HANDOUTS - Module 7 -- Spheres

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  • University of Santo Tomas

    • Motto: "Veritas in Caritate" (Ephesians 4:15)

  • College of Architecture

  • Math and Engineering Cluster

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  • Module 7: Solid Mensuration – Spheres

    • Perform calculations related to spheres.

    • Objectives:

      • Describe spheres and other spherical components.

      • Explain equations for surface area and volume of spheres.

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  • Outline of Module 7:

    • Spheres: Definition and Equations

    • Spherical Segment and Zone

    • Spherical Wedge and Lune

    • Spherical Sector and Cone

    • Spherical Polygon, Pyramid, and Triangle Computations

  • Sphere Definition:

    • A set of points at equal distance (radius, R) from a center in 3D space.

    • Created by rotating a circle around its diameter (d).

    • Source: Earnhart, R.T. (2011). Solid Mensuration: Understanding the 3D Space.

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  • Intersection of Sphere and Plane:

    • Always results in a circle, regardless of intersection location.

      • Great circle: plane intersects the center.

      • Small circle: plane does not pass through the center.

    • Surface Area of a Sphere:

      • Formula: ( SA = 4\pi R^2 )

        • Covers the area of a great circle, which is ( \frac{1}{4} ) of the whole sphere.

    • Source: Earnhart, R.T. (2011).

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  • Volume of a Sphere:

    • Formula: ( V = \frac{4}{3}\pi R^3 )

      • Represents the cubic units inside the sphere.

    • Source: Earnhart, R.T. (2011).

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  • Spherical Segment:

    • Portion formed by two parallel planes through the sphere.

    • Bases are circular sections; distance between bases is altitude.

    • If one plane is tangent, it's a spherical segment of one base (spherical cap).

    • Hemisphere:

      • Defined as a spherical cap with altitude ( h = R ).

    • Source: Earnhart, R.T. (2011).

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  • Spherical Zone:

    • Portion of the surface bounded by two parallel planes.

    • Bases are parallel circular sections; at distance of altitude.

    • Spherical Lune:

    • Portion of surface bounded by two arcs of great circles.

    • Generated by the revolution of a semicircle about its diameter.

    • Angle ( \theta ): spherical angle formed by semicircles.

    • Source: Earnhart, R.T. (2011).

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  • Spherical Wedge:

    • Solid bounded by a lune and planes of two great circles.

    • Surface area and volume computed by ratio of sphere's surface area and volume.

  • Spherical Sector/Cone:

    • Spherical sector: Portion generated by revolving a sector of a great circle about its diameter.

    • Spherical cone: Solid generated by revolving a sector about a diameter aligned with the radius.

    • Source: Earnhart, R.T. (2011).

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  • Spherical Triangle, Polygon & Pyramid:

    • The sum of angles exceeds 180° because sides are arcs.

    • Interior angles sum exceeds planar polygon expectations.

    • Vertices of spherical polygons connect to the sphere's center.