Rhombus(Jump to  Area of a Rhombus  or  Perimeter of a Rhombus )

A Rhombus is a flat shape with 4 equal straight sides.

A rhombus looks like a diamond

.

All sides have equal length

Opposite sides are parallel, and opposite angles are equal (it is aParallelogram).

The altitude is the distance at right angles to two sides

And the diagonals "p" and "q" of a rhombus bisect each other at right angles.

Play with a rhombus:

Area of a Rhombus

The Area can be calculated by:

the altitude times the side length:

Area = altitude × s

the side length squared (s2) times the sine of angle A (or angle B):

Area = s2 sin(A)

Area = s2 sin(B)

by multiplying the lengths of the diagonals and then dividing by 2:

Area = (p × q)/2

Example: A rhombus has diagonals of 6 m and 8 m, what is its Area?

Area = (6 m × 8 m)/2 = 24 m2

If you can draw your Rhombus, try the  Area of Polygon by Drawing  tool.)

Perimeter of a Rhombus

The Perimeter is the distance around the edges.

The Perimeter is 4 times "s" (the side length) because all sides are equal in length:

Perimeter = 4s

Example: A rhombus has a side length of 12 cm, what is its Perimeter?

Perimeter = 4 × 12 cm = 48 cm

Is a Square a Rhombus?

Yes, because a square is just a rhombus where the angles are all right angles.

Other Names

It is more common to call this shape a rhombus, but some people call it a rhomb or even adiamond.

The plural is rhombi or rhombuses, and, rarely, rhombbi or rhombbuses (with a double b).

The name "rhombus" comes from the Greek word rhombos: a piece of wood apparently whirled on a string to make a roaring noise!

Platonic SolidsA Platonic Solid is a 3D shape where:

each face is the same regular polygon the same number of polygons meet at each vertex (corner)

Example: the Cube is a Platonic Solid

each face is the same-sized square

3 squares meet at each corner

There are only five platonic solids.

The Platonic SolidsFor each solid we have two printable nets (with and without tabs). You can make models  with them! Print them on a piece of card, cut them out, tape the edges, and you will have your own platonic solids.

Tetrahedron

3 triangles meet at each vertex 4 Faces 4 Vertices 6 Edges Tetrahedron Net Tetrahedron Net (with tabs) Spin a Tetrahedron

Cube

3 squares meet at each vertex 6 Faces 8 Vertices 12 Edges Cube Net Cube Net (with tabs) Spin a Cube

Octahedron

4 triangles meet at each vertex 8 Faces 6 Vertices 12 Edges Octahedron Net Octahedron Net (with tabs) Spin an Octahedron

Dodecahedron

3 pentagons meet at each vertex

12 Faces 20 Vertices 30 Edges

Dodecahedron Net Dodecahedron Net (with tabs) Spin a Dodecahedron

Icosahedron

5 triangles meet at each vertex 20 Faces 12 Vertices 30 Edges Icosahedron Net Icosahedron Net (with tabs) Spin an Icosahedron

EllipseAn ellipse usually looks like a squashed circle.

It is defined by two special points called foci.

"F" is a focus, "G" is a focus, and together they are called foci.

(pronounced "fo-sigh")

 

The distance from F to P to G is always the same value

In other words, when you go from point "F" to any point on the ellipse and then go on to point "G", you will always travel the same distance.

 

You Can Draw It Yourself

Put two pins in a board, put a loop of string around them, and insert a pencil into the loop. Keep the string stretched so it forms a triangle, and draw a curve ... you will draw an ellipse.

It works because the string naturally forces the same distance from pin-to-pencil-to-other-pin.

A Circle is an Ellipse

In fact a Circle is an Ellipse, where both foci are at the same point (the center).

In other words, a circle is a "special case" of an ellipse. Ellipses Rule!

Definition

An ellipse is the  set of all points  on a plane whose distance from two fixed points F and G add up to a constant.

Major and Minor Axes

The Major Axis is the longest diameter. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. And the Minor Axis is the shortest diameter (at the narrowest part of the ellipse).

The Semi-major Axis is half of the Major Axis, and the Semi-minor Axis is half of the Minor Axis.

Calculations

Area is easy, perimeter is not!

Area

The  area  of an ellipse is:

π × a × b

where a is the length of the Semi-major Axis, and b is the length of the Semi-minor Axis. Be careful:a and b go from the edge to the center.

(Note: for a circle, a and b are equal to the radius, and you get π × r × r = πr2, which is right!

Perimeter Approximation

Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject: read  Perimeter of an Ellipse  for more details.

But a simple approximation that is within about 5% of the true value (so long as a is not more than 3 times longer than b) is as follows:

Remember, this is only a rough approximation!

Tangent

A tangent is a line that just touches a curve at one point, without cutting across it. Here is a tangent to an ellipse:

Here is a cool thing: the tangent line has equal angles with the two lines going to each focus! Try bringing the two focus points together (so the ellipse is a circle) ... what do you notice?

Section of a Cone

You can also get an ellipse when you slice through a cone (but not too steep a slice, or you get a  parabola  or  hyperbola ).

In fact the ellipse is a  conic section  (a section of a cone) with an  eccentricity between 0 and 1.

 

Equation

By placing an ellipse on an  x-y graph  (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve is:

x2/a2 + y2/b2 = 1

(similar to the  equation of the hyperbola : x2/a2 − y2/b2 = 1, except for a "+" instead of a "−")

 

Cubes and Cube Roots

To understand cube roots, first we must understand cubes ...

How to Cube A Number

To cube a number, just use it in a multiplication 3 times ...

Example: What is 3 Cubed?

3 Cubed = = 3 × 3 × 3 = 27

Note: we write down "3 Cubed" as 33

(the little 3 means the number appears three times in multiplying)

Some More Cubes4 cubed = 43 = 4 × 4 × 4 = 645 cubed = 53 = 5 × 5 × 5 = 1256 cubed = 63 = 6 × 6 × 6 = 216

Cube Root

A cube root goes the other direction:

3 cubed is 27, so the cube root of 27 is 3

3 27

The cube root of a number is ...... a special value that when cubed gives the original number.

The cube root of 27 is ...... 3, because when 3 is cubed you get 27.

Note: When you see "root" think

"I know the tree, but what is the root that produced it?"

In this case the tree is "27", and the cube root is "3".

Here are some more cubes and cube roots:

4 645 1256 216

Example: What is the Cube root of 125?

Well, we just happen to know that 125 = 5 × 5 × 5 (if you use 5 three times in a multiplication you will get 125) ...

... so the answer is 5

The Cube Root Symbol

This is the special symbol that means "cube root", it is the"radical" symbol (used for square roots) with a little three to mean cube root.

You can use it like this:   (we say "the cube root of 27 equals 3")

 

You Can Also Cube Negative Numbers

Have a look at this:

When we cube 5 we get 125: 5 × 5 × 5 = 125When we cube −5 we get −125: −5 ×−5 × −5 = −125

So the cube root of −125 is −5

Perfect Cubes

The Perfect Cubes are the cubes of the  whole numbers :

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etcPerfect Cubes: 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 ...

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Example: what is the cube root of 30?

Well, 3 × 3 × 3 = 27 and 4 × 4 × 4 = 64, so we can guess the answer is between 3 and 4.

Let's try 3.5: 3.5 × 3.5 × 3.5 = 42.875 Let's try 3.2: 3.2 × 3.2 × 3.2 = 32.768 Let's try 3.1: 3.1 × 3.1 × 3.1 = 29.791

We are getting closer, but very slowly ... at this point, I get out my calculator and it says:

3.1072325059538588668776624275224...

... but the digits just go on and on, without any pattern. So even the calculator's answer isonly an approximation ! 

(Further reading: these kind of numbers are called  surds  which are a special

type of irrational number ) 

Metric System of Measurement(Correctly called " SI ")

The metric system is a system of measuring. It has three main units:

m the meter for lengthkg the kilogram for masss the second for time

With those three simple measurements we can measure nearly everything in the world!

Examples:

MeterThe length of this guitar When unfolded this ruler 

is about 1 meter: measures 2 meters: 

Kilogram

This gold bar has amass  of 1 kilogram.

A dictionary also has a mass of about 1 kilogram.

Second

1 second is about as long as it takes to say "one thousand and

one"

Larger or Smaller

But what if we want to talk about really big or really small things?

Answer: we can use  Metric Number Prefixes

like "kilo" (a thousand) and "milli" (one thousandth) and so on

Example: something that is 1,000 meters is a "kilometer"

Something that is one thousandth of a second is a "millisecond"

In fact the kilogram already uses this method, as it's a thousand grams, a kilogram.

So one thousandth (1/1000) of a kilogram is simply a "gram"

Here is a quick summary of the special prefixes:

Large NumbersName deca hecto kilo mega giga tera peta exa zetta yotta

Symbol da h k M G T P E Z YFactor 101 102 103 106 109 1012 1015 1018 1021 1024

Example: A million liters is called a megaliter and abbreviated ML

Small NumbersName deci centi milli micro nano pico femto atto zepto yocto

Symbol d c m µ n p f a z yFactor 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 10-21 10-24

Example: A thousandth of a meter is called a millimeter and abbreviated mm

Making Other Units

We can also combine the meter, kilogram and second to make new  Units of Measurement !

Example: Speed

Speed is how far something moves over a period of time

So it can be measured in meters per second

It means: How many meters does something travel in one second

We can write it as meters/second, or simply m/s

Here are a few common units that are based on the meter, kilogram and second:

Area

Square Meter

Area  is length by length, so the basic unit of area is a square that is 1 meter on each side.

The Unit is meters × meters, which is written m2 (square meters).

Volume:

Cubic Meter

Volume  is length by length by length, so the basic unit of volume is a cube that is 1 meter on each side.

The Unit is meters × meters × meters, which is written m3 (cubic meters).

Liter

So, a cube that is 1 meter on each side is a cubic meter (m3) ...

... and that is also equal to 1,000 liters.

1 m3 = 1,000 Liters

Liter is abbreviated L (some people use lowercase l, but that looks too much like 1).

So a liter is actually one-thousandth of a cubic meter.

1 Liter = 1/1000 m3

Another way of thinking about a liter is:

A box that is 0.1 meters (10 cm) on each side, One square meter that is millimeter thick.

Time

Hour

An hour is 60 minutes, and a minute is 60 seconds, so an hour is:

60 × 60 = 3,600 seconds

Day

A day is 24 hours so:

1 day = 24 × 60 × 60 = 86,400 second

Speed

Speed in meters per second (m/s)

This is a combination of two units (meters and seconds) to make a new one (m/s).

If something is traveling at 1 m/s it moves 1 meter every second.

Speed in kilometers per hour (km/h)

A kilometer has 1000 meters, and an hour has 3600 seconds, so a kilometer per hour is:

1000 / 3600 = 1/3.6 = 0.277... m/s

How did I know to make it 1000/3600, and not 3600/1000 (the other way around)? Read how to  Safely Convert From One Unit to Another .

Acceleration

Acceleration is how fast Speed changes.

If something accelerated from a Speed of 5 m/s (5 meter per second) to 6 m/s (6 meters per second) in just one second, it has accelerated by 1 meter per second per second!

That is two lots of "per second" and is written m/s2:

Force

Force is usually measured in the Unit of Newtons, an important measurement in Physics and Engineering.

A Newton is how much force it takes to make 1 kg accelerate at 1 m/s2.

Which is written as 1 kg · m / s2 (one kilogram-meter per second-squared).

So force is actually based on the meter, kilogram and second.

So force is a combination of the three basic units.

SI

The Metric System had its beginnings back in 1670 by a mathematician called Gabriel Mouton.

The modern version, (since 1960) is correctly called "International System of Units" or "SI" (from the French "Système International").

So we should really call it "SI", but mostly people just call it "Metric".

A few special units are also needed to complete the SI System:

ampere for electric current,

kelvin for temperature, mole for the amount of substance, and candela for luminous intensity

So the complete list is:

Quantity Name Symbol

Length meter m

Mass kilogram kg

Time second s

Electrical Current ampere A

Temperature kelvin K

Amount of substance mole mol

Luminous intensity candela cd