Volume+and+Surface+Area

Volume is the amount of space an object takes up.

The volume of a space figure is the number of cubic units the figure contains.
 * You can count the cubes to find the volume of a space figure. ||
 * [[image:http://www.studyzone.org/testprep/math4/d/cube1.gif width="41" height="41" align="center"]] || 1 cubic unit ||
 * [[image:http://www.studyzone.org/testprep/math4/d/cube1.gif width="41" height="41"]][[image:http://www.studyzone.org/testprep/math4/d/cube1.gif width="41" height="41"]] || 2 cubic units ||

volume= =l x w x h=v= ||
 * = You can use a formula to find the ** volume ** of an object. = ||
 * =length x width x height=


 * Example ||
 * [[image:http://www.studyzone.org/testprep/math4/d/volu.gif width="199" height="149" align="center"]]

By counting the cubes you can see that there are 15 cubes. || Volume= 15 cubic units ||


 * [[image:http://www.studyzone.org/testprep/math4/d/book_worm_closeup_blink_sm_wht.gif width="84" height="84"]] || Remember:

You can find the volume of figures by using the formula volume = length x width x height. || = =

=It is important to get the unit for volume correctly. Here are some pointers:=

> Let's say that the width, length and height of a rectangular solid are given as: > Length (** L **) = 5cm and Height (** H **) = 3cm. You must make sure all units are the same. So 20mm becomes 2cm It is important to get the unit for volume correctly.
 * 1) ** First, make sure all the units are the same **
 * Width (** W **) = 20mm **BUT** ||
 * Width (** W **) || = 2cm ||
 * Length (** L **) || = 5cm ||
 * Height (** H **) || = 3cm ||

After ensuring the units are the same, we can calculate the volume, **V =**** W **** xLxH ** = 2×5×3 = 30 cm3
=Watch the following Video on how to calculate the Volume of a Prism=
 * 1) Notice the cm3 (cubic centimeter) above? This 'cubic' comes about because we multiplied the three lengths that are in cm. If ** W **, ** L ** and ** H **. Generally speaking, //unit//3 is the unit for volume where the //unit// can be mm, cm, m, km.

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= = =Volume of a Rectangular Solid= In review to find the volume (**V**) of a rectangular solid, we just multiply the width (** W **), length (** L **) and height (** H **) of the solid together as shown below:

> > =**V =**** W **** xLx **** H **=



The following link to the PDF file provides many practise questions on volume for you to try. You can then check your answers

Now that you have mastered volume in prisms, move onto volume in cylinders.

=Volume of a Cylinder= Now, to find the volume (**V**) of a cylinder, you just need to multiply the area of the base (**π **** R2 **) with the height (** H **) together. This formula is shown below: > =**V =****π x **** R x 2x **** H **= The symbol, **π **, is just a number that is approximately equals to 22/7 or 3.142. A cylinder is a right circular prism, and like all prisms, the volume is found by multiplying the area of one end of the cylinder by its height. = = =As a formula:= //π// is [|Pi], approximately 3.142 //r// is the [|radius] of the circular end of the cylinder //h// height of the cylinder ||
 * [[image:http://www.mathopenref.com/images/solid/cylindervolume.gif width="100" height="21"]] || where:

The following video provides an explanation of how to determine the volume of a cylinder.

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The Powerpoint below contains all of the above formulas as well as Measurement Conversions and various formulas for Volumes of Prisms

Click the button to view as a presentation

The attached PDF document has various volume basic formulae.For some volume problems click on the following PDF. You can check your answers in the next PDF =Calculating Surface Area.= =Below is an overview of all the formulas for surface area.=

**Surface Area of a Cube**
As the diagram below indicates, there are six surfaces to a rectangular prism. There is a front, back, top, bottom, left, and right to every rectangular prism. The surface area of a prism is nothing more than the sum of all the areas of these rectangles.

==Using the labeling of the general prism diagram above, a formula can be created for dealing with the surface area of prisms. You need to calculate the area of each surface and add them together. Remember area is l x w.==
 * [[image:http://www.mathguide.com/lessons/pic-demoprism.gif width="250" height="242" align="center" caption="Six Surfaces of a Rectangular Prism"]] ||
 * **Prism Surface Area Formula** ||
 * Top || : lw ||
 * Bottom || : lw ||
 * Front || : hl ||
 * Back || : hl ||
 * Left || : hw ||
 * Right || : hw ||
 * Total || : lw + lw + hl + hl + hw + hw ||
 * || : 2lw + 2hl + 2hw ||
 * || : 2(lw + hl + hw) ||


 * = **Surface Area of a Rectangular Prism** = ||

(a, b, and c are the lengths of the 3 sides)

The surface area of a rectangular prism is the area of the six sides 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 x 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 x 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 x b. Again, there are two of them, so their combined surface area is 2ab.

Overall the surface area of a reactangle is 2(a x c) + 2 (a x b) + 2 (b x c )
=Surface Area of a Triangular Prism=

Pyramids that have a square base have a total of five surfaces. To determine the shapes of those surfaces, we will start with a pyramid from step one below. If we cut along the lateral edges of the pyramid, we can allow the figure to flatten out in step two below. From step two, the individual figures are easily identified as a square and four triangles. We can use the area formulas for a rectangle and a triangle ; to determine the complete formula for the surface area of the pyramid. To recap, below are the formulas. The square (or base of the solid) has an area that can be calculated by multiplying its length times its width. Since those dimensions are equal, the area is s x s = s2.
 * || [[image:http://www.mathguide.com/lessons/pic-pyramidT.gif width="250" height="200" align="center" caption="General Pyramid"]]
 * General Pyramid** ||
 * [[image:http://www.mathguide.com/lessons/pic-demopyramid3.gif width="340" height="299" caption="Surfaces of a Pyramid"]] ||
 * [[image:http://www.passyworld.com/passyImagesNine/RectangleArea547x488JPG.jpg width="225" height="204" align="right" caption="Rectangle" link="@http://www.passyworld.com/passyImagesNine/RectangleArea547x488JPG.jpg"]][[image:http://www.passyworld.com/passyImagesNine/TriangleArea542x445JPG.jpg width="259" height="195" align="right" link="@http://www.passyworld.com/passyImagesNine/TriangleArea542x445JPG.jpg"]] ||

Now we need to calculate the area of the remaining surfaces. The remaining surfaces happen to form the lateral surface area of the pyramid, which are triangles. The area formula for a triangle is its base times its height divided by two. We determine the area for one triangle and multiple this answer by 4 to get the total surface area of all 4 sides.

The total surface area of the pyramid is equal to the area of the base plus its sides.


 * [[image:http://www.mathguide.com/lessons/pic-pyramidtri.gif width="175" height="137" caption="Internal Right Triangle of a Pyramid"]] ||


 * = **Surface Area of a Sphere** =

**The surface area of a spher is given by the formula 4 x pi x r x 2** || (r is radius of circle)


 * =**Surface area of a cylinder**= ||

A cylinder has a total of three surfaces: a top, bottom, and middle. The top and bottom, which are circles, are easy to visualize. The area of a circle is πr2. So, the area of two circles would be πr2 + πr2 OR 2πr2. To recap the area of a circle is:
 * || [[image:http://www.mathguide.com/lessons/pic-cylinderT.gif width="200" height="137" align="center" caption="General Cylinder"]]
 * General Cylinder** ||



(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
The tops and bottoms are circles. We calculate the area of one circle and double it to get the area of both.

The easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.



You can find the area of the top (or the bottom). That's the formula for area of a circle (//pi// r2). Since there is both a top and a bottom, that gets multiplied by two. The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides.

= We need to determine one side. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. =

= 3 steps =

2. Calculate the length of your retangle. To do this determine the perimeter of the top of your cylinder. To do this we use the formula for circumference. To recap:
= 3. Determine the area of the rectangle using l x w. Use your answer to circumference for one lenght and multiple this by the height. =

4. Add all the areas together.
Watch the video below for a further explanation of this.




 * **Tip! Don't forget the units.** ||

How to Calculate the Surface Area of a Cylinder

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Examples of surface area calculations can be found in the following PDF file. Click the link. with answers here

Extension: Click on the following link for some more examples and problems to calculate

Review all of the concepts covered in this wikipage by viewing the following video.

Revsion

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