Comprehensive Study Guide for Three-Dimensional Geometry and Exam #10

Circle Sectors and Arc Properties

• Definition of a Sector: A sector is a portion of a circle defined by two radii and the arc that connects them. It is often visualized as a "slice of pie."

• Area of a Sector (Degrees): The formula for calculating the area of a sector when the central angle ($\theta$) is provided in degrees is:   $\text{Area} = \frac{\theta}{360} \times \pi \times r^2$   - $\theta$ represents the central angle in degrees.   - $r$ represents the radius of the circle.   - $\pi$ is approximately $3.14159$.

• Area of a Sector (Radians): If the central angle is provided in radians, the formula simplifies to:   $\text{Area} = \frac{1}{2} \times r^2 \times \theta$

• Arc Length Connection: The length of the arc ($s$) subtended by the central angle is related to the sector area. The arc length formula is:   $s = r \times \theta$   - Given arc length, the area can also be expressed as:   $\text{Area} = \frac{1}{2} \times r \times s$

Fundamental Terminology for Solid Geometry

• Faces: The individual flat surfaces of a three-dimensional solid. Faces are typically polygons.

• Edges: The line segments formed where two faces of a solid meet.

• Vertices: The corner points where three or more edges intersect. The plural form is vertices; the singular is vertex.

• Euler's Formula for Polyhedra: For any convex polyhedron, the relationship between the number of faces ($F$), vertices ($V$), and edges ($E$) is defined by:   $V - E + F = 2$

Prisms and Three-Dimensional Geometry

• Prism Definition: A polyhedron with two congruent, parallel bases and lateral faces that are parallelograms (or rectangles in the case of right prisms).

• Types of Prisms:   - Right Prisms: The lateral edges are perpendicular to the bases.   - Oblique Prisms: The lateral edges are not perpendicular to the bases.

• Volume of a Prism: The volume is the product of the area of the base ($B$) and the height ($h$) of the prism:   $V = B \times h$

• Surface Area of a Prism:   - Lateral Area ($LA$): The sum of the areas of the lateral faces.     $LA = P \times h$     - $P$ represents the perimeter of the base.   - Total Surface Area ($SA$): The sum of the lateral area and the area of the two bases.     $SA = 2 \times B + P \times h$

• The Pythagorean Theorem in 3D: To find the length of the space diagonal ($d$) of a rectangular prism (a box) with dimensions length ($l$), width ($w$), and height ($h$), use the extension of the Pythagorean theorem:   $d^2 = l^2 + w^2 + h^2$   $d = \sqrt{l^2 + w^2 + h^2}$

Pyramids and Their Properties

• Pyramid Definition: A polyhedron with one polygonal base and lateral faces that are triangles meeting at a single common point called the apex.

• Regular Pyramid: A pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles.

• Volume of a Pyramid: The volume of a pyramid is exactly one-third the volume of a prism with the same base area and height:   $V = \frac{1}{3} \times B \times h$

• Slant Height vs. Altitude:   - Altitude ($h$): The perpendicular distance from the apex to the plane of the base.   - Slant Height ($l$): The distance from the apex to the midpoint of an edge of the base along the triangular face.   - The relationship between altitude, slant height, and the distance from the center of the base to the edge ($a$) often forms a right triangle:     $l^2 = h^2 + a^2$

Cylinders and Cones

• Cylinders:   - A solid with two congruent, parallel circular bases.   - Volume of a Cylinder:     $V = \pi \times r^2 \times h$   - Surface Area of a Cylinder:     $SA = 2 \times \pi \times r^2 + 2 \times \pi \times r imes h$

• Cones:   - A solid with a circular base and a single vertex (apex).   - Volume of a Cone:     $V = \frac{1}{3} \times \pi \times r^2 imes h$   - Surface Area of a Cone:     $SA = \pi \times r^2 + \pi \times r \times l$     - Where slant height $l = \sqrt{r^2 + h^2}$.

Cavalieri's Principle

• The Principle: If two solid figures have the same height and the same cross-sectional area at every level parallel to their bases, then the two solids have the same volume.

• Application: This principle is used to explain why the volume formulas for right solids ($V = B \times h$ or $V = \frac{1}{3} \times B \times h$) also apply to oblique solids, provided the height ($h$) is measured as the perpendicular distance between the bases or from the apex to the base.

Spheres and Combined Solids

• Sphere Definition: The set of all points in three-dimensional space that are a fixed distance (radius) from a given point (center).

• Volume of a Sphere:   $V = \frac{4}{3} \times \pi \times r^3$

• Surface Area of a Sphere:   $SA = 4 \times \pi \times r^2$

• Combined (Composite) Solids: Problems involving combined solids require calculating the volumes or surface areas of multiple individual shapes (e.g., a cylinder with a hemispherical top) and adding or subtracting them as necessary to find the total.

Mass, Volume, and Density

• Density Definition: Density is the ratio of the mass of a substance to its volume. It describes how much matter is packed into a specific amount of space.

• Density Formula:   $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Derived Formulas:   - To find Mass: $\text{Mass} = \text{Density} \times \text{Volume}$   - To find Volume: $\text{Volume} = \frac{\text{Mass}}{\text{Density}}$

• Units: Density is typically expressed in units such as $g\,cm^{-3}$ or $kg\,dm^{-3}$.

Exam #10 Logistics and Requirements

• Exam Date: Wednesday, April 29, 2026.

• Curriculum Coverage: The exam covers all material taught from March 31 to April 23.

• Formula Responsibility: Students are strictly responsible for memorizing all relevant formulas for this unit. While the instructor may provide a formula sheet as a backup, it is not guaranteed to be comprehensive of all needs.

• Additional Topics: Students must be proficient in density calculations and their application to geometric solids.

• Theorem Sheets: There is no update to the existing Theorem Sheet for this period.