Finding Volume in Math

Finding volume in math article deals with the definition of volume and the formula for the various shapes and the model problems related to volume of various shapes.

Looking out for more help on Units of Volume in algebra by visiting listed websites.

Definition of Volume:

Volume is always measured in cube.units. It is defined as the space that is occupied by the entire three-dimensional shape.

Volume formula for various shapes in math.

Volume of the cube is a3 cubic units

Here “a” is the side of the cube.

Volume of the rectangular prism is L*B*H cubic units

Here L is length, B is breadth and H is the height.

Volume of the cylinder is BH cubic units

Here B is the base area and H is the height

Volume of the pyramid is `(1/3)` BH cubic units.

Here B is the base area and H is the height

Volume of the cone is `(1/3) ` BH cubic units.

Here B is the base area and H is the height

Model problem for finding volume in math.

Problem:

1.  Finding volume of the cube when the side measures about 5 cm in length.

Solution:

Since it is a cube, the formula

Volume of the cube is a3 cubic units

Here “a” is the side of the cube.

Here a= 5cm

The volume of the cube =53

=125 cm3

The volume of the cube is125cm3

2. Finding volume of the rectangular prism when the length = 8cm, breadth = 5cm and the height is 4cm.

Solution:

Since it is a rectangular prism, the formula

Volume of the rectangular prism is L*B*H cubic units

Here L is length, B is breadth and H is the height
L= 8cm

B= 5cm

H = 4cm

Volume of the rectangular prism = (8* 5*4) cubic units.

= 160 cm3

Volume of the trectangular prism is 160cm3

3. Find the volume of the cylinder whose base area is 46m^2 and the height of the cylinder is 10m

Solution:

Base area of the cylinder is 46m2

Height of the cylinder is 10m

Formula:

Volume of the cylinder is BH cubic units

Here B is the base area and H is the height

= 46*10

= 460 m3

the volume of the cylinder is 460m3

LENGTH OF MEASURING SYSTEM

The unit of length is meter. In length of measuring system length can be measured by using various terms such as millimeter, centimeter, decimeter, meter, decameter, hectometer and kilometer.

Some of the length of measuring system

10 millimeter is equal to 1 centimeter

10 centimeter is equal to 1 decimeter

10 decimeter is equal to 1 meter

10 meter is equal to 1 decameter

10 decameter is equal to 1 hectometer

10 hectometer is equal to 1 kilometer

By knowing these values we can convert the measures of the length.

Similarly some other important conversions of length of measuring system is

1000 millimeter is equal to 1 meter

1000 meter is equal to 1 kilometer

Similarly inches, feet, yards and miles are also the measurements of length of measuring system.

CONVERSION TABLES OF LENGTH OF MEASURING SYSTEM

1 feet is equal to 12 inches

1 yard is equal to 36 inches

1 yard is equal to 3 feet

1 mile is equal to 5280 feet

We have to divide when converting a larger unit and similarly when we have to convert a smaller unit we have to multiply it.

EXAMPLES ON LENGTH OF MEASURING SYSTEM

Example 1: Convert 5000 meters to kilometer

Solution

We know that 1000 meter = 1 kilometer

So

5000 meter = 5000/1000 = 5 kilometer



Example 2: How many centimeters equal to 450 meter?

Solution:

1 meter = 100 centimeter
so

450 meter = 450 * 100

= 450000 centimeter

Example 3: Convert 6 feet to yards and inches

Solution

We know that 1 feet = 12 inches

3 feet = 1 yard

So

6 feet = 6/3 = 2 yards

6 feet = 6 * 12 = 72 inches

Example 4: Convert 5 miles into feet’s

Solution

1 mile = 5280 feet

So 5 miles = 5 * 5280

= 26400 feet’s.

PROBLEMS ON LENGTH OF MEASURING SYSTEM

Convert 5 feet to inches

Convert 20 yards into feet

Convert 100 meter to millimeter

Convert 52 kilometer to decameter

Convert 3 kilometer to millimeter

Answers:

60 inches

60 feets

100000 mm

5200 decameter

3000000 mm

Coplanar Geometry

Three or more points, lines or any other geometric shapes that lie on the common plane are knows as Coplanar.

Geometric substance lying in a same plane are said to be coplanar. In a plane three noncollinear points are some extent coplanar. Four points are coplanar, defined by them is 0, lying in the same plane. Example, any set of three points in plane are coplanar. Let us see coplanar geometry in brief.

Conditions for coplanar:

Coplanarity is corresponding to the statement that the pair of lines determined by the four points is not skew, and can be equivalently stated in vector form as

`|[x1,y1,z1,1],[x2,y2,z3,1],[x3,y3,z3,1],[x4,y4,z4,1]| = 0`

(x3 - x1).[(x2 - x1) × (x4 - x3)] = 0

The coplanar not only for four points it is also for two or three points.

An arbitrary number of n  points x1 , ..., xn can be checked for coplanarity by finding the point-plane distances of the points x4, ...,xn from the plane determined by (x1,x2,x3), and checking if they are all zero. Therefore, the points are all coplanar.

A set of n vectors v is coplanar if the nullity of the linear mapping defined by v  has dimension 1, the matrix rank of v (or equivalently, the number of its singular values) is n-1.

Parallel lines in three-dimensional space are said to be coplanar, but skew lines are not.

In this article we see the coplanar of lines on same plane. Using the line equation we find the coplanar for the three lines on a same plane.

Example problem for coplanar:

Example: prove that three lines are coplanar, equation lines are 3x + 2y = 0, 3x + 3y = 3, 2x + 2y = 2

Solution:

Given: 3x + 2y = 0 -------------(1)

3x + 3y = 3 -------------(2)

2x + 2y = 2 --------------(3)

`|[x1,y1,c],[x2,y2,c],[x3,y3,c]| = 0`

`|[3,2,0],[3,3,3],[2,2,2]| = 0`

3[(3×2) - (3×2)] - 2[(3×2) - (3×2)] +0[(3×2) - (3×2)] = 0

3(6 - 6) -2(6 - 6) -0(6 - 6) =0

0 = 0

Hence proved thus the three lines are coplanar.

To Logarithmic Table

Logarithmic Properties plays an important role in complex calculations in math. We can perform big calculations in math,physics , engineering using logarithms. It give accurate answer as the calculator. These calculations are carried out with the help of logarithmic table.We will see the logarithmic table. The ways of reading the logarithmic table and how to use it in calculations.

Any number x in standard form is written as x = m x 10^p where 1 `<=` m<10 p="">Taking log on both sides we get
log₁₀x = log₁₀(m x 10p) = log₁₀m + plog₁₀10
             = log₁₀m + p
Here p is the characteristic of log x and log₁₀m is called the mantissa of logx

How to find the logarithm of a number:

• Step 1: Write the number in the standard form.

For example 431.5 = 4.315 x 10²

• Step 2: Find the characteristic p of the logarithm.

Here p = 2

• Step 3: Find the mantissa from the table.

To find the log of 4.315 from the table. log 4.31 is 0.634473 `~~` 0.6350. We take the approximate value.

log 4.315 = p + logm = 2+0.6350 = 2.6350

Logarithmic table from 1 to 4.99

1.000	0.00000000			2.00	0.3010300	3.00	0.4771213	4.00	0.6020600
1.001	0.00043408			2.01	0.3031961	3.01	0.4785665	4.01	0.6031444
1.002	0.00086772			2.02	0.3053514	3.02	0.4800069	4.02	0.6042261
1.003	0.00130093			2.03	0.3074960	3.03	0.4814426	4.03	0.6053050
1.004	0.00173371			2.04	0.3096302	3.04	0.4828736	4.04	0.6063814
1.005	0.00216606			2.05	0.3117539	3.05	0.4842998	4.05	0.6074550
1.006	0.00259798			2.06	0.3138672	3.06	0.4857214	4.06	0.6085260
1.007	0.00302947			2.07	0.3159703	3.07	0.4871384	4.07	0.6095944
1.008	0.00346053			2.08	0.3180633	3.08	0.4885507	4.08	0.6106602
1.009	0.00389117			2.09	0.3201463	3.09	0.4899585	4.09	0.6117233
1.010	0.00432137	1.10	0.0413927	2.10	0.3222193	3.10	0.4913617	4.10	0.6127839
1.011	0.00475116	1.11	0.0453230	2.11	0.3242825	3.11	0.4927604	4.11	0.6138418
1.012	0.00518051	1.12	0.0492180	2.12	0.3263359	3.12	0.4941546	4.12	0.6148972
1.013	0.00560945	1.13	0.0530784	2.13	0.3283796	3.13	0.4955443	4.13	0.6159501
1.014	0.00603795	1.14	0.0569049	2.14	0.3304138	3.14	0.4969296	4.14	0.6170003
1.015	0.00646604	1.15	0.0606978	2.15	0.3324385	3.15	0.4983106	4.15	0.6180481
1.016	0.00689371	1.16	0.0644580	2.16	0.3344538	3.16	0.4996871	4.16	0.6190933
1.017	0.00732095	1.17	0.0681859	2.17	0.3364597	3.17	0.5010593	4.17	0.6201361
1.018	0.00774778	1.18	0.0718820	2.18	0.3384565	3.18	0.5024271	4.18	0.6211763
1.019	0.00817418	1.19	0.0755470	2.19	0.3404441	3.19	0.5037907	4.19	0.6222140
1.020	0.00860017	1.20	0.0791812	2.20	0.3424227	3.20	0.5051500	4.20	0.6232493
1.021	0.00902574	1.21	0.0827854	2.21	0.3443923	3.21	0.5065050	4.21	0.6242821
1.022	0.00945090	1.22	0.0863598	2.22	0.3463530	3.22	0.5078559	4.22	0.6253125
1.023	0.00987563	1.23	0.0899051	2.23	0.3483049	3.23	0.5092025	4.23	0.6263404
1.024	0.01029996	1.24	0.0934217	2.24	0.3502480	3.24	0.5105450	4.24	0.6273659
1.025	0.01072387	1.25	0.0969100	2.25	0.3521825	3.25	0.5118834	4.25	0.6283889
1.026	0.01114736	1.26	0.1003705	2.26	0.3541084	3.26	0.5132176	4.26	0.6294096
1.027	0.01157044	1.27	0.1038037	2.27	0.3560259	3.27	0.5145478	4.27	0.6304279
1.028	0.01199311	1.28	0.1072100	2.28	0.3579348	3.28	0.5158738	4.28	0.6314438
1.029	0.01241537	1.29	0.1105897	2.29	0.3598355	3.29	0.5171959	4.29	0.6324573
1.030	0.01283722	1.30	0.1139434	2.30	0.3617278	3.30	0.5185139	4.30	0.6334685
1.031	0.01325867	1.31	0.1172713	2.31	0.3636120	3.31	0.5198280	4.31	0.6344773
1.032	0.01367970	1.32	0.1205739	2.32	0.3654880	3.32	0.5211381	4.32	0.6354837
1.033	0.01410032	1.33	0.1238516	2.33	0.3673559	3.33	0.5224442	4.33	0.6364879
1.034	0.01452054	1.34	0.1271048	2.34	0.3692159	3.34	0.5237465	4.34	0.6374897
1.035	0.01494035	1.35	0.1303338	2.35	0.3710679	3.35	0.5250448	4.35	0.6384893
1.036	0.01535976	1.36	0.1335389	2.36	0.3729120	3.36	0.5263393	4.36	0.6394865
1.037	0.01577876	1.37	0.1367206	2.37	0.3747483	3.37	0.5276299	4.37	0.6404814
1.038	0.01619735	1.38	0.1398791	2.38	0.3765770	3.38	0.5289167	4.38	0.6414741
1.039	0.01661555	1.39	0.1430148	2.39	0.3783979	3.39	0.5301997	4.39	0.6424645
1.040	0.01703334	1.40	0.1461280	2.40	0.3802112	3.40	0.5314789	4.40	0.6434527
1.041	0.01745073	1.41	0.1492191	2.41	0.3820170	3.41	0.5327544	4.41	0.6444386
1.042	0.01786772	1.42	0.1522883	2.42	0.3838154	3.42	0.5340261	4.42	0.6454223
1.043	0.01828431	1.43	0.1553360	2.43	0.3856063	3.43	0.5352941	4.43	0.6464037
1.044	0.01870050	1.44	0.1583625	2.44	0.3873898	3.44	0.5365584	4.44	0.6473830
1.045	0.01911629	1.45	0.1613680	2.45	0.3891661	3.45	0.5378191	4.45	0.6483600
1.046	0.01953168	1.46	0.1643529	2.46	0.3909351	3.46	0.5390761	4.46	0.6493349
1.047	0.01994668	1.47	0.1673173	2.47	0.3926970	3.47	0.5403295	4.47	0.6503075
1.048	0.02036128	1.48	0.1702617	2.48	0.3944517	3.48	0.5415792	4.48	0.6512780
1.049	0.02077549	1.49	0.1731863	2.49	0.3961993	3.49	0.5428254	4.49	0.6522463
1.050	0.02118930	1.50	0.1760913	2.50	0.3979400	3.50	0.5440680	4.50	0.6532125
1.051	0.02160272	1.51	0.1789769	2.51	0.3996737	3.51	0.5453071	4.51	0.6541765
1.052	0.02201574	1.52	0.1818436	2.52	0.4014005	3.52	0.5465427	4.52	0.6551384
1.053	0.02242837	1.53	0.1846914	2.53	0.4031205	3.53	0.5477747	4.53	0.6560982
1.054	0.02284061	1.54	0.1875207	2.54	0.4048337	3.54	0.5490033	4.54	0.6570559
1.055	0.02325246	1.55	0.1903317	2.55	0.4065402	3.55	0.5502284	4.55	0.6580114
1.056	0.02366392	1.56	0.1931246	2.56	0.4082400	3.56	0.5514500	4.56	0.6589648
1.057	0.02407499	1.57	0.1958997	2.57	0.4099331	3.57	0.5526682	4.57	0.6599162
1.058	0.02448567	1.58	0.1986571	2.58	0.4116197	3.58	0.5538830	4.58	0.6608655
1.059	0.02489596	1.59	0.2013971	2.59	0.4132998	3.59	0.5550944	4.59	0.6618127
1.060	0.02530587	1.60	0.2041200	2.60	0.4149733	3.60	0.5563025	4.60	0.6627578
1.061	0.02571538	1.61	0.2068259	2.61	0.4166405	3.61	0.5575072	4.61	0.6637009
1.062	0.02612452	1.62	0.2095150	2.62	0.4183013	3.62	0.5587086	4.62	0.6646420
1.063	0.02653326	1.63	0.2121876	2.63	0.4199557	3.63	0.5599066	4.63	0.6655810
1.064	0.02694163	1.64	0.2148438	2.64	0.4216039	3.64	0.5611014	4.64	0.6665180
1.065	0.02734961	1.65	0.2174839	2.65	0.4232459	3.65	0.5622929	4.65	0.6674530
1.066	0.02775720	1.66	0.2201081	2.66	0.4248816	3.66	0.5634811	4.66	0.6683859
1.067	0.02816442	1.67	0.2227165	2.67	0.4265113	3.67	0.5646661	4.67	0.6693169
1.068	0.02857125	1.68	0.2253093	2.68	0.4281348	3.68	0.5658478	4.68	0.6702459
1.069	0.02897771	1.69	0.2278867	2.69	0.

Algebra is widely used in day to day activities watch out for my forthcoming posts on Variance Statistics Formula and Definition of least Common Multiple. I am sure they will be helpful.

Equilateral Square

Equilateral square is a geometry figure, square has four sides all the four sides are equal in length then it is said to be equilateral square. In two dimensions square we find only the area. The area is measured in terms of square units. In three dimensions, square is called cube it is also equilateral for that we find the volume only the figure has length, width and height.

Formula for equilateral square:

Area Formulas for equilateral square

The area of a square can be found by multiply the base times itself

Area of square = side × side or a2

Volume of cube =side x side x side a3



AB = BC = CD = AD

Then it is Equilateral Square

Problem for area of square:

1) Find the area of the square, length of one side id 9cm

Solution:

Given: Side a = 9cm

Area of square = a2

= 9 × 9

Area of square = 81cm2

Problems for volume of square

2) Find the volume of the cube its side is 9cm

Solution:

Given: side =9cm

Volume of cube = a3

= a × a × a

= 9 × 9 × 9

= 729cm3

3) Find the area of the square, length of one side id 4m

Solution:

Given: Side a = 4m

Area of square = a2

= 4 × 4

Area of square = 16m2

Problems for volume of square

4) Find the volume of the cube its side is 12m

Solution:

Given: side = 12m

Volume of cube = a3

= a × a × a

= 12 × 12 × 12

= 1728 m3

Examples for equilateral square:

Prove by distance formula that the square is a equilateral square:



Formula  = √(x2 - x1)2+(y2 - y1)2

A(0,0) B(0,1) C(1,1) D(1,0)

Length of AB = √(x2 - x1)2+(y2 - y1)2

AB = √(0-0)2 + (1-0)2

= √0+(1)2

= 1

Length of BC = √(x2 - x1)2+(y2 - y1)2

BC = √(1-0)2 + (1-1)2

= √(1) + (0)

= 1

Length of CD = √(x2 - x1)2+(y2 - y1)2

CD = √(1-1)2 + (0-1)2

= √(0) +(-1)

= 1

Length of AD = √(x2 - x1)2+(y2 - y1)2

AD =√ (1-0)2 + (0-0)2

= √(1) + 0

= 1

Therefore AB = BC = CD = AD =1

All the sides are equal then the square is equilateral square.

Geometry Area Perimeter

In that geometry, the area and perimeter can be calculated for two dimensional and three dimensional shapes. The quantity of surface occupied by a plane figure is called its area. The unit of area is called as square units. The length of the boundary of any closed figure is called as perimeter. The unit of perimeter is meter units. In this article we shall see about the geometry formula for finding the area and the perimeter of simple closed figures like triangle and rectangle, square, and circle. The geometry area and perimeter example problems and practice problems are given below.

Formulas:

Area of square = a2 square units

Perimeter of square = 4 * a

Area of rectangle = length * Width

Perimeter of rectangle = 2( Length + Width)

Area of circle = `Pi` r2

Circumference of circle = 2`Pi` r

Example problems - Geometry area perimeter:

Example problem 1: Find out the area and perimeter of a geometry circle with 15 cm radius.

Solution:

Given:    Radius = 15 cm

Formula:       Area of Circle = radius2

= 3.14 * 152

= 3.14 * 225

On solving this, we get

= 706.5 cm2.

The Circumference of Circle is equal to the multiplication of 2, and ∏ and the radius of the circle

Formula:              Circumference of the Circle = 2 *Π* radius

= 2 * 3.14 * 15

On solving this, we get

= 94.2 cm.

Example problem 2:

Find the geometry surface area of cube for the side is 9 inches.

Solution:

Area of cube A = 6 (side) 2 = 6 * 92

A = 6 * 81

A = 486 square inches.

Example problem 3:

Find the curved surface area, for hemisphere whose radius is 5

Solution:

Curved surface area = 2 * ∏ * r2

= 2 * 3.14 * (5)2

= 2 * 3.14 * 25

= 157

Practice problems - Geometry area perimeter:

Practice problem 1:

Find the area of  circle, For radius is 8

Answer: 200.96

Practice problem 2:

Determine the area and perimeter of rectangle, for length and width are 7meter and 5 meter respectively.

Answer: Area of rectangle = 35 m2

Perimeter of rectangle = 24 m

About Square Root

Square root of a number it can be represented by `sqrt(y)` . The square root of number is also written as exponent form `sqrt(y)` which is equal to y^1/2
                Square root of a number 25 which is equal to `sqrt(25)` =25^1/2=5.
                In this article we shall discuss about the some square roots of the numbers.

Square root table

                The following table shows the square root of the some numbers.
                Number (x)        square (x²)          Square root (x^1/2)
                1                              1                              1.000
                2                              4                              1.414
                3                              9                              1.732
                4                              16                           2.000
                5                              25                           2.236
                6                              36                           2.449
                7                              49                           2.646 
                8                              64                           2.828 
                9                              81                           3.000 
                10                           100                           3.162

Problem 1:

Find the Square root value of 150
Solution:
                Write prime factors for 150 = 2 * 3 * 5 * 5
                `sqrt(150)` = `sqrt(2 * 3 * 5 * 5)`
                                   = `sqrt(2 * 3 * 5^2)`
             `sqrt(2 * 3 * 5^2) ` we can written as (2 * 3 * 5²)^1/2
                (2 * 3 * 5²)^1/2      By using algebraic property (a*b)^m =a^m * b^m
                        (2 * 3)^1/2 * (5^2*1/2)
                5 (2 * 3)^1/2
                        Square root value of 150= 5 `sqrt(6)`

Problem 2:
Find the Square root value of 300
Solution:
                Write prime factors for 300 = 2 * 2 * 3 * 5 * 5
                `sqrt(300)` = `sqrt(2 * 2 * 3 * 5 * 5)`            
                                   = sqrt(2^2 * 3 * 5^2)
                `sqrt(2^2 * 3 * 5^2)` we can written as (2 * 3 * 5²)^1/2
                (2² * 3 * 5²)^1/2     By using algebraic property (a*b)^m =a^m * b^m
                        (3)^1/2 * (2^2*1/2 * 5^2*1/2)
                5 * 2 (3)^1/2
                        Square root value of 300=10 `sqrt(3)`

Problem 3:
Find the Square root value of 625
Solution:
                Write prime factors for 625 = 5 * 5 * 5 * 5
                `sqrt(625) ` = `sqrt(5 * 5 * 5 * 5)`   
                                   = `sqrt(5^2 * 5^2)`
                `sqrt(5^2 * 5^2)` we can written as (5² * 5²)^1/2          
                (5² * 5²)^1/2            By using algebraic property (a*b)^m =a^m * b^m
                        (5²)^1/2 * (5^2*1/2)
                5 * 5
                        Square root value of 625= 25

Problem 4:
Find the Square root value of 75
Solution:
                Write prime factors for 75 = 3 * 5 * 5
                `sqrt(75)` = `sqrt(3*5*5)`
                                   = `sqrt(3*5^2)`
                `sqrt(3*5^2)` we can written as (3*5²)^1/2 
                (3*5²)^1/2               By using algebraic property (a*b)^m =a^m * b^m
                        (3)^1/2 * (5^2*1/2)
                5 (3)^1/2
                        Square root value of 75= 5 `sqrt(3)`