A glide reflection combines a reflection with a
translation along the direction of the mirror line. Glide reflections are the
only type of symmetry that involve more than one step.
Symmetries Land, Mirror Mirror your sure to see reflections don't get confused with all the symmetry. Fold the landscapes in half and your sure to find your way out.
Symmetry
Rotational Symmetry
With Rotational Symmetry, the shape or image can be rotated and it still looks the same.
Real World Examples
A Dartboard has Rotational Symmetry of Order 10
The US Bronze Star Medal has Order 5
The London Eye has Order ... oops, I lost count!
Point Symmetry
Point Symmetry
It looks the same Upside Down!
(... or from any two opposite directions*)
Point Symmetry is when every part has a matching part:
the same distance from the central point
but in the opposite direction.
Point Symmetry is sometimes called Origin Symmetry, because the "Origin" is the central point about which the shape is symmetrical.
Playing Cards often have Point Symmetry, so that they look the same from the top or bottom.
These Letters have Point Symmetry, too!
Line Symmetry
Another name for reflection symmetry. One half is the reflection of the other half.
The "Line of Symmetry" is the imaginary line where you could fold the image and have both halves match .
Plane Shapes
Not all shapes have lines of symmetry, or they may have several lines of symmetry. For example, a Triangle can have 3, or 1 or no lines of symmetry:
Equilateral Triangle
(all sides equal,
all angles equal)
Isosceles Triangle
(two sides equal,
two angles equal)
Scalene Triangle
(no sides equal,
no angles equal)
Welcome to Oceans Volume. Don't get sunk on this ocean. Knowing the volume to the oceans deep is the only way you'll make it across this ocean. Beware of Cyclone Cones and Cylinders deep your sure to sink if you get stuck in one of these.
Volume Formulas
Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely.
Be sure to use the same units for all measurements.
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a". "b3" means "b cubed", which is the same as "b*b*b.
cube = a 3 rectangular prism = a b c irregular prism = b h cylinder = b h = pi r 2 h pyramid = (1/3) b h cone = (1/3) b h = 1/3 pi r 2 h sphere = (4/3) pi r 3 ellipsoid = (4/3) pi r1 r2 r3
Surface Area is a land that will have you going from plane to plane. The confusion will come when some planes are the exact same as the other. The only way to escape this boldging shaped land is to know the proper formula.
Surface Area Formulas The surface area is the sum of all the areas of all the shapes that cover the surface of the object.
Cube, Rectangle Prism, Prism, Sphere, Cylinder
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements
Surface Area of a Cube = 6 a2
the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.
Since there are six sides, the total surface area, call it SA is:
SA = a2 + a2 + a2 + a2 + a2 + a2
SA = 6 × a2
Example #1:
Find the surface area if the length of one side is 3 cm
Surface area = 6 × a2
Surface area = 6 × 32
Surface area = 6 × 3 × 3
Surface area = 54 cm2
Example #2:
Find the surface area if the length of one side is 5 cm
Surface area = 6 × a2
Surface area = 6 × 42
Surface area = 6 × 4 × 4
Surface area = 96 cm2
Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac
The surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.
Surface Area of Any Prism
(b is the shape of the ends) Surface Area = Lateral area + Area of two ends (Lateral area) = (perimeter of shape b) * L Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)
Surface Area of a Sphere = 4 pi r 2
(r is radius of circle)
Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h
(h is the height of the cylinder, r is the radius of the top) Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2(pi r 2) + (2 pi r)* h
The land of Pythagoras' Theorem will have you seeing square roots but if you remember the formula your sure to make it from a squared to b squared and final destination c squared.
Pythagoras' Theorem
Years ago, a man named Pythagoras found an amazing fact about triangles:
If 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!
The Pythagorean Theorem itself
The theorem is named for a Greek mathematician named Pythagoras. He came up with the theory that helped to produce this formula. The formula is very useful in solving all sorts of problems.
The Pythagorean Theorem helps us to figure out the length of the sides of a right triangle. If a triangle has a right angle (also called a 90 degree angle) then the following formula remains true:
a2 + b2 = c2
Note:
c is the longest side of the triangle
a and b are the other two sides
Where a, b, and c are the lengths of the sides of the triangle (see the picture) and c is the side opposite the right angle. 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 work through a few examples:
1) Solve for c in the triangle below:
In this example a = 3 and b=4. Let's plug those into the Pythagorean Formula.
In this trip don't get lossed in the shapes, knowing the area of these shaped confusing islands will help you travel from one to another.
Areas of Polygons and Circles
Polygons
The first thing we need to know is that the area of a shape is a number that tells how many square units are needed to cover the shape. Area can be measured in different units, such as square feet, square meters, or square inches.
One way you can find an area is by drawing a shape on graph paper, and counting the squares inside the shape.
But we a more practicle way is to use area formulas instead. Note that every polygon and circle has a formula for finding its area.
Area of a Rectangle
A rectangle is a great, easy shape to begin with. The area of a rectangle is equal to the product of the length of its base and the length of its height. The height is perpendicular to the base.
A=bh
Sometimes the base may be called the width(w) A=wh
Area of a Parallelogram
A=bh
To find the area of a parallelogram, we can use the same formula that we used for the area of a rectangle, multiplying the length of the base times the length of the height.
Area of a Square A=s2
A square is a special rectangle, and you can find its area using the rectangle formula. However, since the base and height are always the same number for a square, we usually call them "sides." The area of a square is equal to the length of one side squared.
Area of a Triangle
A=1/2bh
Other polygon formulas
Area of a Trapezoid
Circle
Area of a Circle
Now lets put what we have learned to use
Area of rectangle with base = 14 cm and height = 7 cm
Area of triangle with base = 4 cm and height = 6 cm
Area of parallelogram with base = 15 ft and height = 5 ft
Area of trapezoid with height = 10 cm, bases = 7 cm and 15 cm
To Translate a shape: 
rectangular prism = a b c 
irregular prism = b h 
cylinder = b h = pi r 2 h 
pyramid = (1/3) b h 
cone = (1/3) b h = 1/3 pi r 2 h 
sphere = (4/3) pi r 3 
ellipsoid = (4/3) pi r1 r2 r3 
(h is the height of the cylinder, r is the radius of the top) Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2(pi r 2) + (2 pi r)* h 
