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Dec 5, 2025
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Mastering Surface Area and Volume: Test Prep
Ethan Richards
@ethanrichards_howi
1 / 4
Surface Area & Volume Calculations
Figuring out surface area means adding up all the external faces of a 3D shape. For a hexagonal prism, you'll need to find the lateral area (LA) plus the area of the two bases. The formula typically looks like: SA = LA + 2B, where B represents the base area.
For a cylinder, the surface area calculation combines the curved surface (2πrh) and the two circular ends (2πr²), giving us SA = 2πr. Problems like the one with a 2-foot height and 1-foot radius demonstrate this, resulting in 12π ft².
Volume calculations determine the space inside a 3D shape. For pyramids, the formula is V = ⅓BH, where B is the base area and H is the height. For cylinders, it's V = πr²h, as shown in the problem with a 15 cm diameter cylinder.
💡 When working with composite figures (shapes made of multiple 3D objects), break down the problem by finding the surface area or volume of each part separately, then combine them appropriately.
Complex Surface Area & Volume Problems
When tackling complex figures like a square pyramid, remember that you need to find the slant height to calculate the lateral area. The lateral area is half the perimeter times the slant height , which is then added to the base area for total surface area.
For a cone, the volume formula is V = ⅓πr²h. This is similar to a pyramid's formula but adjusted for a circular base. In problem 8, a cone with radius 12 m and height 22 m has a volume of 264π m³.
Composite figures require breaking the shape into familiar parts. In problem 11, we see a half-cylinder combined with a rectangular prism. The solution involves calculating half the surface area of the cylinder plus the surface area of the prism, and similarly for volume.
🔍 When a problem gives you the volume and asks for surface area (or vice versa), you'll often need to work backwards by finding the radius or height first, then using that measurement in the other formula.
Working Backwards with Volume & Surface Area
Sometimes you'll need to determine dimensions when given volume or surface area. For a cube with volume 148,877 mm³, you can find the edge length by taking the cube root: 53 mm. The surface area is then 6 times the square of this edge length.
When dealing with composite shapes like a cylinder topped with a hemisphere, break down the volume formula into parts. The hemisphere contributes ⅔πr³ (half the volume of a sphere), while the cylinder adds πr²h. Using these together lets you solve for unknown dimensions.
Understanding geometric relationships is crucial. In problem 14, comparing a sphere inside a cylinder reveals an important volume ratio. This type of comparison helps you recognize patterns that can simplify complex calculations.
🧩 When solving for missing dimensions, always identify what you know and what formula connects it to what you need. This systematic approach makes seemingly complex problems manageable.
Application Problems & Similar Figures
Real-world applications often involve regular polygons as bases. A concrete pillar with a hexagonal base demonstrates this: simply multiply the base area (4.5 square feet) by the height (22 feet) to find the volume of 99 cubic feet.
Hollow objects, like metal pipes, require finding the difference between two volumes. For a pipe with outer diameter 4 inches and inner diameter 3.5 inches, calculate the volume of the outer cylinder and subtract the volume of the empty inner space.
When working with similar figures, like the cylinders in problem 18, remember that the ratio of volumes is equal to the cube of the ratio of corresponding linear measurements. For cylinders with heights in a 4:6 ratio, the volumes have a 64:216 ratio.
🔧 Many real-world applications involve composite shapes or hollow objects. Breaking these down into basic geometric components makes them much easier to analyze and calculate.
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?
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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
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
Mastering Surface Area and Volume: Test Prep
Ethan Richards
@ethanrichards_howi
Ready to master surface area and volume calculations? This unit practice covers how to calculate these key geometric measurements for various 3D shapes, from simple prisms to complex composite figures. You'll apply formulas, solve for missing dimensions, and work with... Show more
Surface Area & Volume Calculations
Figuring out surface area means adding up all the external faces of a 3D shape. For a hexagonal prism, you'll need to find the lateral area (LA) plus the area of the two bases. The formula typically looks like: SA = LA + 2B, where B represents the base area.
For a cylinder, the surface area calculation combines the curved surface (2πrh) and the two circular ends (2πr²), giving us SA = 2πr. Problems like the one with a 2-foot height and 1-foot radius demonstrate this, resulting in 12π ft².
Volume calculations determine the space inside a 3D shape. For pyramids, the formula is V = ⅓BH, where B is the base area and H is the height. For cylinders, it's V = πr²h, as shown in the problem with a 15 cm diameter cylinder.
💡 When working with composite figures (shapes made of multiple 3D objects), break down the problem by finding the surface area or volume of each part separately, then combine them appropriately.
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Complex Surface Area & Volume Problems
When tackling complex figures like a square pyramid, remember that you need to find the slant height to calculate the lateral area. The lateral area is half the perimeter times the slant height , which is then added to the base area for total surface area.
For a cone, the volume formula is V = ⅓πr²h. This is similar to a pyramid's formula but adjusted for a circular base. In problem 8, a cone with radius 12 m and height 22 m has a volume of 264π m³.
Composite figures require breaking the shape into familiar parts. In problem 11, we see a half-cylinder combined with a rectangular prism. The solution involves calculating half the surface area of the cylinder plus the surface area of the prism, and similarly for volume.
🔍 When a problem gives you the volume and asks for surface area (or vice versa), you'll often need to work backwards by finding the radius or height first, then using that measurement in the other formula.
Sign up to see the contentIt's free!
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Working Backwards with Volume & Surface Area
Sometimes you'll need to determine dimensions when given volume or surface area. For a cube with volume 148,877 mm³, you can find the edge length by taking the cube root: 53 mm. The surface area is then 6 times the square of this edge length.
When dealing with composite shapes like a cylinder topped with a hemisphere, break down the volume formula into parts. The hemisphere contributes ⅔πr³ (half the volume of a sphere), while the cylinder adds πr²h. Using these together lets you solve for unknown dimensions.
Understanding geometric relationships is crucial. In problem 14, comparing a sphere inside a cylinder reveals an important volume ratio. This type of comparison helps you recognize patterns that can simplify complex calculations.
🧩 When solving for missing dimensions, always identify what you know and what formula connects it to what you need. This systematic approach makes seemingly complex problems manageable.
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
Application Problems & Similar Figures
Real-world applications often involve regular polygons as bases. A concrete pillar with a hexagonal base demonstrates this: simply multiply the base area (4.5 square feet) by the height (22 feet) to find the volume of 99 cubic feet.
Hollow objects, like metal pipes, require finding the difference between two volumes. For a pipe with outer diameter 4 inches and inner diameter 3.5 inches, calculate the volume of the outer cylinder and subtract the volume of the empty inner space.
When working with similar figures, like the cylinders in problem 18, remember that the ratio of volumes is equal to the cube of the ratio of corresponding linear measurements. For cylinders with heights in a 4:6 ratio, the volumes have a 64:216 ratio.
🔧 Many real-world applications involve composite shapes or hollow objects. Breaking these down into basic geometric components makes them much easier to analyze and calculate.
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.
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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
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