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Clara ๐
10/12/2025
Maths
Maths, volume and surface area of 3 d shapes.
518
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10 Dec 2025
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Understanding 3D Shapes: Volume and Surface Area
Clara ๐
@clarapop
1 / 9
Volume of a Cuboid
Volume tells you how much space a 3D shape takes up inside - think of it as how much water you could pour into a box. For cuboids (rectangular boxes), the formula is dead simple: height ร width ร length.
Let's say you've got a box that's 10cm long, 6cm wide, and 4cm high. Just multiply: 10 ร 6 ร 4 = 240cmยณ. The little ยณ symbol means "cubic" - you're measuring in cubic centimetres.
Sometimes you'll get tricky questions where one measurement is missing. Don't panic! If you know the volume is 84cmยณ and two sides are 7cm and 3cm, just work backwards: 84 รท (7 ร 3) = 84 รท 21 = 4cm.
Top Tip: Always check your units match - if some measurements are in metres and others in centimetres, convert them first!
Surface Area of a Cuboid
Surface area is the total area of all the faces wrapped around your 3D shape - imagine painting it and working out how much paint you'd need. Every cuboid has 6 faces, and opposite faces are always identical.
Here's the clever bit: calculate the area of each different face, then double it. For a cuboid with sides 5cm, 10cm, and 4cm, you'd get: (5 ร 10) ร 2 = 100cmยฒ, (4 ร 5) ร 2 = 40cmยฒ, and (10 ร 4) ร 2 = 80cmยฒ.
Add them all up: 100 + 40 + 80 = 220cmยฒ. Remember, surface area is always measured in square units (cmยฒ), not cubic ones.
Quick Check: Count your faces - you should always have exactly 6 for a cuboid!
More Surface Area Practice
Let's tackle some real examples to nail this concept. For a cuboid measuring 6cm ร 15cm ร 3cm, work through each pair of faces systematically.
First pair: 6 ร 15 = 90cmยฒ, doubled gives 180cmยฒ. Second pair: 6 ร 3 = 18cmยฒ, doubled gives 36cmยฒ. Third pair: 15 ร 3 = 45cmยฒ, doubled gives 90cmยฒ.
Total surface area: 180 + 36 + 90 = 306cmยฒ. The key is staying organised - work through each pair methodically and you won't miss any faces.
Memory Trick: Think "opposite twins" - every face has an identical twin on the opposite side!
Volume of Prisms
A prism is any 3D shape with identical ends and the same cross-section all the way through - like a tube of Pringles! The volume formula is brilliant: cross-sectional area ร length.
For triangular prisms, first find the area of the triangular end using ยฝ ร base ร height. If your triangle has a base of 5cm and height of 4cm, that's ยฝ ร 5 ร 4 = 10cmยฒ. Then multiply by the length - if it's 12cm long, volume = 10 ร 12 = 120cmยณ.
Trapezoid prisms use the formula ยฝ ร ร height for the end area. With parallel sides of 8cm and 6cm, height 5cm, and length 12cm: area = ยฝ ร (8 + 6) ร 5 = 35cmยฒ, so volume = 35 ร 12 = 420cmยณ.
Golden Rule: Always find the end area first, then multiply by how long the prism stretches!
Volume of Cylinders
Cylinders aren't technically prisms because they're rounded, but the volume formula follows the same logic: ฯ ร rยฒ ร height. The ฯ (pi) button on your calculator makes this much easier!
For a cylinder with radius 2.5cm and height 6cm: ฯ ร (2.5)ยฒ ร 6 = ฯ ร 6.25 ร 6 = 117.8cmยณ. Remember to square the radius first - it's a common mistake to forget this step.
Working backwards is trickier but totally doable. If volume = 6000cmยณ and radius = 10cm, then 6000 = ฯ ร 100 ร h. Rearranging: h = 6000 รท (ฯ ร 100) = 19.1cm.
Calculator Tip: Use the ฯ button rather than 3.14 for more accurate answers!
Surface Area of Cylinders
Cylinder surface area has two parts: the curved side (like a label wrapped around) and the two circular ends (top and bottom). The formula combines both: 2ฯrยฒ + ฯdh.
The 2ฯrยฒ covers both circular ends - each circle has area ฯrยฒ, and there are two of them. The ฯdh is the rectangular label wrapped around the curved side.
For a cylinder with radius 2cm and height 6cm: circular ends = 2 ร ฯ ร 2ยฒ = 8ฯ cmยฒ, curved side = ฯ ร 4 ร 6 = 24ฯ cmยฒ. Total = 8ฯ + 24ฯ = 32ฯ โ 100.8cmยฒ.
Visual Trick: Imagine peeling the label off a tin can - that's your curved surface area!
Unknown Lengths in Cylinders
Sometimes you'll need to work backwards from the surface area to find missing measurements. This is where your algebra skills shine! Start with the surface area formula and substitute what you know.
If surface area = 137cmยฒ, radius = 4cm, find the height: 137 = 2ฯ(4ยฒ) + ฯ(8)h. This simplifies to 137 = 32ฯ + 8ฯh. Subtract 32ฯ from both sides: 137 - 32ฯ = 8ฯh.
Divide by 8ฯ: h = (137 - 32ฯ) รท 8ฯ โ 1.9cm. The key is taking it step by step and not rushing the algebra.
Success Strategy: Write down the formula first, then substitute known values before rearranging!
Volume of Cones
Cone volume uses the formula V = โ ฯrยฒh - it's exactly one-third of a cylinder with the same base and height. This makes sense because a cone tapers to a point!
For a cone with radius 5cm and height 12cm: V = โ ร ฯ ร 5ยฒ ร 12 = โ ร ฯ ร 25 ร 12 = 100ฯ โ 314.2cmยณ. Always multiply everything together before dividing by 3.
Reverse calculations work brilliantly too. If volume = 12ฯcmยณ and height = 4cm, find the radius: 12ฯ = โ ร ฯ ร rยฒ ร 4. Cancel ฯ from both sides: 12 = โ ร rยฒ ร 4. This gives rยฒ = 9, so r = 3cm.
Memory Hook: Cone volume = cylinder volume รท 3!
Surface Area of Cones
Cone surface area combines the circular base (ฯrยฒ) with the curved side (ฯrl), where l is the slant height from the tip to the edge of the base. Total formula: ฯrยฒ + ฯrl.
The tricky bit is finding the slant height when you only know the radius and vertical height. Use Pythagoras' theorem: lยฒ = rยฒ + hยฒ. For radius 2cm and height 7cm: lยฒ = 4 + 49 = 53, so l = โ53 โ 7.3cm.
Then calculate: base area = ฯ ร 2ยฒ = 4ฯ, curved area = ฯ ร 2 ร 7.3 = 14.6ฯ. Total โ 18.6ฯ โ 58.4cmยฒ.
Pro Tip: Draw a right triangle with the radius and height to visualise the slant height calculation!
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This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because itโs too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm sure you too will see developments.
Xander S
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding 3D Shapes: Volume and Surface Area
Clara ๐
@clarapop
Ready to master 3D shapes? This guide breaks down everything you need to know about calculating volume and surface area for cuboids, prisms, cylinders, and cones. These skills are essential for your maths exams and surprisingly useful in real life... Show more
Volume of a Cuboid
Volume tells you how much space a 3D shape takes up inside - think of it as how much water you could pour into a box. For cuboids (rectangular boxes), the formula is dead simple: height ร width ร length.
Let's say you've got a box that's 10cm long, 6cm wide, and 4cm high. Just multiply: 10 ร 6 ร 4 = 240cmยณ. The little ยณ symbol means "cubic" - you're measuring in cubic centimetres.
Sometimes you'll get tricky questions where one measurement is missing. Don't panic! If you know the volume is 84cmยณ and two sides are 7cm and 3cm, just work backwards: 84 รท (7 ร 3) = 84 รท 21 = 4cm.
Top Tip: Always check your units match - if some measurements are in metres and others in centimetres, convert them first!
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Surface Area of a Cuboid
Surface area is the total area of all the faces wrapped around your 3D shape - imagine painting it and working out how much paint you'd need. Every cuboid has 6 faces, and opposite faces are always identical.
Here's the clever bit: calculate the area of each different face, then double it. For a cuboid with sides 5cm, 10cm, and 4cm, you'd get: (5 ร 10) ร 2 = 100cmยฒ, (4 ร 5) ร 2 = 40cmยฒ, and (10 ร 4) ร 2 = 80cmยฒ.
Add them all up: 100 + 40 + 80 = 220cmยฒ. Remember, surface area is always measured in square units (cmยฒ), not cubic ones.
Quick Check: Count your faces - you should always have exactly 6 for a cuboid!
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More Surface Area Practice
Let's tackle some real examples to nail this concept. For a cuboid measuring 6cm ร 15cm ร 3cm, work through each pair of faces systematically.
First pair: 6 ร 15 = 90cmยฒ, doubled gives 180cmยฒ. Second pair: 6 ร 3 = 18cmยฒ, doubled gives 36cmยฒ. Third pair: 15 ร 3 = 45cmยฒ, doubled gives 90cmยฒ.
Total surface area: 180 + 36 + 90 = 306cmยฒ. The key is staying organised - work through each pair methodically and you won't miss any faces.
Memory Trick: Think "opposite twins" - every face has an identical twin on the opposite side!
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Volume of Prisms
A prism is any 3D shape with identical ends and the same cross-section all the way through - like a tube of Pringles! The volume formula is brilliant: cross-sectional area ร length.
For triangular prisms, first find the area of the triangular end using ยฝ ร base ร height. If your triangle has a base of 5cm and height of 4cm, that's ยฝ ร 5 ร 4 = 10cmยฒ. Then multiply by the length - if it's 12cm long, volume = 10 ร 12 = 120cmยณ.
Trapezoid prisms use the formula ยฝ ร ร height for the end area. With parallel sides of 8cm and 6cm, height 5cm, and length 12cm: area = ยฝ ร (8 + 6) ร 5 = 35cmยฒ, so volume = 35 ร 12 = 420cmยณ.
Golden Rule: Always find the end area first, then multiply by how long the prism stretches!
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Volume of Cylinders
Cylinders aren't technically prisms because they're rounded, but the volume formula follows the same logic: ฯ ร rยฒ ร height. The ฯ (pi) button on your calculator makes this much easier!
For a cylinder with radius 2.5cm and height 6cm: ฯ ร (2.5)ยฒ ร 6 = ฯ ร 6.25 ร 6 = 117.8cmยณ. Remember to square the radius first - it's a common mistake to forget this step.
Working backwards is trickier but totally doable. If volume = 6000cmยณ and radius = 10cm, then 6000 = ฯ ร 100 ร h. Rearranging: h = 6000 รท (ฯ ร 100) = 19.1cm.
Calculator Tip: Use the ฯ button rather than 3.14 for more accurate answers!
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Surface Area of Cylinders
Cylinder surface area has two parts: the curved side (like a label wrapped around) and the two circular ends (top and bottom). The formula combines both: 2ฯrยฒ + ฯdh.
The 2ฯrยฒ covers both circular ends - each circle has area ฯrยฒ, and there are two of them. The ฯdh is the rectangular label wrapped around the curved side.
For a cylinder with radius 2cm and height 6cm: circular ends = 2 ร ฯ ร 2ยฒ = 8ฯ cmยฒ, curved side = ฯ ร 4 ร 6 = 24ฯ cmยฒ. Total = 8ฯ + 24ฯ = 32ฯ โ 100.8cmยฒ.
Visual Trick: Imagine peeling the label off a tin can - that's your curved surface area!
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Unknown Lengths in Cylinders
Sometimes you'll need to work backwards from the surface area to find missing measurements. This is where your algebra skills shine! Start with the surface area formula and substitute what you know.
If surface area = 137cmยฒ, radius = 4cm, find the height: 137 = 2ฯ(4ยฒ) + ฯ(8)h. This simplifies to 137 = 32ฯ + 8ฯh. Subtract 32ฯ from both sides: 137 - 32ฯ = 8ฯh.
Divide by 8ฯ: h = (137 - 32ฯ) รท 8ฯ โ 1.9cm. The key is taking it step by step and not rushing the algebra.
Success Strategy: Write down the formula first, then substitute known values before rearranging!
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Volume of Cones
Cone volume uses the formula V = โ ฯrยฒh - it's exactly one-third of a cylinder with the same base and height. This makes sense because a cone tapers to a point!
For a cone with radius 5cm and height 12cm: V = โ ร ฯ ร 5ยฒ ร 12 = โ ร ฯ ร 25 ร 12 = 100ฯ โ 314.2cmยณ. Always multiply everything together before dividing by 3.
Reverse calculations work brilliantly too. If volume = 12ฯcmยณ and height = 4cm, find the radius: 12ฯ = โ ร ฯ ร rยฒ ร 4. Cancel ฯ from both sides: 12 = โ ร rยฒ ร 4. This gives rยฒ = 9, so r = 3cm.
Memory Hook: Cone volume = cylinder volume รท 3!
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Surface Area of Cones
Cone surface area combines the circular base (ฯrยฒ) with the curved side (ฯrl), where l is the slant height from the tip to the edge of the base. Total formula: ฯrยฒ + ฯrl.
The tricky bit is finding the slant height when you only know the radius and vertical height. Use Pythagoras' theorem: lยฒ = rยฒ + hยฒ. For radius 2cm and height 7cm: lยฒ = 4 + 49 = 53, so l = โ53 โ 7.3cm.
Then calculate: base area = ฯ ร 2ยฒ = 4ฯ, curved area = ฯ ร 2 ร 7.3 = 14.6ฯ. Total โ 18.6ฯ โ 58.4cmยฒ.
Pro Tip: Draw a right triangle with the radius and height to visualise the slant height calculation!
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Students love us โ and so will you.
4.9/5
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4.8/5
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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
Best app on earth! no words because itโs too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm sure you too will see developments.
Xander S
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because itโs too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didnโt even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as Iโm sure you too will see developments.
Xander S
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 ๐๐๐ฒ๐ค๐โจ๐๐ฎ
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user