•

Dec 5, 2025

•

5 pages

Geometry 3D Shapes Unit Summary

Tristan Zoarob

@zerakat

This reference sheet covers essential formulas and concepts for 3D... Show more

1 / 5

Unit 10 Reference Sheet
A polyhedron is a solid that is bounded by polygons called faces, which enclose a single
region of space.
A polyhedr

Polyhedra and Basic Concepts

A polyhedron is a solid shape bounded by polygons called faces that enclose a region of space. Think of shapes like cubes, pyramids, and prisms.

A prism is a polyhedron with two congruent faces called bases that lie in parallel planes. The bases stay the same regardless of how you position the prism.

When working with pyramids or cones, remember that height and slant height are different measurements. The height is the perpendicular distance from the apex to the base, while slant height measures from the apex to the edge of the base.

Helpful Tip: Cavalieri's Principle states that if two solids have the same height and the same cross-sectional area at every level, they have the same volume—even if they look different!

Density and Surface Area Formulas

Density is calculated as mass divided by volume. Similarly, population density equals population divided by area. These concepts help us understand how tightly packed matter or people are in a space.

For rectangular prisms, the surface area formula is SA = 2 $lw + lh + wh$ , where l is length, w is width, and h is height.

For all other prisms and cylinders, the surface area formula is SA = hp + 2B, where h is the height between bases, p is the perimeter of the base, and B is the area of the base.

For pyramids, use SA = $lp/2$ + B, where l is slant height, p is perimeter of base, and B is base area.

Remember: Surface area calculations always include ALL faces of the solid!

Surface Area of Curved Shapes

The surface area of a cone can be calculated using the formula SA = πr $r + l$ , where r is the radius of the base and l is the slant height. Think of this as the circular base plus the curved surface.

For a sphere, the surface area formula is SA = 4πr². You can visualize this as four circles with the same radius as the sphere.

A hemisphere (half a sphere) has a surface area of SA = 3πr². This includes both the curved surface and the circular base.

Visualization Tip: The surface area of a sphere (4πr²) is exactly four times the area of its great circle (πr²), which is any circle that divides the sphere into two equal hemispheres.

Volume of Prisms and Pyramids

For rectangular prisms, the volume formula is V = lwh, where l is length, w is width, and h is height. This is simply the area of the base times the height.

The same principle applies to all prisms and cylinders: V = Bh, where B is the area of the base and h is the height.

For pyramids, the volume formula is V = (Bh)/3, where B is the area of the base and h is the height of the pyramid. Notice pyramids have exactly 1/3 the volume of prisms with the same base and height.

For cylinders, use V = πr²h, where r is the radius of the circular base and h is the height.

Fun Fact: If you fill a pyramid and a prism that share the same base and height with water, you'd need to empty the pyramid exactly three times to fill the prism!

Volume of Curved Solids

The volume of a cone is V = (πr²h)/3, where r is the radius of the base and h is the height. Notice this is exactly 1/3 the volume of a cylinder with the same base and height.

For a sphere, the volume formula is V = (4/3)πr³, where r is the radius. This formula helps calculate the space inside a ball or globe.

A hemisphere (half sphere) has a volume of V = (2/3)πr³, which is exactly half the volume of a complete sphere.

Application Note: These volume formulas are essential for real-world problems—like calculating how much water a tank can hold, how much air is in a balloon, or how much material is needed to make a solid object.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Geometry

•

121

•

Dec 5, 2025

•

5 pages

Geometry 3D Shapes Unit Summary

Tristan Zoarob

@zerakat

This reference sheet covers essential formulas and concepts for 3D shapes (polyhedra). You'll learn how to calculate surface area and volume for various solids like prisms, pyramids, cylinders, cones, and spheres—skills you'll definitely need for geometry tests and real-world applications.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Polyhedra and Basic Concepts

A polyhedron is a solid shape bounded by polygons called faces that enclose a region of space. Think of shapes like cubes, pyramids, and prisms.

A prism is a polyhedron with two congruent faces called bases that lie in parallel planes. The bases stay the same regardless of how you position the prism.

Helpful Tip: Cavalieri's Principle states that if two solids have the same height and the same cross-sectional area at every level, they have the same volume—even if they look different!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Density and Surface Area Formulas

For rectangular prisms, the surface area formula is SA = 2 $lw + lh + wh$ , where l is length, w is width, and h is height.

For all other prisms and cylinders, the surface area formula is SA = hp + 2B, where h is the height between bases, p is the perimeter of the base, and B is the area of the base.

For pyramids, use SA = $lp/2$ + B, where l is slant height, p is perimeter of base, and B is base area.

Remember: Surface area calculations always include ALL faces of the solid!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Surface Area of Curved Shapes

For a sphere, the surface area formula is SA = 4πr². You can visualize this as four circles with the same radius as the sphere.

A hemisphere (half a sphere) has a surface area of SA = 3πr². This includes both the curved surface and the circular base.

Visualization Tip: The surface area of a sphere (4πr²) is exactly four times the area of its great circle (πr²), which is any circle that divides the sphere into two equal hemispheres.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Volume of Prisms and Pyramids

For rectangular prisms, the volume formula is V = lwh, where l is length, w is width, and h is height. This is simply the area of the base times the height.

The same principle applies to all prisms and cylinders: V = Bh, where B is the area of the base and h is the height.

For cylinders, use V = πr²h, where r is the radius of the circular base and h is the height.

Fun Fact: If you fill a pyramid and a prism that share the same base and height with water, you'd need to empty the pyramid exactly three times to fill the prism!

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Volume of Curved Solids

The volume of a cone is V = (πr²h)/3, where r is the radius of the base and h is the height. Notice this is exactly 1/3 the volume of a cylinder with the same base and height.

For a sphere, the volume formula is V = (4/3)πr³, where r is the radius. This formula helps calculate the space inside a ball or globe.

A hemisphere (half sphere) has a volume of V = (2/3)πr³, which is exactly half the volume of a complete sphere.

Application Note: These volume formulas are essential for real-world problems—like calculating how much water a tank can hold, how much air is in a balloon, or how much material is needed to make a solid object.