Years ago, a man named Pythagoras found an amazing fact about triangles: If a the triangle had a right angle (90°) and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together!
Note:
- c is the longest side of the triangle
- a and b are the other two sides
The longest side of the triangle is called the “hypotenuse”, so the formal definition is: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Let’s see if it really works using an example.
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Example A “3,4,5” triangle has a right angle in it. |
Let’s check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works … like Magic! |
Why Is This Useful?
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If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!) |
Now you can use algebra to find any missing value, as in the following examples:
Example:Solve this triangle. |
a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169 c = √169 c = 13 |
You can also read about Squares and Square Roots to find out why √169 = 13
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Example: Solve this triangle |
a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides: b2 = 144 b = √144 b = 12 |
It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled
1.3.2 Mass
Mass is the measurement for the quantity of substance present in a body. Mass is different from weight. Weight is a measure of the attraction of the earth for a given mass and is measured in Newton. The common measuring instruments for mass are scales. Scales can take many forms and the more an item weighs, the larger the scale. The SI unit of mass is the kilogram (kg).In everyday usage, mass is often referred to as weight, the units of which are often taken to be kilograms (for instance, a person may state that their weight is 75 kg). In scientific use, however, the term weight refers to a different, yet related, property of matter. Weight is the gravitational force acting on a given body — which differs depending on the gravitational pull of the opposing bodies (e.g. a person’s weight on Earth vs. on the Moon) — while mass is an intrinsic property of that body that never changes. In other words, an object’s weight depends on its environment, while its mass does not. On the surface of the Earth, an object with a mass of 50 kilograms weighs 491 Newtons; on the surface of the Moon, the same object still has a mass of 50 kilograms but weighs only 81.5 Newtons. Restated in mathematical terms, on the surface of the Earth, the weight W of an object is related to its mass m by W = mg, where g is the Earth’s gravitational field strength, equal to about 9.81 m s−2. In the International System of Units (SI), mass is measured in kilograms (kg). The gram (g) is 1⁄1000 of a kilogram.
Other units are accepted for use in SI:
- The tonne (t) is equal to 1000 kg.
- The electron volt (eV) is primarily a unit of energy, but because of the mass-energy equivalence it can also function as a unit of mass. In this context it is denoted eV/c2, or simply as eV. The electron volt is common in particle physics.
- The atomic mass unit (u) is defined so that a single carbon-12 atom has a mass of 12 u; 1 u is approximately 1.66×10−27 kg.[note 1] The atomic mass unit is convenient for expressing the masses of atoms and molecules.
In classical mechanics, mass has a central role in determining the behaviour of bodies. Newton’s second law relates the force F exerted in a body of mass M to the body’s acceleration α:
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Additionally, mass relates a body’s momentum p to its velocity v:
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and the body’s kinetic energy K to its velocity:
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In special relativity, relativistic mass is a formalism which accounts for relativistic effects by having the mass increase with velocity.
Since energy is dependent on reference frame (upon the observer) it is convenient to formulate the equations of physics in a way such that mass values are invariant (do not change) between observers, and so the equations are independent of the observer. For a single particle, this quantity is the rest mass; for a system of bound or unbound particles, this quantity is the invariant mass. The invariant mass M of a body is related to its energy E and the magnitude of its momentum p by
where c is the speed of light.
1.3.3 Area
Area is the size that a plane surface or two-dimensional shape takes up between its boundary lines (length and breadth). Area is calculated by multiplying the length with the breadth. The blocks in the figure below shows
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (m2), which is the area of a square whose sides are one metre long.[1] A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.
For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Every unit of length has a corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measure in square metres (m2), square centimetres (cm2), square millimetres (mm2), square kilometres (km2), square feet (ft2), square yards (yd2), square miles (mi2), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units.
The SI unit of area is the square metre, which is considered an SI derived unit.
The conversion between two square units is the square of the conversion between the corresponding length units.
Since 1 foot = 12 inches,
The relationship between square feet and square inches is
1 square foot = 144 square inches,
where 144 = 122 = 12 × 12. Similarly:
- 1 square kilometre = 1,000,000 square meters
- 1 square meter = 10,000 square centimetres = 1,000,000 square millimetres
- 1 square centimetre = 100 square millimetres
- 1 square yard = 9 square feet
- 1 square mile = 3,097,600 square yards = 27,878,400 square feet
In addition,
- 1 square inch = 6.4516 square centimetres
- 1 square foot = 0.09290304 square metres
- 1 square yard = 0.83612736 square metres
- 1 square mile = 2.589988110336 square kilometres
There are several other common units for area. 1 are = 100 square metres
Though they are fallen out of use, the hectare is still commonly used to measure land:
- 1 hectare = 100 acres = 10,000 square metres = 0.01 square kilometres
Other uncommon metric units of area include the tetrad, the hectad, and the myriad.
The acre is also commonly used to measure land areas, where
- 1 acre = 4,840 square yards = 43,560 square feet.
An acre is approximately 40% of a hectare.
On the atomic scale, area is measured in units of barns, such that,
- 1 barn = 10−28 square meters.
The barn is commonly used in describing the cross sectional area of interaction in nuclear physics.
Basic area formula
The area of this rectangle is length x width.
The most basic area formula is the formula for the area of a rectangle. Given a rectangle with length l and w, the formula for the area is
A = lw (rectangle).
That is, the area of the rectangle is the length multiplied by the width. As a special case, the area of a square with side length s is given by the formula
A = s2 (square).
The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom. On the other hand, if geometry is developed before arithmetic, this formula can be used to define multiplication of real numbers.
Most other simple formulae for area follow from the method of dissection. This involves cutting a shape into pieces, whose areas must sum to the area of the original shape.
For an example, any parallelogram can be subdivided into a trapezoid and a right triangle, as shown in figure to the left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle:
A = bh (parallelogram).
Two equal triangles
However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of the parallelogram:
(triangle)
Similar arguments can be used to find area formulae for the trapezoid and the rhombus, as well as more complicated polygons.
A circle can be divided into sectors which rearrange to form an approximate parallelogram.
Main article: Area of a circle
The formula for the area of a circle is based on a similar method. Given a circle of radius r, it is possible to partition the circle into sectors, as shown in the figure to the right. Each sector is approximately triangular in shape, and the sectors can be rearranged to form and approximate parallelogram. The height of this parallelogram is r, and the width is half the circumference of the circle, or πr. Thus, the total area of the circle is r × πr, or πr2:
A = πr2 (circle).
Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of the areas of the approximate parallelograms is exactly πr2, which is the area of the circle.
This argument is actually a simple application of the ideas of calculus. In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus. Using modern methods, the area of a circle can be computed using a definite integral:
Sphere
Archimedes showed that the surface area and volume of a sphere is exactly 2/3 of the area and volume of the surrounding cylindrical surface.
Most basic formulae for surface area can be obtained by cutting surfaces and flattening them out. For example, if the side surface of a cylinder (or any prism) is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone, the side surface can be flattened out into a sector of a circle, and the resulting area computed.
The formula for the surface area of a sphere is more difficult: because the surface of a sphere has nonzero Gaussian curvature, it cannot be flattened out. The formula for the surface area of a sphere was first obtained by Archimedes in his work On the Sphere and Cylinder. The formula is
A = 4πr2 (sphere).
where r is the radius of the sphere. As with the formula for the area of a circle, any derivation of this formula inherently uses methods similar to calculus.
List of formulae
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Common formulae for area: |
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Shape |
Formula |
Variables |
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Regular triangle (equilateral triangle) |
is the length of one side of the triangle. |
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Triangle |
is half the perimeter, , and are the length of each side. |
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Triangle |
and are any two sides, and is the angle between them. |
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Triangle |
and are the base and altitude (measured perpendicular to the base), respectively. |
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Square |
is the length of one side of the square. |
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Rectangle |
and are the lengths of the rectangle’s sides (length and width). |
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Rhombus |
and are the lengths of the two diagonals of the rhombus. |
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Parallelogram |
is the length of the base and is the perpendicular height. |
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Trapezoid |
and are the parallel sides and the distance (height) between the parallels. |
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Regular hexagon |
is the length of one side of the hexagon. |
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Regular octagon |
is the length of one side of the octagon. |
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Regular polygon |
is the side length and is the number of sides. |
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Regular polygon |
is the perimeter and is the number of sides. |
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Regular polygon |
is the radius of a circumscribed circle, is the radius of an inscribed circle, and is the number of sides. |
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Regular polygon |
is the apothem, or the radius of an inscribed circle in the polygon, and is the perimeter of the polygon. |
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Circle |
is the radius and the diameter. |
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Circular sector |
and are the radius and angle (in radians), respectively. |
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Ellipse |
and are the semi-major and semi-minor axes, respectively. |
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Total surface area of a Cylinder |
and are the radius and height, respectively. |
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Lateral surface area of a cylinder |
and are the radius and height, respectively. |
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Total surface area of a Cone |
and are the radius and slant height, respectively. |
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Lateral surface area of a cone |
and are the radius and slant height, respectively. |
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Total surface area of a Sphere |
and are the radius and diameter, respectively. |
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Total surface area of a Pyramid |
is the base area, is the base perimeter and is the slant height. |
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Square to circular area conversion |
is the area of the square in square units. |
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Circular to square area conversion |
is the area of the circle in circular units. |
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The above calculations show how to find the area of many common shapes.
The surface area of a right prism is 2 · B + P · h, where B is the area of the base, h the height, and P the base perimeter. The surface area of a right prism whose base is a regular n-sided polygon with side length s and height h is therefore:
The volume of a prism is the product of the area of the base and the distance between the two base faces, or the height (in the case of a non-right prism, note that this means the perpendicular distance).
The volume is therefore:
where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side length s is therefore:
1.3.4 Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.[1] Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces.
Volume Formulas
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Cube = a 3 |
Rectangular prism = a b c |
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Irregular prism = b h |
Cylinder = b h = pi r 2 h |
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Pyramid = (1/3) b h |
Cone = (1/3) b h = 1/3 pi r 2 h |
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Sphere = (4/3) pi r 3 |
Ellipsoid = (4/3) pi r1 r2 r3 |
Volume is measured in “cubic” units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.
Volume of a cube = side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.
If a square has one side of 4 inches, the volume would be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic inches can also be written in3.)
Be sure to use the same units for all measurements. You cannot multiply feet times inches times yards, it doesn’t make a perfectly cubed measurement.
The volume of a rectangular prism is the length on the side times the width times the height. If the width is 4 inches, the length is 1 foot and the height is 3 feet, what is the volume?
1.3.5 Time
Time is what we use to measure the period that an action or event takes to occur.Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects. The temporal position of events with respect to the transitory present is continually changing; events happen, then are located further and further in the past. Time has been a major subject of religion, philosophy, and science, but defining it in a non-controversial manner applicable to all fields of study has consistently eluded the greatest scholars.
Time is one of the seven fundamental physical quantities in the International System of Units. Time is used to define other quantities — such as velocity — so defining time in terms of such quantities would result in circularity of definition. In day-to-day life, the clock is consulted for periods less than a day, the calendar, for periods longer than a day. Increasingly, personal electronic devices display both calendars and clocks simultaneously. The most accurate timekeeping devices are atomic clocks, which are accurate to seconds in many millions of years, and are used to calibrate other clocks and timekeeping instruments. Atomic clocks use the spin property of atoms as their basis, and since 1967, the International System of Measurements bases its unit of time, the second. From the second, larger units such as the minute, hour and day are defined, though they are “non-SI” units because they do not use the decimal system, and also because of the occasional need for a leap second. They are, however, officially accepted for use with the International System.
1.3.5 Speed and Acceleration
Speed is the rate of motion, or the rate of change of position. It is expressed as distance moved (d) per unit of time(t). Speed is a scalar quantity with dimensions distance/time. Speed is measured in the same physical units of measurement as velocity, but does not contain an element of direction. Speed is thus the magnitude component of velocity. Velocity contains both the magnitude and direction components.
Common speeds of moving objects
For human beings, an average walking speed is about 3 mph (~5 km/h, 1.39m/s),
The speed of long distance jogging for average persons is about 6 mph (~10 km/h, 2.7 m/s).
Top athletic sprinters can run at 23.03 mph (~36.85 km/h, 10.24 m/s) within a short distance such as a 200 meters dash.
Cycling can average 12 mph (~20 km/h, 5.56 m/s)
Car can average 65 mph (~104 km/h, 28.9 m/s ) on highway
A 747 Airplane has an average speed 565 mi/hr
Speed(S)=distance travelled(d) devide the amount of time it took (t).S=D/T.
Acceleration, (symbol: a) is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time². In SI units, acceleration is measured in meters/second². To accelerate an object is to change its velocity, which is accomplished by altering either its speed or direction (like in case of uniform circular motion) in relation to time. Acceleration can have positive and negative values. Any time that the sign (+ or -) of the acceleration is the same as the sign of the velocity, the object will speed up. If the signs are opposite, the object will slow down. Acceleration is a vector quantity. When either velocity or direction changes, there is acceleration (or deceleration).
To accelerate an object requires the application of a force.
The graph of velocity (m/sec.) vs. time (sec.) is a straight line for accelerating objects.
acceleration = velocity / time
What is the slope of this graph?
The graph of velocity (m/sec.) vs. time (sec.) is a straight line for accelerating objects.
Acceleration = velocity / time
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The graph of distance (x) vs. time (t) is a curve where the equation of motion is: x = 1/2 at². What value of (a) would you expect for this graph? Hint: For each value of t (in s), there is only one value for x (in m). Pick any ordered pair for t and x and substitute them into the above equation. Then solve for a. |
In a qualitative manner, we can describe the temperature of an object as that which determines the sensation of warmth or coldness felt from contact with it.
It is easy to demonstrate that when two objects of the same material are placed together (physicists say when they are put in thermal contact), the object with the higher temperature cools while the cooler object becomes warmer until a point is reached after which no more change occurs, and to our senses, they feel the same. When the thermal changes have stopped, we say that the two objects (physicists define them more rigorously as systems) are in thermal equilibrium . We can then define the temperature of the system by saying that the temperature is that quantity which is the same for both systems when they are in thermal equilibrium.
Calculating speed, distance and time
Speed, Time and Distance are related and can be calculated using the following formulas.
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Example Jane drives at an average speed of 45 mph on a journey of 135 miles. |
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Example Chris cycles at an average speed of 8 mph. |
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Example Nikki has to travel a total of 351 miles. She travels the first 216 miles in 4 hours. Her average speed is the same for the whole of her journey.
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1.3.7 Temperature
In a qualitative manner, we can describe the temperature of an object as that which determines the sensation of warmth or coldness felt from contact with it.
It is easy to demonstrate that when two objects of the same material are placed together (physicists say when they are put in thermal contact), the object with the higher temperature cools while the cooler object becomes warmer until a point is reached after which no more change occurs, and to our senses, they feel the same. When the thermal changes have stopped, we say that the two objects (physicists define them more rigorously as systems) are in thermal equilibrium . We can then define the temperature of the system by saying that the temperature is that quantity which is the same for both systems when they are in thermal equilibrium.
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Exercise 1 Convert the following units from SI to Imperial: a) 34cm to inches b) 22 liters to gallons c) 70 kilometers to miles d) 78 kilograms to pounds e) 144 square meters to square yards f) 56 meters to feet and yards 2.Convert the following: a) Convert 600C to Fahrenheit b) Convert 1000F to degrees Celsius c) Calculate the area of a field of 50m length and 15m breadth. d) A tractor drives 48 km in 2 hours. What is its velocity? e) A train travels at 120km/hr for 3 hours. How far did it travel? 3.A car’s engine has 4 cylinders, with a total capacity of 2000 cc. 4. A plastic container is 50cm long, 20cm wide and 10cm deep. What amount of liquid is required to fill the container? 5. An oil tank contains 1,050 L of oil, which is being used at the rate of 25 L per hour. How many days until the tank is empty? 6. West view School has a track in the school yard. To run 2 km every day, estimate then calculate how many times you have to go around the track. Compare your estimated and calculated answers. Show your work.
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