Exhaustive Study Notes: Geometry of Pyramids, Cones, and Spheres

Polyhedron Definition: A three-dimensional solid comprised of flat, polygonal faces. The term originates from the Greek roots poly (meaning "many") and hedron (meaning "faces").
Pyramid Definition: A polyhedron formed by connecting every point of a polygonal base to a single point called the apex. The apex is located on a plane different from the flat surface containing the base.
Lateral Faces: These are the triangular faces of a pyramid. They are formed by the segments extending from the apex to the endpoints of a side of the base, as well as the side of the base itself.
Pyramid Types by Base Shape: * Triangular Pyramid: A pyramid with a triangular base. * Square Pyramid: A pyramid with a square base. * Rectangular Pyramid: A pyramid with a rectangular base. * Hexagonal Pyramid: A pyramid with a hexagonal base.
The Tetrahedron: A special type of pyramid where the base is a triangle. In a tetrahedron, any of the four faces can act as its base. It is also synonymous with a triangular-based pyramid.

Volume Comparison: Through visual demonstration, it is established that the volume of a pyramid is directly related to the volume of a prism that shares the same base area ( $B$ ) and the same vertical height ( $h$ ).
Observation from Experimentation: * The bases of the compared solids must be identical. * The vertical heights ( $h$ ) of the solids must be identical. * The liquid volume of exactly three pyramids is required to completely fill the corresponding prism.
Fundamental Formula for Pyramid Volume: * $V = rac{1}{3}Bh$

Cone Definition: A three-dimensional solid with a circular base and an apex. It functions as the circular analogue to a pyramid.
Relationship to Cylinders: Similarly to the pyramid-prism relationship, a cone has one-third the volume of a cylinder with the same base radius and height.
Slant Height ( $l$ ): The distance from the apex down the side to the edge of the base. This is distinct from the vertical height ( $h$ ).
Volume of a Cone Formula: $V = rac{1}{3} ext{Base Area} imes ext{height} = rac{1}{3} imes ext{Base Area} imes h = rac{1}{3} imes ext{Base Area} imes h$
Variables for Surfaces: * Perimeter of base is often denoted as $P$ or calculated via circumference $C = 2 imes ext{pi} imes r$ . * Lateral Area involves the slant height $l$ .

Sphere Definition: The locus of all points in space that are a fixed distance (the radius) from a given point (the center).
Radius: A segment connecting the center of the sphere to any point on its surface.
Hemisphere: Exactly one half of a sphere.
Great Circle: A circle on the surface of a sphere that has the same circumference as the sphere itself and divides the sphere into two equal hemispheres. * Named Example: The Equator is a real-world example of a great circle.
Relationship to Cylinders: A sphere's volume is documented as being equal to two-thirds of the volume of a cylinder that has a height equal to the diameter of the sphere ( $h = 2r$ ).
Sphere Formulas: * Volume of a Sphere: $V = rac{4}{3} imes ext{pi} imes r^3$ * Surface Area of a Sphere: $A = 4 imes ext{pi} imes r^2$

Lateral Surface Area: The sum of the areas of all faces of the solid, excluding the base or bases.
Total Surface Area: The cumulative sum of the areas of all faces, including the base(s).
Height Confusion: The term "height" can refer to two different measurements in a pyramid: 1. Vertical Height ( $h$ ): The perpendicular distance from the apex to the center of the base. 2. Slant Height ( $l$ ): The height of the triangular lateral faces.

The TransAmerica Building (San Francisco): * Shape: A square-based pyramid. * Dimensions: Edge of square base = $96\,m$ ; Vertical height of building = $220\,m$ . * Problem: Power-washing requires $1\,gallon$ of solution for every $250\,square meters$ of surface area. To solve for the volume of solution needed, students must first calculate the slant height ( $l$ ) of the triangular faces using the Pythagorean theorem with half the base edge ( $48\,m$ ) and the vertical height ( $220\,m$ ).
The Ice Cream Cone (Lekili's Problem): * Dimensions: Diameter = $4.0\,inches$ (Radius $r = 2.0\,inches$ ); Slant height ( $l$ ) = $6.0\,inches$ . * Goal: Determine the volume of the melted ice cream (filling the cone) and the surface area of the waffle material.
Giant Squid Eye Comparison: * Giant Squid Eyeball: Approximately a sphere with diameter = $25\,cm$ . * Human Eyeball: Approximately a sphere with diameter = $2.5\,cm$ . * Volume Factor: Because the diameter is $10\times$ larger, the volume is $10^3 = 1000\times$ greater.
Effects of Changing Dimensions: * If the radius of a sphere is tripled, the volume increases by a factor of 27 ( $3^3$ ). * If the radius of a sphere is divided by 3, the surface area is reduced by a factor of 9 ( $3^2$ ).

Calculating Height from Volume: A pyramid with $V = 108\,cubic inches$ and Base Area = $27\,square inches$ . * Calculation: $108 = rac{1}{3} imes 27 imes h \implies 108 = 9h \implies h = 12\,inches$ .
Similar Cylinders: A cylinder with $V = 500 ext{pi}\,cm^3$ is similar to one with $V = 4 ext{pi}\,cm^3$ . * Volume ratio: $rac{500 ext{pi}}{4 ext{pi}} = 125$ . * Linear ratio (heights): The cube root of the volume ratio ( $ext{sqrt}[3]{125} = 5$ ). The ratio of heights is $5:1$ .
Algebraic Systems and Coordinate Geometry: * System: * $y = -x$ * $x^2 + y^2 = 8$ * Substitution Procedure: $x^2 + (-x)^2 = 8 \implies 2x^2 = 8 \implies x^2 = 4 \implies x = +/-\,2$ . * Solutions: $(2, -2)$ and $(-2, 2)$ .

Prompt: What did you notice about the relationship between a pyramid and a similarly-based prism based on the video?
Response: The lesson highlights that the bases and heights were the same, and the pyramid's liquid volume fits into the prism exactly three times.
Prompt: What is a real-world example of a great circle?
Response: The Equator.
Prompt: What did you notice about the relationship between a cone and a cylinder based on the video?
Response: The cone follows a similar rule to the pyramid, containing one-third the volume of its corresponding cylinder.