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Distance, Time, Volume, Mass

Distance

Definition

Distance is the total length we cover when we move from one place to another. Distance is a scalar quantity or the object’s movement that does not specify direction. There are a lot of units of distance measurement like kilometre (km), centimetre (cm), or millimetre (mm), but the SI unit is meter (m).

Let us say, for example, the three points ( A, B, and C ) below:

The distance from point A to point B is 3 meters.
The distance from point B to point C is 5 meters.
The distance from point C to point A is 4 meters.

Formula

If we calculate the total distance covered from one point to another just like the diagram above, we may use:

Total Distance=d1+d2+d3

Thus, for example, if a person starts walking from any point ( A, B, or C ) and follows the path as suggested by the triangle and that person ends at the same point, then the total distance covered would be,

Total Distance = 3 meters + 5 meters + 4 meters

           Total Distance = 12 meters

Hence, the total distance covered is 12 meters.

Some illustrations also show distances on a number line. The image below shows two points, D and F.

To calculate their distance, we must subtract the values at the two locations, D and F, on this number line.

Distance (DF) = 7 – 2 = 5 units

Hence, the distance between points D and F is five ( 5 ) units.

In this case, if we let the value at point D as p and the value at point F as q, then the formula to get the distance between two points on a number line would be,

Distance = | p – q | or Distance = | q – p |

Remember that distance is absolute, so the answer is always positive.

To get the distance between two points on a coordinate system, we will use this formula,

d= $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

where; x1,  y1 and ( x2,  y2 ) are the coordinates of the two given points 

The topic of the distance formula will be covered as you move to your higher math subjects covering the distance between two points.

Examples

Example 1: A particle follows the yellow path as shown below from point D to F. Find the total distance covered by the particle if the side of the square measures 15 cm.

Solution: Let us find the distance between the following points DW, WS, ST, and TF. Considering that the side of the square is 15 cm, the distance from point D to point W is 45 cm. The distance from point W to point S is 60 cm. The distance from point S to point T is 30 cm. And the distance from point T to F is 45 cm. To compute the distance, we can have this equation, 

Total distance = DW + WS + ST + TF
Total distance = 45 cm. + 60 cm. + 30 cm + 45 cm.
Total distance = 180 cm.

Therefore, the total distance from point D to F that the particle followed is 180 cm.

Example 2: What is the distance between 17 and 54 on the number line?

Solution:  To solve this, let us subtract the two numbers giving us 37. Hence, the distance between 17 and 54 is 37 units.

Time 

Definition

We use the time to see how long it takes to do something. Let us say, for example, you are waiting for your mother to pick you up at school; you would know how long you have waited by checking the time. There are a lot of units for time, but the SI unit of time is second (s).

How to measure time

There are two methods of measuring time: the olden methods and the modern methods. The olden methods include using hourglasses, water or candle clocks, shadow clocks, and even the sound of certain animals. The modern methods include using modern clocks like wrist watches, wall clocks, stopwatches, and built-in clocks in our cellphones or computers.

A 12- or 24-hour clock can be used to keep track of the time. A 12-hour clock uses A.M. and P.M.

A.M ( Ante Meridiem ) means before noon or from midnight to just before noon. P. M. ( Post Meridiem ) means afternoon or noon to just before midnight. 

A 24-hour clock does not use A.M. or P.M.; instead, it uses the digits from 1 to 24, where 1 to 12 means before noon while 13 to 24 means afternoon. So, for example, in a 12-hour clock, if it says 4 P.M., in a 24-hour clock, it would be 16:00.

We usually start counting time using seconds; then, it becomes a minute, then hours, days, weeks, months, years, and so on. 

Units of Time Measure

1 millennium = 1000 years
1 century = 100 years
1 decade = 10 years
1 year = 12 months
1 year = 365 ¼ days
1 month = 4 weeks
1 week = 7 days
1 days = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds

Operations with Time

Operations with time are similar to the operations of whole numbers but with the application of time conversions. To convert from one unit of time to another, we must multiply it by a unit fraction. A unit fraction is always equal to 1, for example, $\frac{60 seconds}{1 minute}$, $\frac{1 hour}{60 minutes}$, or $\frac{1 day}{24 hours}$ and so on.

Examples

Example 1: How many minutes are there in a day?

Solution:  Since we need to do a conversion of 1 day to minutes, we need to multiply 1 day by the product of $\frac{24 hours}{1 day}$ and $\frac{60 minutes}{1 hour}$.

1 $\cancel{day}$  x  $\frac{24 \cancel{hours}}{\cancel{1 day}}$ x $\frac{60 minutes}{\cancel{1 hour}}$ = $\frac{1 x 24 x 60}{1 x 1}$ = 1440 minutes

Example 2: Convert 5 hours and 30 minutes into minutes.

Solution: To answer this, multiply 5 hours by $\frac{60 minutes}{1 hour}$ convert to minutes, and then add the product to 30 minutes.

5 hours + 30 minutes = 5 $\cancel{hours}$ x $\frac{60 minutes}{\cancel{1 hour}}$ + 30 minutes = (5 x 60) + 30 = 300+25 = 330 minutes

Example 3: Find the total of 4 minutes 20 seconds and 10 minutes 35 seconds.

Solution: Let us list the numbers vertically and add them.

                           4 minutes 20 seconds
                       + 10 minutes 35 seconds
                           ————————-
                           14 minutes 55 seconds 

Example 4:     5 months 90 days 70 hours
                   + 5 months 20 days 10 hours
                   ———————————-

Solution: Let us add the given time and perform conversion when needed.

                          5 months 90 days 70 hours
                     +  5 months 20 days 10 hours
                      ———————————
                      10 months 110 days 80 hours

The answer must then be simplified since, in 1 day, we have 24 hours, so for 80 hours, there are three 24 hours ( 3 x 24 ) and 8 hours. 

There are 30 days a month, so for 110 days, there are three 30 days ( 3 x 30 ) and 20 days. 

Since we have computed three days from 80 hours, three days will be added to the full days, so 20 days will be 23 days already. 

Since there are three months in 110 days, it will be added to 10 months giving us 13 months. 

Hence, we can have 13 months, 23 days 8 hours. 

The conversion of thirteen months to year and months is one year and one month.

Thus, the total time is 1 year 1 month 23 days 8 hours.  

Example 5:  Multiply 5 years 15 months, and by 12.

Solution:   If we multiply five years and 15 months by 12, we have ( 5 years x 12 ) 60 years and ( 15 months x 12 ) 180 months. So, we need to work on 60 years and 180 months. We need 12 months to complete a year, so we can say that 180 months is 15 years. Adding that 15 years to 60 years, we have 75 years.

Therefore, if we multiply five years one month by 12, the product is 75 years or 7 ½ decades

Volume

Definition

Volume is a geometry quantity that all three-dimensional ( 3D ) objects have. The volume uses cubic units as its measure. There are a lot of units for volume, like cubic centimetre ( cm3), cubic inches ( in3 ), and cubic foot ( ft3 ) but cubic meter ( m3) is the SI unit for volume. Multiplying three 1-dimensional units together will give us a cubic unit.

For example, in the image below, if each edge measures 1 cm, we can have 1 cm x 1 cm x 1 cm. We can abbreviate the product using exponent notation as centimetres cubed or centimetres to the 3rd power.

How to Calculate Volume

If you take a two-dimensional ( 2D ) shape and extend it along a third dimension, you can form a three-dimensional ( 3D ) shape.

For example, extending a rectangle along a third dimension can get a rectangular prism. If we have a triangle and extend it along the third dimension, we get a 3D shape called a triangular prism. If you have a circle and extend it along the third dimension, you get a cylinder.

Thus, we can get the volume of these 3D shapes by multiplying the area of the 2D shape that got extended and then multiplying by the length or distance it was extended. In most cases, the 2D shape is the base of the object. 

The volume of a Rectangular Prism

For us to calculate the volume of a rectangular prism, we need to multiply the length that the rectangle was extended by the area of the rectangle.

Volume =  Area of the Rectangle x Extended Length (height)

To simply this formula for the volume of a rectangular prism, we have 

VolumeRectangular Prism=length x width x height

where ( length x width ) is the area of the rectangle and height is the extended length.

Examples

Example 1:  The dimensions of a rectangular prism are: length of 5.4 cm. , width of 4.5 cm. and height of 2. 2 cm. Find the volume.

Solution:  Let us find the volume by multiplying the given dimensions.

Volume ( Rectangular Prism ) = 5.4 cm x 4.5 cm x 2.2 cm

Hence, the volume of the rectangular prism is 53.46 cm3. 

Example 2:  If these are the dimensions of a rectangular prism: length of 6.7 ft., a width of 8.5 ft. and height of 9. 1 ft. Calculate its volume.

Solution:  Let us find the volume by multiplying the given dimensions.

Volume ( Rectangular Prism ) = length x width x height
Volume ( Rectangular Prism ) = 6.7 ft x 8.5 ft x 9.1 ft
Volume ( Rectangular Prism ) = 518.245 ft3
Therefore, the rectangular prism’ volume is 518.245 ft3. 

The volume of a Triangular Prism

In a triangle prism, the product of the area of the triangle and the length it was extended gives its volume. 

Volume= Area of the triangle x Extended Length
VolumeTriagular Prism=($\frac{1}{2}$bh)(l)
where 12bh is the area of the triangle and l is the length the triangular prism was extended.

Examples

Example 1: Find the volume of the triangular prism below whose height is 6 cm.

Solution:  Let us use the formula  VolumeTriangular Prism=($\frac{1}{2}$bh)(l) to find the volume. Let us substitute 8 cm. for the base of the triangle, 6 cm for the height and 10 cm. for the extended length of the triangular prism.

Volume ( Triangular Prism ) = $\frac{1}{2}$ ( 8 ) ( 6 ) ( 10 )

Volume ( Triangular Prism ) = 240 cm3

Hence, the volume of the given triangular prism is 240 cubic centimetres.

Example 2

Find the volume of a triangular prism with the following dimensions: base and height of the triangle are seven ( 7 ) ft. and 8.5 ft., respectively, and an extended length of 6 ft. 

Solution:  Let us use the formula  VolumeTriangular Prism=($\frac{1}{2}$bh)(l) to find the volume by substituting seven ( 7 ) ft. for the base of the triangle, 8 ft. for the height and 6 cm. for the extended length of the triangular prism.

Volume ( Triangular Prism ) = $\frac{1}{2}$ ( 7 ) ( 8.5 ) ( 6 )

Volume ( Triangular Prism ) = $\frac{1}{2}$ ( 357 )

Volume ( Triangular Prism ) = 178.5 ft3

Therefore, the volume of the given triangular prism is 178.5 ft3.

The volume of a Cylinder

The volume of a cylinder can be calculated by multiplying the area of the circle by the length it was extended.

Volume = Area of the Circle x Extended Length

Volume=(πr2)(h)  

where r2 is the area of the circle and h is the length the circle was extended.

Examples

Example 1: Find the volume of the cylinder with a radius of 16 cm. and a height of 30 cm.

Solution:  Let us solve for the volume using the formula Volume ( Cylinder ) =r2h. We need to substitute 16 cm for r  and 30 cm for h.

Volume (Cylinder )  =(πr2)(h)  

Volume (Cylinder ) = 162 (30)

Volume (Cylinder) = 7680π cm3  or

Volume ( Cylinder )  24, 127.43 cm3

Therefore, the volume of the cylinder is approximately 24,127.43 cubic centimetres.

Example 2: Find the volume of the cylinder with a radius of 12 inches and a height of 14 inches.

Solution:  We need to substitute 12 in. for r  and 14 in. for h.

Volume (Cylinder )  =(πr2)(h)  

Volume (Cylinder ) = 122 (14)

Volume (Cylinder ) = 2016 in3 or

Volume ( Cylinder )  6333.45 in3

Therefore, the volume of the cylinder is approximately 6333.45 in23.

The volume of a Sphere

We will use this formula to find the volume of a sphere,

VolumeSphere=$\frac{4}{3}$πr3

where; the radius of the circle is r.

Examples

Example 1: Find the volume of the sphere whose radius measure 12 inches.

Solution: Using the formula VolumeSphere=$\frac{4}{3}$πr3, let us substitute 12 inches for the radius.

Volume ( Sphere ) = $\frac{4}{3}$ π( 12 )3

Volume ( Sphere ) =  $\frac{4}{3}$ π ( 1728 )

Volume ( Sphere ) = 2304 π in3

Volume ( Sphere ) 7238.23 in3

Thus, the volume of the sphere is approximately 7238.23 cubic inches.

Example 2: Find the volume of the sphere illustrated below.

Solution: Since 8 cm is the sphere’s radius, let us substitute 8 cm. for r.

Volume ( Sphere ) = $\frac{4}{3}$π( 8 )3

Volume ( Sphere ) =  $\frac{4}{3}$ π ( 512 )

Volume ( Sphere ) 682.6 π cm3

Volume ( Sphere ) 2144.66 cm3

Thus, the volume of the sphere is approximately 2144.66 cubic centimetres.

The volume of a Cone

We will use this formula to find the volume of a cone,

VolumeCone=$\frac{1}{3}$ πr2h

where; r is the circle’s radius, and h is the height from the centre of the circular base straight up to the tip of the cone.

Example

Find the volume of the cone below.

Solution: Using the formula VolumeCone=$\frac{1}{3}$ πr2h, let us find the volume by substituting 6 cm. for the radius and 11 cm for the height of the cone.

Volume ( Cone ) = $\frac{1}{3}$ π ( 6 )2 ( 11 )

Volume ( Cone ) = $\frac{1}{3}$ π ( 396 )

Volume ( Cone ) = 132 π cm3

Volume ( Cone ) 414.69 cm3

Hence, the volume of the cone is 414.69 cubic centimetres. 

Mass

Definition

Mass is a quantity of matter that a body contains. For example, a book, a pencil, a notebook, a chair, a table, a pillow, or a tumbler has mass. Thus, because of mass, objects are light or heavy. It is a scalar quantity since it only has magnitude. The most common units of mass are gram and kilogram.

Mass is often confused with weight. These two terms are different since weight is the force experienced by an object due to gravity, so if you are on a distant planet, your weight will be different, but your mass stays the same. 

Formula

Mass is the fundamental quantity of matter. There are a lot of units for mass, but the SI unit of mass is the kilogram ( kg ).

The formula for calculating mass can be written as:

Mass = Density x Volume

Summary

  • Distance is the total length we cover when we move from one place to another.

Formula to calculate the total distance

Total Distance=d1+d2+d3

Distance between Two Points on a Number Line ( Formula )

Distance = | p – q | or Distance = | q – p |

Distance Between Two Points ( Formula )

d= $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

  • We use the time to see how long it takes to do something.

Units of Time Measure

1 millennium = 1000 years
1 century = 100 years
1 decade = 10 years
1 year = 12 months
1 year = 365 ¼ days
1 month = 4 weeks
1 week = 7 days
1 days = 24 hours
1 hour = 60 minutes
1 minute = 60 seconds

  • Volume is a geometry quantity that all three-dimensional ( 3D ) objects have. Cubic units are used to measure volume. Cubic units are made by multiplying three 1-dimensional units together.

Formulas: 

The volume of a Cube

Volume = a3, where a is the length of the side of a cube

The volume of a Rectangular Prism

Volume = length x width x height

The volume of a Triangular Prism

Volume = ½ base x height x length

The volume of a Cylinder

Volume=(πr2)(h)

The volume of a Sphere

Volume=$\frac{4}{3}$ πr3

The volume of a Cone

 Volume=$\frac{1}{3}$ πr2h

  • Mass is a quantity of matter that a body contains.

The formula for calculating mass can be written as:

Mass = Density x Volume

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Spatial Skill: Volume of Solid Figures (Boxing Day Themed) Math Worksheets

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