FINDING THE VOLUME OF A [RIGHT] RECTANGULAR PRISM/CUBE:
The formula for this is V=Bh or Volume is equal to the base times the height of the figure. To find the area of the base, first you must multiply the width times the length of the cube. Let's see an example;
As you can see (if you can see, that is), the width of the prism is 6 inches, the length is 4 inches, and the height is 3 inches. So to find the area of the base, we need to multiply 6 (w) times 4 (l) to get 24 cubic inches as (B) the base. Pretty simple right? Now that we have the area of the base, we can multiply it by the height to find the volume. Now multiply (B) 24 by (h) 3 to get 72 as the volume (V). Tada!
Hopefully, you now understand how to find the volume of a rectangular prism or a cube.
FINDING THE VOLUME OF [RIGHT] TRIANGULAR PRISMS:
We still must find the area of the base first for this. Our formula to find the volume is the same V=Bh, But, the formula for the base is not the same. For cube's it is B=wl or width times length, for Triangular Prisms the formula for finding the base is B=½ bH or the area of the Base is equal to ½ of the base (or leg) of the triangle times H the altitude to the base.
Here we (might) see both the 3D triangular prism and it's Base. So you can tell the difference between the two bases in this, The actual base of the prism I will write with a capital B like this; Base. And the other base, which is more like a leg I will write like this; base. And though that was most likely not needed for me to specify, I did. So let's get back to the problem. To find the area of the Base we must multiply the base (leg) times the altitude. So, 6 times 9 is equal to 54, but unfortunately it cannot be that easy. So let's take 54 and divide it in half. 54 divided by 2 equals 27. We now know that the area of the Base is equal to 27 cubic centimeters. Onto the Volume, we will use the same formula as we did for the cube. V=Bh We know the Base (B) is 27 and the height (h) is 5, 27 times 5 is equal to 135. So our volume for this particular right triangular prism, is 135 cubic centimeters.FINDING THE VOLUME OF A [RIGHT] CONE
As always, first we must find the area of the Base before we can calculate the Volume. Our formula for finding the Base of a cone is B = π r2, or the area of the Base is equal to pi times the radius of the circle to the second power. Let's take a look at a sample. So for this cone, we have a radius (r) of 5 m
eters, raise that to the second power, and we have 25 meters! (In case anyone has forgotten, when you raise a number to the second power, you multiply it times itself. So in this case, 5 times 5 to get 25.) So, the area of our base is 25 π m2. Now to find our Volume, we need to use the formula V = ⅓ Bh. Or the Volume equals the area of the Base times the height of the cone divided by 3. Let’s try it. We’ve determined that the area of the base is 25 and the height is 6. 25 times 6 equals 150, and 150 divided by 3 is 50. So the volume of our cone is 50 cubic meters. Not too hard, huh?
eters, raise that to the second power, and we have 25 meters! (In case anyone has forgotten, when you raise a number to the second power, you multiply it times itself. So in this case, 5 times 5 to get 25.) So, the area of our base is 25 π m2. Now to find our Volume, we need to use the formula V = ⅓ Bh. Or the Volume equals the area of the Base times the height of the cone divided by 3. Let’s try it. We’ve determined that the area of the base is 25 and the height is 6. 25 times 6 equals 150, and 150 divided by 3 is 50. So the volume of our cone is 50 cubic meters. Not too hard, huh?FINDING THE VOLUME OF A [RIGHT] PYRAMID
Let’s move on to pyramids. Pyramids are similar to cones; the biggest difference is the shape of its base. A pyramid’s base is always a polygon; we’ll start with a right rectangular pyramid. We will use the same formula for this type of pyramid as we did for cones. So our formula will be V
= ⅓ Bh. And to find the area of the base, we will use the formula for cubes, B = wl or the area of the base is equal to the width times the length. The width is 6 and the length is 5, so let’s multiply 5 times 6 to get 30. Now, let’s calculate in the height, 30 times 8 is 240, divide 240 by 3 and our Volume is 80. Now let's see a different kind of pyramid, This one is called a Right Triangular Pyramid. The formula is a little different for the base, But our Volume formula is still V = ⅓ Bh, I'm not explaining that in words again, if you've already forgotten it, please read the first two instructions again. Our formula for finding the Base is B = ½ bH just like for the triangular prisms! So to refresh your memory, The area of the base is equal to the base (b/leg) times the altitude (H) divided by two. What is that? You want to see a sample?Well I don't have one yet. Okay, okay, I'll draw one up for you. Just stop begging.
FINDING THE VOLUME OF A [RIGHT] CIRCULAR CYLINDER
Just like finding the area of anything that involves a circle, you must use the useless pi symbol. So let's take a gander at our two formulas for this particular three dimensional shape. Our Volume formula is V = Bh or the Volume is the area of the Base times the height. And our Base formula is B = π r2 or the area of the Base is equal to pi times the radius to the second power. Hey kids, guess what time it is? Sample tiime♪ !
Can you see that one? Kind of? Well, the radius of this one is 4 meters, so 4 to the second power is 16, so the area of our base is 16π m2, now let's multiply that times the height and we'll have our Volume. 16 times 5 is 80. 80 cubic meters is our Volume for this Right Circular Cylinder. Not too hard, huh? Now promise me something, Don't forget this before your test.What is that? What about variables? What about them? Oh! You don't remember how to do them, gotcha. Then read on!
FINDING THE MISSING HEIGHT OF A CUBE / RECTANGULAR PRISM
Example: A cube has a Volume of 4352 cubic feet, its sides are both 16 feet, what is the height of this box?
To find the answer to this question, we'll multiply the values we know, which is the sides. So 16 times 16 is equal to 256. To find the height of this box, we will now divide 4352 by 256, which gives us 17. So the missing variable, which is the height, is equal to 17 feet.
The formula for this is: not currently remembered.
FINDING THE MISSING LENGTH/WIDTH OF A CUBE / RECTANGULAR PRISM
Example: A right rectangular prism has a volume of 240 cubic inches. The height, h, of the prism is 8 inches and the width, w, of the rectangular base is 6 inches. What is l, the length of the base?
To start off, take the volume, 240, and divide it by the height, 8. 240 divided by 8 is equal to 30. Now divide 30 by the width, 6. 30 divided by 6 is 5. So the length of the base is 5 inches.
The formulas for this is V = B times h and the other formula is B = l times w.
Note: If you are trying to find the width rather than height, I'm fairly certain this works both ways. So instead of dividing the product of the volume divided by the height by the width, divide the product by the height to get the width.
FINDING THE MISSING RADIUS OF A CONE
Example: A right circular cone has a volume of 32π inches3 and a height of 6 inches. What is the radius of this cone?
To start out, Let's take the Volume, which is 32 and bring it to the third power. So our Volume is now 96. Now take our height, which is 6 and divide it into 96. So 96 divided by 6 is equal to 16. To get your final answer for the radius, we'll find the square root of 16, and then we will know the radius of this cone. The square root of 16 is 4. So the radius of this cone is 4.
The formulas for this problem is V = 1/3 times B times h. And our other one is B = π r2 .
FINDING THE MISSING HEIGHT OF A CYLINDER
Example: A right circular cylinder has a volume of 16π cubic meters and a radius of 2 meters. What is the height?
First, let's take the radius and turn it into the area of the base. Remember how to do this? We need to take it to it's second power, 2 to the second power is 4, so 4 is our area. Now for our volume which is 16, let's divide that by 4. Oh look! The answer is 4, that isn't a surprise considering 4 is the square root of 16. So did you see what I did? First I found the area of the base, then I divided the volume by it, to find our missing variable, the height.
Our formulas; V = B times h and our other is B = π radius to the second power.
FINDING THE MISSING HEIGHT OF A TRIANGULAR PRISM
Example: A right triangular prism has a volume of 21 cubic units. The base is a right triangle with legs that has a length of 2 units and another length of 3 units. What is the height of this prism?
Since we have the lengths, we need to find the area of the Base first. So 2 times 3 is 6, but since it's a triangle, we must divide this in half. So now let's divide 3 into 21, and we should get 7. Meaning 7 is out height!
Formulas: V = B times h and B = b times H divided by 2
FINDING THE MISSING HEIGHT OF A CYLINDER
Example: A right circular cylinder has a volume of 16π cubic meters and a radius of 2 meters. What is the height?
First, let's take the radius and turn it into the area of the base. Remember how to do this? We need to take it to it's second power, 2 to the second power is 4, so 4 is our area. Now for our volume which is 16, let's divide that by 4. Oh look! The answer is 4, that isn't a surprise considering 4 is the square root of 16. So did you see what I did? First I found the area of the base, then I divided the volume by it, to find our missing variable, the height.
Our formulas; V = B times h and our other is B = π radius to the second power.
FINDING THE MISSING HEIGHT OF A TRIANGULAR PRISM
Example: A right triangular prism has a volume of 21 cubic units. The base is a right triangle with legs that has a length of 2 units and another length of 3 units. What is the height of this prism?
Since we have the lengths, we need to find the area of the Base first. So 2 times 3 is 6, but since it's a triangle, we must divide this in half. So now let's divide 3 into 21, and we should get 7. Meaning 7 is out height!
Formulas: V = B times h and B = b times H divided by 2
FINDING THE MISSING HEIGHT OF A RECTANGULAR PYRAMID
Example: A right rectangular pyramid had a volume of 108 units, a length of 9 units and a width of 6 units. What is the height of this pyramid?
Let's start with finding the area of the base, 6 times 8 is equal to 48. Now take 48 and divide it by 3 so we can accurately find the height. 48 divided by 3 equals 16. It's time to bring the volume into play. The volume, 80, divided by 16 is equal to 5. So 5 is our height of this pyramid.
Formulas: B = w times h. V = 1/3 Bh
If I missed something, Please comment and let me know what. I'll get it up as soon as possible. I hope this helps! Also sorry there's no pictures, but you couldn't really see them anyways, or at least... I couldn't.
