All About Cylinders

A cylinder is a three-dimensional shape in geometry. A cylinder is round and has a top and bottom in the shape of a circle. The top and the bottom are flat and always the same size. I think the best way to describe the shape of a cylinder is to think of a can of soup!

Can of Soup

Volume of a Cylinder

The volume of a cylinder tells us how much space it has on the inside of it. If you had a can of pop, the volume would be equal to how much pop fills the entire can.

In the formula to find the volume of a cylinder, you will need to know the height and radius of the cylinder. The height is basically how tall it is. The radius is equal to half of the length of the diameter of its circular top or bottom. Remember also that the number pi can be rounded to 3.14.

Volume of a Cylinder

Surface Area of a Cylinder

The surface area of a cylinder tells us how much space covers the entire surface of the cylinder. If you were to wrap a can of Pringles with wrapping paper for a birthday gift, the amount of wrapping paper that is used would be equal to the surface area of the Pringles can. To find the surface area of a cylinder, you will need to know the height and radius of a cylinder.

Surface Area of Cylinder

Wrapping Food

Now image that you need to cover a cylindrical piece of cheese with foil. After measuring its dimensions, you figured out that the height is 3 inches and the radius is 1 inch. In order to figure out how much foil you need to cover the entire piece of cheese, you need to calculate the surface area!

Angles

When your child is learning about angles, it may be difficult for them to distinguish between different angle sizes. There are a variety of mathematical lessons associated with angles, and it is important for children to learn some simple patterns that can help them learn about the different types of angles.

Angles are measured in degrees, and there are three different types of angles: right, acute, and obtuse. Also, angles are measured by the space that exists in between its two connected lines, as shown in the images below. Right angles are the most basic point of reference for students when they learn about acute and obtuse angles, as they measure to be 90 degrees.

Acute angles are angles that measure between 0 and 90 degrees. These types of angles can be identified by their narrowness; the space in between the two lines of the angle is smaller in comparison to that of a right angle, as depicted by the image below. Compare this 54-degree angle to the 90-degree angle image. Notice how the 54-degree angle has a space that is smaller than the 90-degree angle.

Obtuse angles are angles that exclusively measure between 90 and 180 degrees. The reason why a measurement of 0 to 180 degrees is not applicable to obtuse angles is because angles that measure from 0 to 90 degrees are considered to be acute angles. These angles look wider and broader than acute angles, and any angle larger than a right angle is considered obtuse. Compare the 130-degree angle to the 90-degree angle, and then compare it to the 54-degree angle. You will notice that the 130-degree angle is much larger than both.

Make sure to use a right angle as a point of reference when trying to distinguish between different angle types. Any angle that is wider than 90 degrees, no matter how small the gap, is considered obtuse; any angle that is smaller than 90 degrees is considered acute. Use this easy tip to help your child along as they learn more about their angles.

De Morgan's Laws | Venn Diagrams | Proofs Maths | Sets

De Morgan's Laws | Venn Diagrams | Proofs Maths | Sets

          De Morgan’s father (a British national) was in the service of East India Company, India. Augustus De Morgan (1806-1871) was born in Madurai, Tamilnadu, India. His family moved to England when he was seven months old. He had his education at Trinity college, Cambridge, England. De Morgan’s laws relate the three basic set operations Union, Intersection and Complementation.

Venn diagram, graphical method of representing categorical propositions and testing the validity of categorical syllogisms, devised by the English logician and philosopher John Venn (1834–1923). Long recognized for their pedagogical value, Venn diagrams have been a standard part of the curriculum of introductory logic since the mid-20th century.

Venn introduced the diagrams that bear his name as a means of representing relations of inclusion and exclusion between classes, or sets. Venn diagrams consist of two or three intersecting circles, each representing a class and each labeled with an uppercase letter. Lowercase x’s and shading are used to indicate the existence and nonexistence, respectively, of some (at least one) member of a given class.

Two-circle Venn diagrams are used to represent categorical propositions, whose logical relations were first studied systematically by Aristotle. Such propositions consist of two terms, or class nouns, called the subject (S) and the predicate (P); the quantifier all, no, or some; and the copula are or are not. The proposition “All S are P,” called the universal affirmative, is represented by shading the part of the circle labeled S that does not intersect the circle labeled P, indicating that there is nothing that is an S that is not also a P. “No S are P,” the universal negative, is represented by shading the intersection of S and P; “Some S are P,” the particular affirmative, is represented by placing an x in the intersection of S and P; and “Some S are not P,” the particular negative, is represented by placing an x in the part of S that does not intersect P.

What is an Equation

An equation says that two things are equal. It will have an equals sign "=" like this:

x

+

2

=

6

That equation says: what is on the left (x + 2) is equal to what is on the right (6)

So an equation is like a statement "this equals that"

Parts of an Equation

So people can talk about equations, there are names for different parts (better than saying "that thingy there"!)

Here we have an equation that says 4x − 7 equals 5, and all its parts:

A Variable is a symbol for a number we don't know yet. It is usually a letter like x or y.

A number on its own is called a Constant.

A Coefficient is a number used to multiply a variable (4x means 4 times x, so 4 is a coefficient)

Variables on their own (without a number next to them) actually have a coefficient of 1 (x is really 1x)

Sometimes a coefficient is a letter like a or b instead of a number:

Example: ax² + bx + c

• x is a variable
• a and b are coefficients
• c is a constant

An Operator is a symbol (such as +, ×, etc) that shows an operation (ie we want to do something with the values).

A Term is either a single number or a variable, or numbers and variables multiplied together.

An Expression is a group of terms (the terms are separated by + or − signs)

So, now we can say things like "that expression has only two terms", or "the second term is a constant", or even "are you sure the coefficient is really 4?"

Exponents

The exponent (such as the 2 in x2) says how many times to use the value in a multiplication. Examples: 82 = 8 × 8 = 64 y3 = y × y × y y2z = y × y × z

Exponents make it easier to write and use many multiplications

Example: y⁴z² is easier than y × y × y × y × z × z, or even yyyyzz

Polynomial

Example of a Polynomial: 3x² + x - 2

A polynomial can have constants, variables and the exponents 0,1,2,3,...

But it never has division by a variable.

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary.
Follow the links for more, or go to Trigonometry Index

Trigonometry ... is all about triangles.

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle. The right angle is shown by the little box in the corner:

Another angle is often labeled θ, and the three sides are then called:

• Adjacent: adjacent (next to) the angle θ
• Opposite: opposite the angle θ
• and the longest side is the Hypotenuse

Why?

Why is this triangle so important?
Imagine we can measure along and up but want to know the direct distance and angle:

Or we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.
And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle "θ":

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan.)

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = OppositeHypotenuse = 2.84.9 = 0.57...

Calculators have sin, cos and tan, let's see how to use them:

Example: What is the missing height here?

• We know the Hypotenuse
• We want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse:

sin(45°) = OppositeHypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

0.7071... is the ratio of the side lengths: in other words the Opposite is about 0.7071 times as long as the Hypotenuse.
Maybe you can figure out the height now?
But let's do it formally using some algebra:

Start with:sin(45°) = OppositeHypotenuse

Put in what we know:0.7071... = Opposite20

Swap sides:Opposite20 = 0.7071...

Multiply both sides by 20:Opposite = 0.7071... × 20

Calculate:Opposite = 14.14 (to 2 decimals)

Done!

Try Sin Cos and Tan

What are Real Numbers?

When both the rational and irrational numbers combined, the combination is defined as the real numbers. Real numbers can be both positive or negative, and they are denoted by the symbol “R”. Numbers like a natural number, decimals, and fraction comes under the real number.

Classification of Real Numbers

• Natural Numbers– It includes all the counting numbers such as 1, 2, 3, 4,…
• Whole Numbers– Numbers starting with zero are called whole numbers, like 0, 1, 2, 3, 4,…
• Integers– Whole numbers and negative of all natural numbers are collectively known as integers, for example -3, -2, -1, 0, 1, 2,
• Rational Numbers– All the numbers that can be written in the form of p/q, where $q\ne 0$ are known as Rational numbers.
• Irrational Numbers– The numbers which cannot be written in the form of p/q (simple fraction) are known as irrational numbers. Irrational numbers are non-terminating and non-repeating in nature.

Properties of real numbers

• Commutative property- If we have real numbers m,n. The general form will be m + n = n + m  for adaddition andmn = nm for multiplication
• Associative property- If we have real numbers m,n,r. The general form will be  m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication
• Distributive property- If we have real numbers m,n,r. The general form will be – m (n + r) = mn + mr and (m + n) r = mr + nr
• Identity property- For addition: m + (- m) = 0