where r is the radius and h the height You know pi" already: pi = 3.14159 pi=3.14 [approximately] Surface area of a closed cylinder SA = base area + lid area + lateral (side) surface area SA= pi x r x r + pi x r x r + 2x pix r x h SA = pi [ 2 r.r + r.h] For open cylinder:

SA = pi [ r.r +r.h]

[no lid here!]

1 A soup can is in the shape of a cylinder.Its diameter is 8 in and height is 6 in. Find its volume and surface area Volume V = pi x 4 x 4 x 6 V= 3.14* 16*6 301.44 cu in Surfae area= pi[4x4 + 4 x6] pi[40] 125.6 sq in Applied Problems 2 Cookie boxes come in different types of cylinders. let us take three types : 1. height /diameter = 1 2 height/diameter = 2 3 height/diameter = 0.5 Linda wants to try these beoxes: boxes: flat boxes to tall boxes. She needs a volume of 200 cu in. Find the daimeters and heights for the three shapes and find out hich one would require less material to make ---that is less surface area.

Type 1

Type 2

Volume = 200 cuin V= pi x dia x dia x ht/4 200 = 3.14 x dia^3/4 200 = (9.42/4)xdia^3 dia^3 = 84.93 dia= 4.39 in dia = 4.4 in radius = 2.2 in Surface area = SA= 3.14(2*2.2*2.2 + 2.2*2.2) 45.59 sq in h/d = 2 Volume = 200 cuin 200 = 3.14xdiaxdiax2xdia/4 dia^3 = 800/6.28 127.39 dia= 5.02 in height= 10.04 in h/d=1

SA = 3.14(2.56x2.56 + 2.56x5.02) 60.93 sq in Type 3

h/d=0.5

Volume = 200 = 3.14Xdiaxdiax).5Xdia/4 V= 200 = (3.14/8) dia^3 dia ^3 = 509.55 dia= 7.99 in radius= 3.99 height = 3.99 SA= 3.14(2x 3.994x3.994 + 3.994 x 3.994) 150.27 sqin You see that a flat box with height is equal to diameter/2 is costlier to build as the surface area is more. Expensive cookies come in such boxes. When h =d height = diameter , the ost is minimum…. A chunky cylinder is least expensive. 2

Sphere

A sphere is taken as a perfect figure by ancient Greek Planets and the earth is almost a sphere. The ancients The Music of the spheres". The volume of a sphere: V= 4 x pi x r xr xr /3 Surafce area

mathematicians. spoke of

SA= 4 x pi x r x r

A sphere gives the smallest surface area for a given volume , compared to other shapes. next comes a cylinder with h/d ratio of 1, that is; height is equalt ot diameter. 1 Find the volume and surface area of hemispherical dome of a sports stadium with diameter of 200 feet. radius r = 100 feet Volume V = 4x 3.14 x 100 x 100 x100/3 ### cuft Surface area = 4 x 3.14 x 100 x 100 SA= 125600 sq ft 2 Engineers often combine the shapes of cylinder and spheres to construct vessels to reduce the surface area and have higher volume..that is to optimise the thespae space inside. inside. If the surface area is reduced ,less materila s material is required is required to construct. Also such shapes with low surface area will help to reduce heat losses from the vessels,if the vessels have to be heated for chemical reactions. What is more, if the surface area is reduced, we would need less paint material or other coatings to protect the surface. Find the volume and surface area of a chemical reactor vessel with cylindrial midsection of diameter 12 feet and length 20 feet and two hemispherical domes at the two ends with the same diameter. Volume:

Cylindrical section:

V-cyl = pi x r x r xh Vcyl = 3.14 x 6 x 6 x 20 3391.2 cu ft V _sph = 4 x 3.14 x 6 x6x6/3

904.32 cu ft Total volume = 3391.2+904.32 4295 cu ft Surface area SA-cyl = 2xpi x r x h SA-cyl = 2 x 3.14 x 6 xx12 20 753.6 SA-sph = 4 x pi x r x r SA-sph =4 x 3.14 x 6 x 6 452.16 Total surface area SA= 1205.8 sq in 3 A spherical the diameter d.

{lateral surface only}

ball of diameter d is kept tightly in a cubical box of side equal to What is the % utilisation of the box volume inpacking this way? What is the volume of wasted space?

Volume of the cubical box V = d^3 V = 8 r ^3 where r is the radius Volume of the ball = 4 * pi * r ^3/3 4.19 r^3 Ratio of vol of ball/vol of box =

4.1867/8 0.52 Therefore when we pack balls of same diameter in a cubical box, we have volume efficiency of 52.3% only. 47.3% or nearly half the space of the box is wasted/! Do-It -Yourself problems 4 A submarine is in the shape of a cigar, which can be approximated to a cylinder plus two hemispherical ends. If the diameter of the cylindrical portion is 20 feet and length 100 feet, the dia of the dome like also 20 feet, find the totaltotal surfae surface area area of theofsubmarine. the submarine

5 A grain silo is a cylinder with a hemispherical bottom. If the daimeter of the cylinder is 20 feet and the height 50 feet ,and the daimeter of the heispherical section is also the same find the volume and surface area of the silo. 6 The volume to SA of a sphere is 4 x pi x r x r xr/( 3 x 4 x pi x r xr ) V /SA r/3 Compare two spheres of diameter 6 in and 12 in. V/SA = 1 and 2 respectively But the volume of 12 in sphere is 8 times that of 6 in sphere. Therefore it is beneficial to use large spherical bodies instead of several small spheres to have the same volume. 7 The earth has a radius of 3600 km. Find its volume. 8 A light bulb has a with neon gas,

shape of sphere, with diameter 4 inches.If it filled find the volume of the gas inside the bulb.

y cubical boxes

formulas to

daily

dimensions are:

s 60 cents per sq ft.

with width: height:length in the ratios of

erial or surface area to make the

2 is costlier to build as the

mathematicians.

compared to equalt ot diameter. of a sports stadium

to construct vessels to reduce the

es from the vessels,if the

nt material or other coatings to

with cylindrial rical domes at the two ends

{lateral surface only}

n a cubical box of side equal to ox volume inpacking this way?

x, we have volume efficiency of

n be approximated to

gth 100 feet, the dia of the dome like ends are

If the daimeter of the cylinder is heispherical section is also the same,

stead of several small spheres

eter 4 inches.If it filled