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Master Surface Area and Volume of Prisms: Key Geometry Skills

Geometry: Master Shapes, Proofs & Spatial Reasoning

Measuring Systems

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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

Metric systems in measuring 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 in measuring systems

Imperial systems

Imperial Length Conversion

Conversions involves squares and cubes

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

Surface Area and Volume

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

Points, Lines, Segments, Midpoints and Chords

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

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>

Slope-Y-intercept form: y = mx + b

Slope intercept form: y = mx + b

Graph using a single point and slope

Solve linear equations

Introduction to linear equations

Circles, plane and coordinate

Angles in a circle

Chord properties in circle geometry

Tangent properties in circle geometry

Circles and circumference 

Circle properties

Arcs of a circle

Areas and sectors of circles

Central angles and proofs

Inscribed angles and proofs

Central and inscribed angles in circles

Circle chord, tangent, and inscribed angles proofs

Triangles and special triangles

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Pythagorean theorem

Pythagorean Theorem

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

Isosceles and equilateral triangles

Angles and trigonometry

Use sine ratio to calculate angels and side (Sin = \(\frac{o}{h})\)

Use cosine ratio to calculate angles and side (Cos =  \(  \frac{a}{h}\)  )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Use tangent ratio to calculate angles and side (Tan =  \(  \frac{o}{a}\)  )

Transformations

Line symmetry

Rotation symmetry and transformations

Similarity

Scale diagrams

Similar triangles

2D figures and 3D figures

Surface area of three-dimensional (3D) objects

Nets of three-dimensional objects

Tessellations using rotations

Probability

Fundamental counting principle

Fundamental Counting Principle Involving Restrictions

Permutations

Solve problems involving both permutations and combinations

Problems involving both permutations and combinations

Probability of independent events 

Probability with Venn diagrams

Vectors

Word problems on vectors

Introduction to vectors

Magnitude of a vector

Direction angle of a vector

Equivalent vectors

Adding and subtracting vectors in component form

Set Theory

Set notation

Understanding How to Use Set Notation

Set builder notation

Intersection and union of 2 sets

Intersection and Union of 2 Sets

Intersection and union of 3 sets

Intersection and Union of 3 Sets

Interval notations

Sine Rule and Cosine Rule

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Applications of the sine law and cosine law

Bearings

Introduction to bearings

Bearings and direction word problems

Bearings and Direction Word Problems

Angle of elevation and depression

Logic

Inductive reasoning

Statements

Negations

Conjunctions and disjunctions

Truth tables

Conditionals

Inverses, converses, and contrapositives

Biconditionals

Deductive reasoning

Algebraic proofs

Reasoning

Consecutive integers

Number sequences

Finding the missing digit

Angles, lines and transversals

Parallel line equations

Parallel line equation

Parallel and perpendicular lines

Combination of both parallel and perpendicular line equations

Pairs of lines and angles

Parallel lines and transversals

Parallel line proofs

Perpendicular line proofs

<p>Get better math marks with our complete Geometry help, whether it&apos;s <a href="http://www.corestandards.org/Math/Content/HSG/introduction/">Geometry common core standards</a>, Geometry <a href="http://www.fldoe.org/accountability/assessments/k-12-student-assessment/end-of-course-eoc-assessments/index.stml">EOC</a>, or <a href="https://www.engageny.org/resource/geometry-module-1">EngageNY Geometry</a> (Regents Examination). We've got you covered!</p>
<p>Keeping with your class or textbook, our thorough help for Geometry includes topics such as Similar triangles, Geometry proofs, right triangle trigonometry, 3D shapes, Triangle geometry, Midpoint, Surface area of circles, Translations, and more. Learn the concepts with our video tutorials that show you step-by-step solutions to even the hardest geometry problems. Then, strengthen your understanding with tons of geometry practice.</p> 
<p>All our lessons are taught by experienced Geometry teachers. Let&apos;s finish your homework in no time, and ACE that final.</p>


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math

<h2>What is Geometry?</h2>
<p>Understanding geometry can be difficult. With so many sub-topics under the umbrella of Geometry it can be difficult to grasp what exactly is Geometry, <a href="https://www.studypug.com/blog/">why and when could ever be useful?</a> Generally speaking, Geometry is study of shapes &ndash; how they can be transformed, scaled, measuring/calculating the angles found within them, as well as other characteristics such as area, volume, and so forth. The study of Geometry also involves the study of lines, the building blocks of shapes, and their properties. Here at StudyPug, Geometry will also introduce you to the realm of statistics where we look at probability, and how this is to shapes (i.e. Venn diagrams).</p>

<h2>Who is known as the father of geometry?</h2>
<p>The honour and prestige of this title was bestowed to the one and only Euclid of Alexandria. He was an incredible Mathematician who studied at Plato&apos;s Academy in Greece and the first to study Geometry.  He went on to publish a variety of textbooks used by mathematicians to this day. He is most known for his work on a publication titled "The Elements" that feature ground-breaking content on algebra, number theories, and of course, geometry. It is said that this masterpiece was what earned him the title of "Father of Geometry" (<a href="https://en.wikipedia.org/wiki/Euclid">https://en.wikipedia.org/wiki/Euclid</a>).</p>

<h2>Is geometry hard?</h2>
<p>Studying Geometry can have its challenges and learning curves, but that&apos;s why we&apos;re here! Here at StudyPug, we offer help for all - those who fall under the category of Geometry for dummies or those who are looking for more practice and/or advanced questions. At StudyPug we strive to make learning geometry easy, accessible, and engaging. With over 1000&apos;s of lessons, Step-by-step examples, and 24/7 help, we are prepared to help students with endless opportunities for easy practice, be it for Geometry exam preparation or to simply to reinforce learning. Here at StudyPug we cover a wide variety of questions you may encounter and we also use a variety of Geometry textbooks (e.g. Prentice Hall Geometry; Glencoe Geometry; McDougal Littell Geometry; Big Ideas Math Geometry) to cater comprehensively to the content you may be learning. Rest assured, we&apos;re well equipped to help with the fundamental concepts and basic problems that will allow you to progress confidently to more complex, challenging questions at your own pace.</p>

<h2>What is Euclidean geometry?</h2>
<p>Euclidean geometry is the branch of geometry that is focused on the study of 2-dimension solid geometry and planes. Some examples include basic facts such as the interior angles of triangles should equal to 180 degrees and that the shortest distance between point A and B is a straight line. Euclidean geometry was built on the basis of Euclid mathematical theories and postulates, most notably composed and synthesized in his publication "The Elements". In a broader sense, Euclidean geometry is also referred to the importance of utilizing cognitive abilities in Math. This encompasses the learner to defer techniques of memorization and formulas, and to rely on a strong, holistic understanding of the subject that allows application of skills and techniques to solve Math problems. While you may have begun studying Euclidean geometry early on in school, you will soon begin exploring concepts of non-euclidean geometry (e.g. properties of 3D shapes, such as a sphere).</p>

<h2>How to pass geometry?</h2>
<p>Whether you&apos;re preparing for a Geometry test, exam, or simply require Geometry help, StudyPug is here to help you succeed and pass your assessments through our Geometry tutoring and Geometry lessons.  Our Geometry tutors will cover content, study tips, strategies, and practice questions that will provide you with as much help with Geometry as you need. We recommend starting with the difficult sub-topics/chapters you are struggling with, follow our Geometry tutorials as we take you step-by-step through the process of understanding a concept or problem you may have. Some concepts or problems are trickier than others so you may need to revisit theses areas more than once, which shouldn&apos;t be a problem given your unlimited, 24/7 access. Once you are feeling confident, we encourage you to practice, practice, practice! This enables you to be prepared for different styles of questions as well as different levels of difficulty. These questions will supplement your learning and truly test your understanding.</p>

<h2>What grade is geometry?</h2>
<p>Students begin learning Geometry in 10th Grade. Students should have a strong understanding of <a href="[[region_textbook.html|{"textbook_key":"algebra-1"}]]">Algebra 1</a> (completed in 9th Grade) before beginning to tackle topics covered in Geometry. For a refresher, please see our <a href="[[region_textbook.html|{"textbook_key":"algebra-1"}]]">Algebra 1 page</a>.  Following the completion of Geometry, students may proceed to beginning <a href="[[region_textbook.html|{"textbook_key":"algebra-2"}]]">Algebra 2</a>. Ahead of the game? Wonder what comes after Geometry? Visit our <a href="[[region_textbook.html|{"textbook_key":"algebra-2"}]]">Algebra 2 page</a> for more learning.</p>

https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-geometry.png

Ex. 1

Intro

<p>- 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?<br />
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<br />
"S", okay?<br />
So that is a right triangle right here. That means you could use the C-square equals A-square plus the B-square. Okay?<br />
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?<br />
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?<br />
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?<br />
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?<br />
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?<br />
That's it for this question. Thanks for watching.</p>


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

Surface area of prisms

Volume of prisms