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Eva Morgan
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This guide covers essential geometry concepts, focusing on circles, cylinders, cones, and spheres. It explains formulas for calculating various measurements and provides practical examples.
07/05/2023
196
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°) × 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 / 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.
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:
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:
Example: Calculate the surface area of a cylinder with a diameter of 8cm and a height of 10cm.
Highlight: When calculating the surface area of a cylinder, remember to use the radius (half 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.
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.
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 = (1/3)πr²h, 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 = (1/3) × 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.
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:
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.
Highlight: The slant height (l) 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.
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 (the 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.
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:
Example: Calculate the arc length AB for a circle with a diameter of 8cm and an arc angle of 45°.
Highlight: The formula for arc length can be expressed as: Arc length = (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.
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 = (4/3)πr³, 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 = (4/3) × 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.
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Area and Circumference of Circles
here you can find how to workout the atea and circumference of circles
8
287
10/11
20 KEY GCSE FACTS
Areas, volumes, Pythagoras theorem, speed, mean, mode, range, median, square/prime numbers, metric lengths; weights, capacity and imperial conversions
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1453
11
Formula Sheet
For higher math students in Year11, formula sheet to revise. Inclues circle theorums, probability, graphs, bidmas, etc
8
241
11/10
gcse maths revision
maths formulas
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Eva Morgan
@evamorgan_nvvp
·
1 Follower
Follow
This guide covers essential geometry concepts, focusing on circles, cylinders, cones, and spheres. It explains formulas for calculating various measurements and provides practical examples.
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°) × 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 / 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.
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:
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:
Example: Calculate the surface area of a cylinder with a diameter of 8cm and a height of 10cm.
Highlight: When calculating the surface area of a cylinder, remember to use the radius (half 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.
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.
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 = (1/3)πr²h, 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 = (1/3) × 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.
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:
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.
Highlight: The slant height (l) 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.
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 (the 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.
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:
Example: Calculate the arc length AB for a circle with a diameter of 8cm and an arc angle of 45°.
Highlight: The formula for arc length can be expressed as: Arc length = (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.
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 = (4/3)πr³, 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 = (4/3) × 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.
Maths - Area and Circumference of Circles
here you can find how to workout the atea and circumference of circles
11
624
0
Maths - 20 KEY GCSE FACTS
Areas, volumes, Pythagoras theorem, speed, mean, mode, range, median, square/prime numbers, metric lengths; weights, capacity and imperial conversions
8
287
0
Maths - Formula Sheet
For higher math students in Year11, formula sheet to revise. Inclues circle theorums, probability, graphs, bidmas, etc
53
1453
1
Maths - gcse maths revision
maths formulas
8
241
0
Average app rating
Pupils love Knowunity
In education app charts in 12 countries
Students have uploaded notes
iOS User
Philip, iOS User
Lena, iOS user
