3-D+Shapes+-+Round

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

//__** What is a 3D shape? **__// 3D means that the shape has three dimensions, that can be measured in; length, width, and height. All 3D shapes have depth and are solid. **//__ Cylinder __//** A cylinder by definition is: For example:
 * 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

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 V =πr 2 h  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 =πr 2 h  This means - Volumepi*radius(squared)*height.
 * 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:

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

Find the surface area of the cylinder** Sa= 2   πr²+ 2πrh Sa= 2π(9)²+ 2π(9)(20)

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=1/3 p r²h

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 ** 

__ 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=



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

 Surface area Sa= πrs+πr² Sa=   π(3)(10)+π(3)² Sa= <span style="font-size: 110%; font-family: Verdana, Geneva, sans-serif;">  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= <span style="font-size: 110%; font-family: Verdana, Geneva, sans-serif;">  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= <span style="font-size: 110%; font-family: Verdana, Geneva, sans-serif;">  π(30)+π(9) Step 5: Times the <span style="font-size: 110%; font-family: Verdana, Geneva, sans-serif;">  π with the numbers Step 6: Your final answer is 122.46 cm² The surface area of the cone is 122.46 cm²

**Volume** <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">V= πr²h/3 <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;"> 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 <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">Since you dont know one of the sides of the triangle you use the pythagorean theorem. C    <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">²=a²+b² <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">  10 ²=3²+b² 100=9+b  <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;"> ² 100-9= <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;"> b² 91=b  <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">² <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">    √91=√b² 9.54=b <span style="font-size: 110%; color: #a816c0; font-family: Verdana, Geneva, sans-serif;">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



<span style="font-size: 140%; color: #000000; font-family: Verdana, Geneva, sans-serif;"> <span style="font-size: 120%; color: #da1616; font-family: Verdana, Geneva, sans-serif;"> <span style="font-size: 90%; color: #e40c0c; font-family: Verdana, Geneva, sans-serif;">-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=4 p r2

Solving for volume of the sphere:

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

**<span style="font-size: 130%; color: #ec1313; font-family: Verdana, Geneva, sans-serif;"><span style="font-size: 90%; color: #e01010; font-family: Verdana, Geneva, sans-serif;">__<span style="font-size: 130%; font-family: Verdana, Geneva, sans-serif;">Surface area __

<span style="font-size: 90%; color: #0038ff; font-family: Verdana, Geneva, sans-serif;"> SA= 4   ** <span style="font-size: 90%; color: #0038ff; font-family: Verdana, Geneva, sans-serif;">**<span style="font-size: 130%; color: #ec1313; font-family: Verdana, Geneva, sans-serif;"><span style="font-size: 90%; color: #e01010; font-family: Verdana, Geneva, sans-serif;"> 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  ** <span style="font-size: 120%; color: #e81111; font-family: Verdana, Geneva, sans-serif;"> **__ 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=**  <span style="font-size: 120%; color: #e81111; font-family: Verdana, Geneva, sans-serif;">  **4π(14)³/3 Step 4: Do the calculations for the brackets first, (14)³=2744 Step 5: Calculate,**  <span style="font-size: 120%; color: #eb0fad; font-family: Verdana, Geneva, sans-serif;"><span style="font-size: 120%; color: #e81111; font-family: Verdana, Geneva, sans-serif;"> **π times 2744 times 4, then divide by 3 Step 6: The volume of the sphere is 11488.21 cm³** __** 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.
 * HINT: if they give you the diamter of the sphere, you divide it by two to find the radius.**

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

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 ** **  c 2=3² +7² c **** 2=9+49 c ** ** 2=58 ** **    √c²=√58 c=7.62 **

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

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

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

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- [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