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Master Surface Area and Volume of Prisms: Key Geometry Skills

MA.912.GR.1.1

Pairs of lines and angles

Parallel lines and transversals

Parallel line proofs

Perpendicular line proofs

MA.912.GR.1.2

Classifying triangles

Congruence and congruent triangles

Triangles congruent by SSS proofs

Triangles congruent by SAS and HL proofs

Triangles congruent by ASA and AAS proofs

MA.912.GR.1.3

Isosceles and equilateral triangles

MA.912.GR.1.6

Similar triangles

Similar polygons

MA.912.GR.2.1

Rotational symmetry and transformations

MA.912.GR.2.2

Enlargements and reductions with scale factors

MA.912.GR.2.3

Scale diagrams

MA.912.GR.2.5

Introduction to transformations

Horizontal and vertical distances

MA.912.GR.2.6

Draw on coordinate planes

MA.912.GR.2.8

Cartesian plane

Cartesian Plane — Coordinates

MA.912.GR.3.1

auto-gen

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

MA.912.GR.3.2

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Slope equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Slope intercept form:  y = mx + b

Slope intercept form: y = mx + b

General form:  Ax  + By  + C = 0

General form: Ax + By + C = 0

Point-slope form: y - y_1 = m(x - x_1)

MA.912.GR.3.3

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and slope

Word problems of graphing linear functions

MA.912.GR.4.1

Surface area of 3-dimensional shapes

MA.912.GR.4.2

Line symmetry

MA.912.GR.4.3

Introduction to surface area of 3-dimensional shapes

Nets of 3-dimensional shapes

MA.912.GR.4.5

Surface area and volume of prisms

Surface area and volume of pyramids

Surface area and volume of cylinders

Surface area and volume of cones

Surface area and volume of spheres

MA.912.GR.4.6

Surface area of prisms

Surface Area of Prisms Rectangle

Calculate the Surface Area

Surface area of cylinders

MA.912.GR.5.1

Metric systems

Add and Subtract Mixed Units

Metric Length Conversion

Add or Subtract Different Metric Units Convert to CM

Add or Subtract Different Metric Units Convert to M

Add or Subtract Different Metric Units Convert to MM

Imperial systems

Imperial Length Conversion

MA.912.GR.5.2

Conversions between metric and imperial systems

Metric to Imperial Conversions

Imperial to Metric Conversions

Conversions Between Metric and Imperial Systems Speed KM to MI

Conversions Between Metric and Imperial Systems Speed  MI to M

Metric-Imperial Length Conversion Inch To Cm

Metric-Imperial Length Conversion MM/CM/M/KM To FT

Metric-Imperial Length Conversion MM/CM/M/KM To MI

Conversions involving squares and cubic

Imperial to Metric Conversions — Squares and Cubes

Imperial Conversion — Squares and Cubes

Metric to Imperial Conversion — Squares and Cubes

Metric Area Conversion FT to CM

Metric Area Conversion Foot to Inch

Metric Area Conversion M/CM to Inch

MA.912.GR.5.3

Circles and circumference

Circle properties

MA.912.GR.6.1

Arcs of a circle

MA.912.GR.6.2

Areas and sectors of circles

Central angles and proofs

Inscribed angles and proofs

MA.912.GR.6.3

Central and inscribed angles in circles

Circle chord, tangent, and inscribed angles proofs

MA.912.GR.6.4

Angles in a circle

Chord properties

Tangent properties

MA.912.GR.7.2

Conics - Circle

MA.912.GR.7.3

Conics - Parabola

Conics - Ellipse

Conics - Hyperbola

MA.912.LT.4.10

Biconditionals

Deductive reasoning

Algebraic proofs

MA.912.LT.4.3

Inductive reasoning

Statements

Negations

Conjunctions and disjunctions

Truth tables

Conditionals

Inverses, converses, and contrapositives

MA.912.T.1.1

Use sine ratio to calculate angles and sides (Sin = o / h)

Use cosine ratio to calculate angles and sides (Cos = a / h)

Combination of SohCahToa questions

Use tangent ratio to calculate angles and sides (Tan = o / a)

MA.912.T.1.2

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Pythagorean theorem

Pythagorean Theorem

Estimating square roots

Using the pythagorean relationship

Applications of pythagorean theorem

math

Ex. 1

Intro

- Hi, so welcome to this question here. Well, we're trying to find the surface area and the volume for this question here. It looks like a simple one but it's not, okay. Because it's different shapes, different lengths and the width for the different dimensions here. So we'll start off by figuring out the surface area, okay? 
So to figure out the surface area, that means you get the front and the back, you get the bottom piece, you get the slanting side right here. You also get the side over here which is the left side, okay? So a couple of things we should check before we start the question, it's to check all the dimensions to see that they all have the same units for all of them. We get the millimeter, millimeter and the centimeter. So I'm going to change this centimeter into millimeter first. But we know that for every centimeter, it's just a 10 millimeter, right? So I'm just going to change this number to 12 millimeters right away. So just let me change a color panel here, so I'm going to change it into 12 millimeter here so you won't forget, okay? Let me erase that quickly, 12 millimeters. Okay. So coming back over here. So then once now we've changed it, we need one more dimension that's not given to us. Okay, because if we're going to calculate the surface area, eventually we&apos;ve got to calculate this side right here. So we need the dimension of this side right here, this slanting side, okay? So we need a little bit of math which is called the pythagoras. So we need to figure out this slanting side here which we called the 
"S", okay? 
So that is a right triangle right here. That means you could use the C-square equals A-square plus the B-square. Okay? 
Let me see if we can find a little space right here to show you the work, so let me sign in red. So coming down over here, let's just kind of do the equation S-square equals to A-square plus the B-square, okay? So to find the S, you're going to square root whatever you have for the two sides. So looking for this one right here, so we&apos;ve got the 16 and 12, 16 and the 12 okay? So if you do the math, you're going to figure out the slanted side is going to turn out to be-- it's approximately 20, okay? So let's just use 20 over here, and let me just put equals to 20 over here as well. Okay? So now once we have all the dimensions, we're now ready to go ahead and calculate all the surface area that we have. All right? 
So to start off is really up to you. You can calculate area here, the top, the front, the back, it doesn't matter. I'm just going to follow my work right here. I'm going to do the front and the back which is this side right here and the ones in the back. Okay? So I'm going to call in the F and the B. So the front part of triangular prism here is actually triangular shape, right? So you're going to use the base times the height divided by two equation. So we're going to do the base times the height divided by two to find the front part. But since there's two identical pieces, so you're going to multiply it by two anyway. So to save yourself a work, two divided by two, that part is gone. So just take the base which is the 12, okay? Just going to multiply it by the 16, okay? So we&apos;ve got a nice number on the side right here, it's going to give us 192, okay. It's the millimeter square, that's the 192 over here. Then we're going to look at what other shapes that we have. So we still have the bottom piece. So let's say we're going to do the bottom piece right now. So let's come down to here for the bottom piece. Let me fill in the whole work here for you. Okay? Now if you look at the bottom piece, it's actually going to be a rectangular shape. We can actually draw a little bit of lines in there to show you that the shape is going to look like something like this, right? Okay, and I believe this one is down this way. Okay, and that's it. So it's a rectangular piece. So that means you just going to take the length times the width for the area calculation. Okay? 
So coming back over here, it's just going to be the length times the width. So we&apos;ve got the length right here, it's actually 12 times 12. So it's actually a cube. So it's 12 times 12, let's just square that number right there. So we&apos;ve got 144 millimeter square. Okay, the 144. Then what else do we have? We still got the left side which is nice and easy to calculate because we do have the dimension already. If this is 12, that's going to be 12 right here, and that's 16. So let's say we do the one on the left. So this is the left, okay? To find the surface area, again it's just the length times the width. So we&apos;ve got the 12 and the 16. So let's put down 12 times 16. A little messy here, let me get it erased right there, 12 times 16 and my answer to that part right there would be 192. Okay, so 192 meter square. Okay? 
So coming down to the last side, okay it's a little bit of more work because every side is a little different. So this slanted side right here. Now remember the reason we'll be calculating the S is so that we can get the surface area because the length is 12 here and then the height will be 20, okay? So let's call it a slanted side right there, the slanted. We&apos;ve got the length times the width again. You can put in any number you like in any order because we're doing multiplication anyway, 20 times the 12. We&apos;ve got a nice number right here, it's 240 centimeter square. So at the end, to get all the surface area all you have to do is just add everything together. So if you punch it in a calculator, you should get a nice number which going to give you 768, okay? Don't forget the unit, square. Okay? Now, coming back to the very last part of the question, it's to find the volume. Now, the volume it's much, much easier because there's-- use the equation, right? 
So I'm going to put a little space right here. To figure the volume of a rectangular prism is equation is just the base of the area, the base area actually. I'm just going to call it the base area, and multiply it by the height of that triangle which is going to be the 16, okay? So multiply it by the height. So the base area is going to be 12 times the 12, and the height is going to be the 16, okay? So we&apos;ve got to take-- Oh sorry, it's not the base. Sorry, my bad. It's actually the triangular part is your base, okay? It's the 12 times the 16 divided by two. So we&apos;ve got 12 times the 16 divided by two because you're doing a triangular prism. The base is not the rectangular part, it's the triangular part. My bad. So it's the 12 times 16 divided by two and then you multiply it by the height. Now the height, if you would treat this side as a base, then the height is going this way, okay? Then the 12 is the height in this question, okay? So the volume for this one at the end is going to be 1,152 millimeter cube. Okay? 
That's it for this question. Thanks for watching.

So, we've learned the basics in the "Introduction to volume" lesson. Now, let's practice some harder questions on how to find out the surface area and volume of prisms and composite solids. 

Finding the surface area and volume of prisms

Volume of prisms