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Почему площадь сферы в четыре раза больше её тени? [3Blue1Brown]

Vert Dider・13 minutes read

The text explores the relationship between the surface area of a sphere and four circles, demonstrating various methods to understand this connection through cylinders, rectangles, and triangles. It discusses practical tasks to visualize the shadow of rings on a sphere, highlighting the area of a sphere as a unique case related to the average area of shadows of convex figures.

Insights

  • The area of a sphere is intricately linked to the areas of circles, cylinders, and other geometric shapes, demonstrating a profound interconnectedness between seemingly disparate objects.
  • By using triangles, rectangles, and practical tasks, complex mathematical concepts such as the relationship between shadows on spheres and their actual surface area can be visualized and understood in a tangible manner.

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Recent questions

  • How is the area of a sphere calculated?

    The area of a sphere is calculated using the formula 4πr², where r is the radius of the sphere. This formula is derived from the relationship between the surface area of a sphere and the area of four circles.

  • What is the connection between a sphere and a cylinder?

    The surface area of a sphere is compared to the area of a cylinder without bases to show a connection between the two shapes. Understanding this relationship can help visualize the similarities and differences between spheres and cylinders.

  • How are rectangles used to approximate the surface area of a sphere?

    Rectangles are used as an approximation for the surface area of a sphere by converting circles into triangles to fit into rectangles. This method showcases a direct connection between spheres and circles, providing a practical way to estimate the surface area of a sphere.

  • What practical tasks can help understand the relationship between shadows on a sphere?

    Practical tasks such as projecting rectangles onto a cylinder can help understand the relationship between the shadows of rings on a sphere and the sphere itself. By visualizing these projections, one can gain insights into how shadows are cast on curved surfaces.

  • What is the significance of the ratio of width and height in rectangles projected onto a cylinder?

    The ratio of the width and height of rectangles projected onto a cylinder is explained through triangles and trigonometry. Understanding this ratio can provide insights into how shapes are distorted when projected onto curved surfaces, highlighting the geometric properties of cylinders and their projections.

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Summary

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Sphere Area: Connections with Circles and Shadows

  • The area of a sphere is 4πr², which is equivalent to the area of a circle multiplied by 4.
  • The relationship between the surface area of a sphere and the area of four circles is explored.
  • Two methods are presented to demonstrate the connection between the sphere and its shadow.
  • The surface area of a sphere is compared to the area of a cylinder without bases, showing a connection between the two.
  • The projection of rectangles onto a cylinder helps understand the relationship between the width and height during projection.
  • The ratio of the width and height of rectangles projected onto a cylinder is explained through triangles and trigonometry.
  • The use of rectangles as an approximation for the surface area of a sphere is discussed.
  • Circles can be converted into triangles to fit into rectangles, showcasing a direct connection between spheres and circles.
  • Practical tasks are suggested to understand the relationship between the shadows of rings on a sphere and the sphere itself.
  • The area of a sphere being a special case of a general rule regarding the average area of shadows of convex figures is mentioned.
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