Subjects

Careers

Open the App

Subjects

Easy Math: Circle Arcs, Cylinder Areas, and Sphere Volumes

Open

12

0

E

Eva Morgan

07/05/2023

Maths

revision m3

Easy Math: Circle Arcs, Cylinder Areas, and Sphere Volumes

A comprehensive guide to geometric calculations covering circles, pressure, and various 3D shapes. The material explains essential formulas and practical applications for calculating arc length of a circle, formula for surface area of a cylinder, and volume calculation for a sphere.

  • Introduces fundamental circle components including radius, diameter, arc, segment, and tangent
  • Details pressure calculations with clear relationships between force, pressure, and area
  • Covers arc length and sector area calculations with step-by-step examples
  • Explains surface area calculations for cylinders, cones, and spheres
  • Concludes with volume calculations for cones and spheres with practical examples
...

07/05/2023

217

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Calculating Arc Length

This page focuses on calculating arc length of a circle, which is a crucial skill in geometry and practical applications.

The process for calculating arc length is broken down into three steps:

  1. Calculate the entire circumference of the circle using the formula C = π × d, where d is the diameter.
  2. Divide the circumference by 360° to find the length of one degree of the circle.
  3. Multiply this value by the angle of the arc you want to measure.

Example: Calculate the arc length AB for a circle with a diameter of 8cm and an arc angle of 45°.

  1. Circumference = π × 8 = 25.13cm
  2. Length of 1° = 25.13 / 360 = 0.0698cm
  3. Arc length = 0.0698 × 45 = 3.14cm

Highlight: The formula for arc length can be expressed as: Arc length = θ/360°θ / 360° × πd, where θ is the angle of the arc in degrees.

This method allows for accurate calculation of any arc length when the diameter and angle are known.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Area of a Sector

This page explains how to calculate the area of a sector, which is a portion of a circle enclosed by two radii and an arc.

The formula for the area of a sector is derived from the area of a full circle:

Definition: Area of a sector = θ/360°θ / 360° × πr², where θ is the angle of the sector in degrees and r is the radius of the circle.

This formula can be understood as taking the fraction of the full circle's area that corresponds to the angle of the sector.

Example: Calculate the area of a sector with a radius of 8cm and an angle of 21°.

Area of sector = 21/36021 / 360 × π × 8² = 11.73cm²

The page provides a step-by-step solution to this example, demonstrating how to apply the formula in practice.

Highlight: This method can be used to calculate the area of any sector, regardless of its size or the dimensions of the circle it's part of.

Understanding how to calculate sector areas is essential for more advanced geometric problems and real-world applications involving circular segments.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Surface Area of a Cylinder

This page introduces the formula for surface area of a cylinder and demonstrates its application through a practical example.

The surface area of a cylinder consists of three parts:

  1. The top circular face
  2. The bottom circular face
  3. The curved lateral surface

Definition: Total surface area of a cylinder = 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder.

This formula can be broken down as follows:

  • Area of top and bottom circles: 2πr²
  • Area of curved surface: 2πrh

Example: Calculate the surface area of a cylinder with a diameter of 8cm and a height of 10cm.

  1. Area of top circle: πr² = π × 4² = 50.27cm²
  2. Area of bottom circle: 50.27cm² sameastopsame as top
  3. Curved surface area: 2πrh = 2π × 4 × 10 = 251.33cm²
  4. Total surface area: 50.27 + 50.27 + 251.33 = 351.86cm²

Highlight: When calculating the surface area of a cylinder, remember to use the radius halfthediameterhalf the diameter for the circular faces, but the full height for the curved surface.

This method allows for accurate calculation of the surface area of any cylinder, which is useful in various fields including engineering, architecture, and manufacturing.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Surface Area of a Cone

This page explains how to calculate the surface area of a cone, which consists of a circular base and a curved lateral surface.

The total surface area of a cone is the sum of:

  1. The area of the circular base
  2. The area of the curved surface

Definition: Surface area of a cone = πr² + πrl, where r is the radius of the base and l is the slant height of the cone.

The page provides a step-by-step example for calculating the surface area of a cone:

Example: Calculate the surface area of a cone with a base radius of 6cm and a slant height of 10cm.

  1. Area of the base: πr² = π × 6² = 113.10cm²
  2. Area of the curved surface: πrl = π × 6 × 10 = 188.50cm²
  3. Total surface area: 113.10 + 188.50 = 301.60cm²

Highlight: The slant height ll is different from the vertical height of the cone. It's the distance from the apex to any point on the circumference of the base.

Understanding how to calculate the surface area of a cone is crucial for various applications in geometry, engineering, and design. This formula allows for accurate measurements of conical structures and objects.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Surface Area of a Sphere

This page introduces the formula for calculating the surface area of a sphere and demonstrates its application.

Definition: The surface area of a sphere is given by the formula: S.A. = 4πr², where r is the radius of the sphere.

This elegant formula encapsulates the entire surface area of a sphere in a simple expression.

Example: Calculate the surface area of a sphere with a radius of 4cm.

Surface Area = 4π × 4² = 201.06cm²

Highlight: The surface area of a sphere is exactly four times the area of its great circle thelargestcirclethatcanbedrawnonthespheressurfacethe largest circle that can be drawn on the sphere's surface.

This formula is derived from advanced calculus, but its simplicity makes it easy to apply in various practical situations. Understanding how to calculate the surface area of a sphere is crucial in fields such as astronomy, physics, and engineering, where spherical objects or concepts are common.

The page also includes a helpful diagram illustrating a sphere and its radius, which aids in visualizing the concept.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Volume of a Cone

This page explains how to calculate the volume of a cone and provides a practical example.

Definition: The volume of a cone is given by the formula: V = 1/31/3πr²h, where r is the radius of the base and h is the height of the cone.

This formula can be understood as one-third of the volume of a cylinder with the same base radius and height.

Example: Calculate the volume of a cone with a base radius of 4cm and a height of 12cm.

Volume = 1/31/3 × π × 4² × 12 = 201.06cm³

The page provides a step-by-step solution to this example, demonstrating how to apply the formula in practice.

Highlight: The height used in this formula is the perpendicular height from the center of the base to the apex of the cone, not the slant height used for surface area calculations.

Understanding how to calculate the volume of a cone is essential for various applications in geometry, engineering, and design. This formula allows for accurate measurements of conical containers, structures, and natural formations.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Volume of a Sphere

This final page covers the volume calculation for a sphere, providing the formula and a practical example.

Definition: The volume of a sphere is given by the formula: V = 4/34/3πr³, where r is the radius of the sphere.

This formula represents the amount of space enclosed within the spherical surface.

Example: Calculate the volume of a sphere with a radius of 5cm.

Volume = 4/34/3 × π × 5³ = 523.6cm³

The page includes a clear diagram of a sphere with its radius labeled, which helps in visualizing the concept.

Highlight: The volume of a sphere increases cubically with its radius, meaning a small increase in radius results in a large increase in volume.

Understanding how to calculate the volume of a sphere is crucial in many fields, including physics, astronomy, and engineering. This formula is used to determine the volume of planets, stars, and various spherical objects or containers in everyday life.

The simplicity of this formula belies its power in describing the volume of any perfect sphere, regardless of its size.

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

View

Sphere Volume

The final page covers the calculation of sphere volume.

Formula: Volume = 4/34/3πr³

Example: For a sphere with radius 5cm: Volume = 4/34/3π × 5³ = 523.6cm³

Highlight: This formula is one of the most fundamental in three-dimensional geometry.

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

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

21 M

Pupils love Knowunity

#1

In education app charts in 17 countries

950 K+

Students have uploaded notes

Still not convinced? See what other students are saying...

iOS User

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

Philip, iOS User

The app is very simple and well designed. So far I have always found everything I was looking for :D

Lena, iOS user

I love this app ❤️ I actually use it every time I study.

 

Maths

217

7 May 2023

9 pages

Easy Math: Circle Arcs, Cylinder Areas, and Sphere Volumes

E

Eva Morgan

@evamorgan_nvvp

A comprehensive guide to geometric calculations covering circles, pressure, and various 3D shapes. The material explains essential formulas and practical applications for calculating arc length of a circle, formula for surface area of a cylinder, and volume calculation... Show more

Radius
chard
Parts of a Circle
Diameter
Arc
Segment
Ba
O
Circumference
о
Tangent
C
Sector F
"PA
Pressure =
Pressure
Force
Area
Force = Press

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

Calculating Arc Length

This page focuses on calculating arc length of a circle, which is a crucial skill in geometry and practical applications.

The process for calculating arc length is broken down into three steps:

  1. Calculate the entire circumference of the circle using the formula C = π × d, where d is the diameter.
  2. Divide the circumference by 360° to find the length of one degree of the circle.
  3. Multiply this value by the angle of the arc you want to measure.

Example: Calculate the arc length AB for a circle with a diameter of 8cm and an arc angle of 45°.

  1. Circumference = π × 8 = 25.13cm
  2. Length of 1° = 25.13 / 360 = 0.0698cm
  3. Arc length = 0.0698 × 45 = 3.14cm

Highlight: The formula for arc length can be expressed as: Arc length = θ/360°θ / 360° × πd, where θ is the angle of the arc in degrees.

This method allows for accurate calculation of any arc length when the diameter and angle are known.

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

Area of a Sector

This page explains how to calculate the area of a sector, which is a portion of a circle enclosed by two radii and an arc.

The formula for the area of a sector is derived from the area of a full circle:

Definition: Area of a sector = θ/360°θ / 360° × πr², where θ is the angle of the sector in degrees and r is the radius of the circle.

This formula can be understood as taking the fraction of the full circle's area that corresponds to the angle of the sector.

Example: Calculate the area of a sector with a radius of 8cm and an angle of 21°.

Area of sector = 21/36021 / 360 × π × 8² = 11.73cm²

The page provides a step-by-step solution to this example, demonstrating how to apply the formula in practice.

Highlight: This method can be used to calculate the area of any sector, regardless of its size or the dimensions of the circle it's part of.

Understanding how to calculate sector areas is essential for more advanced geometric problems and real-world applications involving circular segments.

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 a Cylinder

This page introduces the formula for surface area of a cylinder and demonstrates its application through a practical example.

The surface area of a cylinder consists of three parts:

  1. The top circular face
  2. The bottom circular face
  3. The curved lateral surface

Definition: Total surface area of a cylinder = 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder.

This formula can be broken down as follows:

  • Area of top and bottom circles: 2πr²
  • Area of curved surface: 2πrh

Example: Calculate the surface area of a cylinder with a diameter of 8cm and a height of 10cm.

  1. Area of top circle: πr² = π × 4² = 50.27cm²
  2. Area of bottom circle: 50.27cm² sameastopsame as top
  3. Curved surface area: 2πrh = 2π × 4 × 10 = 251.33cm²
  4. Total surface area: 50.27 + 50.27 + 251.33 = 351.86cm²

Highlight: When calculating the surface area of a cylinder, remember to use the radius halfthediameterhalf the diameter for the circular faces, but the full height for the curved surface.

This method allows for accurate calculation of the surface area of any cylinder, which is useful in various fields including engineering, architecture, and manufacturing.

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 a Cone

This page explains how to calculate the surface area of a cone, which consists of a circular base and a curved lateral surface.

The total surface area of a cone is the sum of:

  1. The area of the circular base
  2. The area of the curved surface

Definition: Surface area of a cone = πr² + πrl, where r is the radius of the base and l is the slant height of the cone.

The page provides a step-by-step example for calculating the surface area of a cone:

Example: Calculate the surface area of a cone with a base radius of 6cm and a slant height of 10cm.

  1. Area of the base: πr² = π × 6² = 113.10cm²
  2. Area of the curved surface: πrl = π × 6 × 10 = 188.50cm²
  3. Total surface area: 113.10 + 188.50 = 301.60cm²

Highlight: The slant height ll is different from the vertical height of the cone. It's the distance from the apex to any point on the circumference of the base.

Understanding how to calculate the surface area of a cone is crucial for various applications in geometry, engineering, and design. This formula allows for accurate measurements of conical structures and objects.

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 a Sphere

This page introduces the formula for calculating the surface area of a sphere and demonstrates its application.

Definition: The surface area of a sphere is given by the formula: S.A. = 4πr², where r is the radius of the sphere.

This elegant formula encapsulates the entire surface area of a sphere in a simple expression.

Example: Calculate the surface area of a sphere with a radius of 4cm.

Surface Area = 4π × 4² = 201.06cm²

Highlight: The surface area of a sphere is exactly four times the area of its great circle thelargestcirclethatcanbedrawnonthespheressurfacethe largest circle that can be drawn on the sphere's surface.

This formula is derived from advanced calculus, but its simplicity makes it easy to apply in various practical situations. Understanding how to calculate the surface area of a sphere is crucial in fields such as astronomy, physics, and engineering, where spherical objects or concepts are common.

The page also includes a helpful diagram illustrating a sphere and its radius, which aids in visualizing the concept.

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 a Cone

This page explains how to calculate the volume of a cone and provides a practical example.

Definition: The volume of a cone is given by the formula: V = 1/31/3πr²h, where r is the radius of the base and h is the height of the cone.

This formula can be understood as one-third of the volume of a cylinder with the same base radius and height.

Example: Calculate the volume of a cone with a base radius of 4cm and a height of 12cm.

Volume = 1/31/3 × π × 4² × 12 = 201.06cm³

The page provides a step-by-step solution to this example, demonstrating how to apply the formula in practice.

Highlight: The height used in this formula is the perpendicular height from the center of the base to the apex of the cone, not the slant height used for surface area calculations.

Understanding how to calculate the volume of a cone is essential for various applications in geometry, engineering, and design. This formula allows for accurate measurements of conical containers, structures, and natural formations.

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 a Sphere

This final page covers the volume calculation for a sphere, providing the formula and a practical example.

Definition: The volume of a sphere is given by the formula: V = 4/34/3πr³, where r is the radius of the sphere.

This formula represents the amount of space enclosed within the spherical surface.

Example: Calculate the volume of a sphere with a radius of 5cm.

Volume = 4/34/3 × π × 5³ = 523.6cm³

The page includes a clear diagram of a sphere with its radius labeled, which helps in visualizing the concept.

Highlight: The volume of a sphere increases cubically with its radius, meaning a small increase in radius results in a large increase in volume.

Understanding how to calculate the volume of a sphere is crucial in many fields, including physics, astronomy, and engineering. This formula is used to determine the volume of planets, stars, and various spherical objects or containers in everyday life.

The simplicity of this formula belies its power in describing the volume of any perfect sphere, regardless of its size.

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

Sphere Volume

The final page covers the calculation of sphere volume.

Formula: Volume = 4/34/3πr³

Example: For a sphere with radius 5cm: Volume = 4/34/3π × 5³ = 523.6cm³

Highlight: This formula is one of the most fundamental in three-dimensional geometry.

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

Parts of a Circle and Pressure Calculations

This page introduces the fundamental parts of a circle and basic pressure calculations. Understanding these concepts is crucial for more advanced geometric calculations.

Vocabulary: Radius - A line segment from the center of a circle to any point on its circumference.

Vocabulary: Chord - A line segment connecting two points on the circumference of a circle.

Vocabulary: Diameter - A chord that passes through the center of the circle, equal to twice the radius.

Vocabulary: Arc - A portion of the circumference of a circle.

Vocabulary: Segment - The region of a circle bounded by a chord and an arc.

Vocabulary: Sector - A region of a circle bounded by two radii and an arc.

Vocabulary: Tangent - A line that touches the circle at exactly one point.

The page also introduces the relationship between pressure, force, and area:

Definition: Pressure = Force / Area

This formula can be rearranged to calculate force or area when the other two variables are known:

Force = Pressure × Area Area = Force / Pressure

Highlight: Units of measurement are crucial in these calculations. Force is measured in Newtons, pressure in Newtons/m² or cm², and area in m² or cm².

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

Best app on earth! no words because it’s 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 didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m 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 😍😁😲🤑💗✨🎀😮

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 it’s 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 didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m 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 😍😁😲🤑💗✨🎀😮

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