GEOMETRY

TRIANGLES Equilateral, Isosceles and Scalene There are three special names given to triangles that tell how many sides (or angles) are equal. There can be 3, 2 or no equal sides/angles: Equilateral Triangle Equilateral Triangle Three equal sides Three equal angles, always 60° Isosceles Triangle Isosceles Triangle Two equal sides Two equal angles Scalene Triangle Scalene Triangle No equal sides No equal angles What Type of Angle? Triangles can also have names that tell you what type of angle is inside: Acute Triangle Acute Triangle All angles are less than 90° Right Triangle Right Triangle Has a right angle (90°) Obtuse Triangle Obtuse Triangle Has an angle more than 90° Combining the Names Sometimes a triangle will have two names, for example: Right Isosceles Triangle Right Isosceles Triangle Has a right angle (90°), and also two equal angles Can you guess what the equal angles are? Play With It ... Try dragging the points around and make different triangles: Needs Flash Player View Larger You might also like to play with the Interactive Triangle. Perimeter The perimeter is the distance around the edge of the triangle: just add up the three sides: Needs Flash Player Area triangle b h The area is half of the base times height. "b" is the distance along the base "h" is the height (measured at right angles to the base) Area = ½ × b × h The formula works for all triangles. Note: another way of writing the formula is bh/2 Example: What is the area of this triangle? Triangle (Note: 12 is the height, not the length of the left-hand side) Height = h = 12 Base = b = 20 Area = ½ × b × h = ½ × 20 × 12 = 120 The base can be any side, Just be sure the "height" is measured at right angles to the "base": Needs Flash Player (Note: You can also calculate the area from the lengths of all three sides using Heron's Formula.) Why is the Area "Half of bh"? Imagine you "doubled" the triangle (flip it around one of the upper edges) to make a square-like shape (it would be a "parallelogram" actually), THEN the whole area would be bh (that would be for both triangles, so just one is ½ × bh), like this: triangle area By slicing the new triangle and moving the sliced part to the other side you get a simple rectangle, whose area is bh.

Solid Geometry

Solid Geometry is the geometry of three-dimensional space, the kind of space we live in ...

Three Dimensions

It is called three-dimensional, or 3D because there are three dimensions: width, depth and height.

Simple Shapes

Let us start with some of the simplest shapes:

Common 3D Shapes

Properties

Solids have properties (special things about them), such as:

• volume (think of how much water it could hold)
• surface area (think of the area you would have to paint)
• how many vertices (corner points), faces and edges they have

Polyhedra and Non-Polyhedra

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

Polyhedra : (they must have flat faces)	Cubes and Cuboids (Volume of a Cuboid) Platonic Solids Prisms Pyramids
Non-Polyhedra: (if any surface is not flat)	Sphere Torus Cylinder Cone

Euler's Theorem

Algebra - Basic Definitions

What is an Equation

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

x

+

2

=

6

That equations 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) An Operator is a symbol (such as +, ×, etc) that represents an operation (ie you 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 you never have division by a variable.

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same.
In other words, terms that are "like" each other. (Note: the coefficients can be different)

Example:

(1/3)xy2

-2xy2

6xy2

Are all like terms because the variables are all xy²

Algebra - Basic Definitions

What is an Equation

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

x

+

2

=

6

That equations 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) An Operator is a symbol (such as +, ×, etc) that represents an operation (ie you 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 you never have division by a variable.

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

Like Terms

Like Terms are terms whose variables (and their exponents such as the 2 in x²) are the same.
In other words, terms that are "like" each other. (Note: the coefficients can be different)

Example:

(1/3)xy2

-2xy2

6xy2

Are all like terms because the variables are all xy²