Marked 01/02/09
Created By: Dominik, Luke, Maryam, Mitch


What is a 3D shape?
3D means that the shape has three dimensions, that can be measured in; length, width, and height.
external image 1.JPG
All 3D shapes have depth and are solid.

Cylinder
A cylinder by definition is:

  • a cylinder is a long solid tube with straight sides and two equal-sized circular ends
  • a solid bounded by a cylindrical surface and two parallel planes(the bases)
  • a surface generated by rotating a parallel line around a fixed point
For example:

external image cylinder.gif

The cylinder is somewhat a prism, it has parallel congruent bases, but the bases are circles. You find the volume of the cylinder the same way you find the volume of the prism.
Base area times the height of the cylinder
V=bh
*But since the base of the cylinder are circles, you substitue the formula for area of the circle into the formula of volume which then equals this:
V =πr2h
There are different formulas for finding the volume and surface area of a cylinder.
To find the volume of a cylinder the formula is;
V =πr2h This means - Volumepi*radius(squared)*height.


TIP:


Just to let you know, Pi( p) can be rounded hundreds,thousands, and even millions past the decimal point, without having a pattern or stopping(3.14159265358979323846264...)

How to find the surface area and the volume of the cylinder
Height: 20 Radius: 9


Cylinderr.gif

Find the surface area of the cylinder


Sa= 2
πr²+ 2πrh
Sa= 2π(9)²+ 2π(9)(20)

Sa= 2π(81)+2π(180)
Sa= 508.68+ 1130.4
Sa=1639.08 cm²
Step 1: First you write the formula down for surface area for cylinder
Step 2: Find the radius and the height
Step 3: Put the numbers in the formula Sa= 2π(9)²+ 2π(9)(20) the radius is 9 and the height is 20
Step 4: Calculate the brackets first, Sa= 2π(81)+ 2π(180)
Step 5: After you do that do the calculations for the remaining brackets Sa=508.68+1130.4
Step 6: Do the rest and your answer is Sa=1639.08cm² DONT FORGET THE UNITS!!
The surface area of the cylinder is 1639.08cm²
Thats how you find the surface area of the cylinder
Find the volume of the cylinder
Volume
V=πr²h
V=π(9)²(20)
V=π(81)(20)
V=5086.8 cm³
The Volume of the cylinder is 5086.8 cm³
Step 1: Write down the formula of the cylinder for volume
Step 2: Find the radius and the height
Step 3: sub in the numbers in the formula V=πr²h which then equals V=π(9)²(20)
Step 4: Calculate the brackets first, V= π(81)(20)
Step 5: After you do that do the calculations for the remaining brackets V= 5086.8
Step 6: Your final answer is V=5086.8 cm³ DONT FORGET THE UNITS!!
The volume of the cylinder is 5086.8 cm³
Thats how you find out the surface area and the volume of the cylinder.
Cone
A cone is:
  • a shape whose base is a circle and whose sides taper up to a point
  • a three dimensional geometric shape that tapers smoothly from a flat base to a point called a apex or vertex




Even though cones are awesome they are sometimes a pain in the but becasue u have to do a little thing called pothagarean theorem(which is explained later on the page)

Volume of a cone
V=1/3pr²h

This formula is used to find the volume of a cone.

H= Height
S= Side
R= Radius of base
D= Diameter
S= surface area
V= volume
T= total surface area



conerr.jpg



VOLUME OF A CONE
The volume of a cone
A total = alteral surfaces + A base
SA =pi rs +pi r2

Volume
V= (Abase)(height)
V= 1/3 (pi)(r)2 (h) or
V=pi r2 h /3


Below are some examples of how to do the surface area and the volume of the cone.

Find the surface area and the volume of the cone below:
height=


rccone.gif


How to find the volume and the surface area of the cone below:

tamriko1_1.gif
Surface area

Sa= πrs+πr²
Sa=
π(3)(10)+π(3)²
Sa=
π(30)+π(9)
Sa=94.2+28.26
Sa=122.46 cm²

The surface area for the cone is 122.46 cm²
Step 1: Write the formula for surface area for cone
Step 2: Sub in the numbers Sa=
π(3)(10)+π(3)²
Step 3: After you do that calculate the brackets first
Step 4: Once you've calculated the brackets this is how its going to look like Sa=
π(30)+π(9)
Step 5: Times the
π with the numbers
Step 6: Your final answer is 122.46 cm²
The surface area of the cone is 122.46 cm²


Volume
V=πr²h/3

V=π(3)²(9.54)/3
V=π(9)(9.54)/3
V=π(85.86)/3
V=269.60/3
V=89.87 cm³

the volume of the cone is 89.87 cm³

Since you dont know one of the sides of the triangle you use the pythagorean theorem.
C
²=a²+b²
10²=3²+b²
100=9+b
²
100-9=

91=b
²
√91=√b²
9.54=b

The side length of the triangle is 9.54
Step 1: Write down the formula for the volume of the cone
Step 2: Sub in the numbers V=π(3)²(h)/3
Step 3: Since you dont know the height of the triangle you have to use the pythagorean theorem
Step 4: Use c²=a²+b², this is the formula for the pythagorean theorem
Step 5: sub in the numbers for c²=a²+b² it will look like this 10²=3²+b²
Step 6: Once you do that it will look like this 100=9+b²
Step 7: You do 100-9 which equals 91 then you square root both sides and thats your answer fort he pythagorean theorem
Step 8: Your answer for the pythagorean theorem is 9.54, sub that in into the formula, V=π(3)²(9.54)/3
Step 9: Calculate the brackets first, in the end your answer will be V=89.87 cm³

The volume of the cone is 89.87 cm³













spherer.gif

-A sphere is a three-dimensional figure. The term is used to refer either to a round ball or to its two-dimensional surface
-a solid geometric figure
-


Radius: r
Diameter: d
Surface area: S
Volume: V

S = 4 Pi r2 = Pi d2
V = (4 Pi/3)r3 = (Pi/6)d3
Formulas for finding the surface area and the volume of the sphere
Solving for surface area of the sphere:
SA=4pr2

Solving for volume of the sphere:

sphere volume
sphere volume




EX: Find the volume and the surface area of the sphere below:



f-545-7-1.gif
Surface area

SA= 4
πr²
SA= 4π(14)²
SA= 4π(196)
SA= 2461.76 cm²

The surface area of the sphere is 2461.76cm²

((REMEMBER: DONT FORGET TO PUT THE SQUARED THAT COMES AFTER THE CM/M, THE UNIT!!!!!!!))

Step 1: Write the Surface area formula for sphere
Step 2: Find the radius, the radius is 14 cm
Step 3: Put the radius in the formula SA=4π(14)²
Step 4: Do the calculations brackets first, (14)²=196
Step 5: Calculate, π times 196 times 4=2461.76
Step 6: the surface area of the sphere is 2461.76cm²


Volume

V=4πr³/3
V=4π(14)³/3
V=4π(2744)/3
V=34464.64/3
V=11488.21 cm³

The volume of the sphere is 11488.21 cm³

((REMEMBER: DONT FORGET TO PUT THE CUBE AFTER THE CM/M THE UNIT!!!!!!!!!))

Step 1: Write the Volume formula for sphere
Step 2: Find the radius, the radius is 14 cm
Step 3: Put the radius in the formula V=
4π(14)³/3
Step 4: Do the calculations for the brackets first, (14)³=2744
Step 5: Calculate,
π times 2744 times 4, then divide by 3
Step 6: The volume of the sphere is 11488.21 cm³


HINT: if they give you the diamter of the sphere, you divide it by two to find the radius.


Pythagorean Theorem

Pothagarean theorem isn't something we learn in grade nine, but is very much needed and helpful. You will need this helpful tool in finding the missing side lenght of a triangle, cone, triangular prism etc. But it only works if the triangle has a side a 90 degrees ( or a right angle triangle). The longest side of the triangle is called the hypotenuse. In a right angle triangle the square of the hypotenuse is equal to the sum of the squares of the two sides.

the square of a (a²) plus the square of b (b²) is equal to the square of c (c²)

The formula for the pythagorean theorem is

pic-pythagorasT.gif
c2=a2 + b2
its the same both ways.

Why is it so useful?

if you know the lengths of the two sides of a triangle you will find the length of the third missing side by using the pythagorean theorem( But remember it only works on right angled triangles)

How do you use it
c2=a2 + b2
c2=3² +7²
c
2=9+49
c
2=58
√c²=√58
c=7.62


if the numbers too long you round to two decimal places:)
The missing side of the triangle has a length of 7.62.

Step 1: Write down the formula for the pythagorean theorem
Step 2: Put in the numbers c²=3²+7²
Step 3: Calculate the exponents, you will get c²=9+49
Step 4: Once you've done that it will look like this c²=58
Step 5: Square root both sides so the c² will be just c
Step 6: Your final answer for the missing side length of the triangle is c=7.62


More examples

right angled triangle
right angled triangle

a2 + b2 = c2
52 + 122 = c2
25 + 144 = 169
c2 = 169
c = √169
c = 13

right angled triangle
right angled triangle

a2 + b2 = c2
92 + b2 = 152
81 + b2 = 225
Take 81 from both sides
b2 = 144
b = √144
b = 12









Making connections

Making connections to the real world 3D round shapes are used everywhere. At home, school, anywhere place or thing you could think of has 3D round shapes in it:

For example:
Sports - soccer ball, basket ball, tennis ball, volleyball, beachball
Kitchen -glasses, icecream cone, lollipop, lindor chocolates
home uses - candles, gas cylinders, straws
school -globe, erasers, shapes
outside - traffic cone, poles, pipes

These are just some of the examples of 3D round shapess but there are many many more, as you can see we use 3D round shapes in our everyday lives, whatever you see or touch might be something 3D.

Surface area and volume is also used in almost everything including, school ex: in math class, jobs ex: artists, architects, building and many more, home ex: if you want to make a cabinet at home or something you will need to know the surface area and the volume to construct it.





























Citations
Note Book
3-D Shapes- http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/3d/index.htm
Surface Area Formulas- http://www.math.com/tables/geometry/surfareas.htm
Area and Volume Formulas- http://www.science.co.il/formula.asp
picture of sphere-http://z.about.com/d/math/1/5/I/F/spherer.gif
sphere - http://mathforum.org/dr.math/faq/formulas/faq.sphere.html
formulas - http://math.about.com/library/blmeasurement.htm
picture of sphere with radius- [[http://education.yahoo.com/homework_help/math_help/solutionimages/miniprealggt/9/1/1/miniprealggt_9_1_1_29_80/f-545-7-1.gifpicture of cylinder - http://z.about.com/d/math/1/5/F/F/Cylinderr.gif|http://education.yahoo.com/homework_help/math_help/solutionimages/miniprealggt/9/1/1/miniprealggt_9_1_1_29_80/f-545-7-1.gifpicture of cylinder - http://z.about.com/d/math/1/5/F/F/Cylinderr.gif]]
pythagorean theorem - http://www.mathguide.com/lessons/pic-pythagorasT.gif