Feel The Math

Welcome ! My name is Sheldon. I will be your guide throughout this mathematical experience. You are about to enter a parallel universe. You must successfully navigate through this world, mastering the challenges that it presents. Good luck, be careful, and trust no one.

What is this Place? Three-dimensional Geometry is full of mystery, adventure, and misfortune. It is extremely complex, yet straight forward. Take the area formulas from grade 7 and 8 and used cautiously you can find Volume and Surface Area of solids

Where do you want to go today? Mission 1 – Surface Area of Prisms and Cylinders Mission 2 – Surface Area of Pyramids and Cones Mission 3 – Volume of Cubes & Rectangular Prisms Mission 4 – Volume of Prisms and Cylinders Mission 5 – Volume of Prisms and Cones Mission 6 – Volume of a Sphere Mission 7 – Surface Area of a Sphere

Mission 1 Mission 1: To successfully find the Surface Area of prisms and cylinders. Vocabulary: Total (surface) Area, Lateral Area Formulas: L.A. = Pb H T.A. = L.A. + Abases

Terminology Surface area is the total area of the surface of a 3-dimensional figure. In other words, surface area is the amount of cardboard it takes to make a box, or the amount of wrapping it takes to cover a present.

Terminology Lateral area is the area of the lateral sides of a 3-dimensional figure. In other words, lateral area is just the area of the sides of a figure. For example, lateral area is the surface I would paint if I had to paint the sides of a bucket.

Formula for L.A. The formula for the lateral area (L.A.) of a right prism or cylinder is the product of its height H and the perimeter P of its base. L.A. = Pb  H

Example Find the lateral area of a right cylinder with a height of 5 and a base radius of 3. LA = Pb H = 2  r  H = 2  (3) (5) = 94.2 units2 Remember… area is measured in square units

You try it! What’s the lateral area of a rectangular prism to the left? 14 252 154 308 50 2 9

The answer you have selected is incorrect. The correct answer was 308 The answer you have selected is incorrect. The correct answer was 308. Click on “show me” to see how this problem should be done.

Here is your Answer... LA = Pb H 14 LA = Pb H = [2l + 2w]  H = [(2)(9) + (2)(2)]  14 = [18 + 4]  14 = 308 units2 2 9

You got it. Onto to the next concept!

T.A. = L.A. + Abases Formula for Total Area The formula for the surface (T.A.) of a right prism or cylinder is the sum of its lateral area and twice the area of its base. T.A. = L.A. + Abases

example Find the total area of a right cylinder with a height of 5 and a base radius of 3. T.A. = L.A. + Abases = 2r H + 2 r2 = 2 (3)(5) + 2 (3)2 = 94.2 + 56.5 = 150.7 units2

You try it! What’s the surface area of the right cylinder to the left?

Not bad hot shot! Click next to move on.

The answer you have selected is incorrect. The correct answer was 351 The answer you have selected is incorrect. The correct answer was 351.8. Click on “help me” to see how this problem should be done.

Here is your Answer... T.A. = L.A. + Abases = 351.86 units2 = 2r H + 2 r2 = 2 (4)(10) + 2 (4)2 = 351.86 units2

One more for mission 1 What is the surface area of an open box with a 12 X 17 base and a height of 7? T.A. = L.A. + A base [2l + 2w]  H + l  w = [2(12)+2(17)]  7 + (12) (17) = (58) (7) + 204 = 610 units2

Mission 2 Mission 2: To successfully find the total area of pyramids and cones. Vocabulary: Total (surface) Area, Lateral Area, Slant height Formulas: L.A. = T.A. = L.A. + A base

Formula for L.A. The formula for the lateral area (L.A.) of a regular pyramid or cone is the half product of its slant height s and the perimeter p of its base. L.A. =

T.A.= L.A.+ Abase Formula for S.A. The formula for the surface area (S.A.) of a regular pyramid or cone is the sum of its lateral area and the area of its base (A). T.A.= L.A.+ Abase

Click on “Next” after you’ve finished the problem. Let’s Try one Find the surface area of a cone with a radius of 10 and a slant height of 13. Click on “Next” after you’ve finished the problem.

Here is your Answer... T.A. = L.A. + A base = 408.41 + 314.16 = + (10)2 = 408.41 + 314.16 = 722.57 units2

Here is another one! Find the lateral area of a square pyramid with a slant height of 14 and a base edge of 25.

The answer you have selected is incorrect. The correct answer was 700. Don’t worry. . . The solution lies ahead!

Here is your Answer... LA = = = 700 units2

Pretty slick rick. Keep on truckin’

Now let’s pump up the VOLUME!

Mission 3 Mission 3: To successfully find the volume of boxes and cubes. Vocabulary: Volume, Capacity Formulas: V = A base  H

Volume Volume is the measure of how much space something takes up. Volume is measured in terms of units3. Capacity is the measure of how much something can hold Capacity is measured in litres E.g.: The measure of how much this Pepsi cup holds is its capacity.

V=AbaseH =l  w  h h l w Volume To find the volume of a box, one should use the formula listed below: V=AbaseH =l  w  h l h w

Let’s do one! Find the volume of a box with the dimensions to the right.

You’re a human calculator! Keep on rockin’!

I’m sorry. I’m sure you tried hard. The correct answer was 308 I’m sorry! I’m sure you tried hard. The correct answer was 308. Click on the thumb for an explanation

Here is your Answer... V = l  w  h = (4)(7)(11) = 308 units3 Remember… volume is measured in cubic units

Volume of A Cube What happens if the length, the width, and the height of the box are all the same? Well, you have a cube on your hands. The formula for the volume of a cube is:

Let’s do one! Find the volume of a cube with the dimensions to the right.

Raise the roof. You are a mathlete in training!

Even a cool dude like Raj got that one right. You must be too square.

Here is your Answer... V = s3 = (4)3 = 64 units 3

Cube root Suppose we know the volume of a sugar cube is 512 mm3. Could we find the length of the side of a cube? Yes, we can. But we must use something called a cube root.

Cube root V = s3 512 = s3 = s 8 = s Each side is 8 mm long So if the cube has a volume of 512 mm3, then use the Volume formula to solve for the length of a side. V = s3 512 = s3 = s 8 = s Each side is 8 mm long

Let’s do one! B.C. Plastics Incorporated is offering a special on plastic bins which each hold 2127 cubic centimetres of “stuff.” What is the length of an edge of the cube?

2 Legit! 2 Legit 2 Quit! Good work!

Check Please. The correct answer is 12. 6 cms Check Please. The correct answer is 12.6 cms. Click on your guide to see how it’s done!

Here is your Answer... = 12.86

Now let’s think outside the box

Mission 4 Mission 4: To successfully find the volumes of various prisms and cylinders. Vocabulary: Volume, Prism, Cylinder Formula: V = A base  H

Volumes of Prisms & Cylinders Recall that a prism is a 3-dimensional figure which has two congruent polygonal bases. Also recall that a cylinder is a 3 -dimensional figure with two circular bases.

Two Shapes . . . Same Formula The formula for the volume of both these figures is the same: V=AbaseH where A is the area of the base and H is the height of the figure.

Let’s do one! Find the volume of a cylinder with a radius of 4 and a height of 10. Recall that V = Abase  H. V is what we have to solve for, h = 10, A is the area of the base: V = Abase  H = 50.3(10) = 503 Abase =  r 2 = (4)2 = 50.3

Your Turn! Suppose a cylinder shaped silo is 50 metres tall and has a diameter of 10 metres. What is the volume of the silo?

Raise the roof!

That’s Ok. The correct answer is 3927.

Here is your Answer... V = Abase  H = (r 2) H = (5)2  50 = 3926.99 m3

Remember, if you can find the volume, you can also find the capacity

Let’s move on to POINTY volumes

Mission 5 Mission 5: To successfully find the volumes of pyramids and cones. Vocabulary: Volume, Pyramid, Cone Formula: V = Abase  H

Pyramid-Cone volume Recall that a cone is a 3 -D figure with a circular base and meets at a point called a vertex. Also recall that a pyramid has a polygonal base and meets at a point called a vertex.

V= Abase  H Do you see a pattern? The formula for the volume of both these figures is the same: V= Abase  H where A is the area of the base and H is the height of the figure.

Let’s see it! V = Abase  H, H = 35. Find the volume of a square pyramid with a height of 35 m and base length of 44 m. V = Abase  H, H = 35. The base is square A = s2 = 442 = 1936 m2 35 V = Abase  H = 1936  35 = 22 587 m3 44

Your turn! Suppose you are going to the carnival, and you buy a snow cone which has a height of 6 cm. and a volume of 40 cm3. What’s the radius of the base?

Do you feel the love? Way to go. You got that one right!

Don’t give up. We’re all cheering for you Don’t give up! We’re all cheering for you. Press on the math nerd for an explanation.

Here is your Answer... V = 1/3 Abase  H 40 = 1/3 ( r 2) H

Mission 6 Mission 6: To successfully find the volume of a sphere. Vocabulary: Volume, Sphere Formula: V =  r3

What’s a sphere? Basketballs. Volleyballs. Baseballs. Marbles. Gum Balls. Bubbles. All of these are examples of spheres. A sphere is a three-dimensional figure which is a circle in “all directions.”

Formula for a Sphere The formula for the volume of a sphere is where V is the volume and r is the radius of the sphere.

Practice makes perfect! Find the volume of a sphere with radius 12. V = 4/3  r3 We know that the radius is 12, so we can simply plug and play. V = 4/3  (12)3 V = 7238.23 44

Just do it! A junior bowling ball has a radius 4.3 cms. What is the volume of such a ball?

Time for you to continue. Good job on getting that one right!

Hey. That’s ok. These spheres are tricky. The correct answer is 333 cm3. Click on the nerd ball for some help.

Here is your Answer... V = 4/3  r 3 = 4/3 () (r)3 = 4/3 () (4.3)3 = 333.04 cm3

Move on!

Mission 7 Mission 7: To successfully find the surface area of a sphere. Vocabulary: Surface Area, Sphere Formula: A = 4  r2

Surface area Review Recall that surface area is the measure of the total surface of a 3-dimensional shape. For example, if you were hired to paint golf balls green, you would be painting the surface of the golf ball. The measure of the area that you painted is surface area.

Formula for a Sphere The formula for the surface area of a sphere is where S.A. is the surface area and r is the radius of the sphere.

This is how we do it! Find the surface area of a beach ball with radius of 50 cm. The formula for surface area of a sphere (beach ball) is : SA = 4  r2 = 4  (50)2 Therefore, the surface area is 31415.7 cm2 Notice that the answer is in square units …even though its round, and rounded.

Mi Problem es su Problem What is the radius of a bowling ball with a surface area of 225 cm2?

Right On! Wow. . . And I thought I had all the answers! I guess you’re just a pro now!

Even Leonard had trouble with that one. Click on him for an explanation

Here is your Answer... SA = 4  r 2 225 = 4  r 2 17.9 = r 2 4.23 = r

Looks like we made it!

Click on this handsome stud if you love Geometry Nerds !

Exit! I hope you’ve enjoyed the show. Now go 4th and Multiply