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Honors Geometry
Table of Contents
Unit 12 Overview: Surface Area and Volume
12.1 Three-dimensional figures and their properties
12.2 Surface area of prisms, cylinders, pyramids, and cones
12.3 Volume of prisms, cylinders, pyramids, and cones
12.4 Surface area and volume of spheres

🔷honors geometry review

12.1 Three-dimensional figures and their properties

Citation:

Three-dimensional figures come in various shapes and sizes. From everyday objects like soup cans to more complex structures, these figures are all around us. Understanding their properties helps us make sense of the world we live in.

This topic covers the classification and characteristics of 3D figures, including polyhedra like prisms and pyramids, and non-polyhedra like cylinders and spheres. We'll also learn how to sketch these figures and explore their cross-sections, giving us a deeper understanding of their structure.

Three-Dimensional Figures

Classification of 3D figures

  • Polyhedra
    • Prisms
      • Consist of two congruent, parallel bases connected by rectangular faces (triangular prism, rectangular prism)
      • Named according to the shape of their bases (pentagonal prism has pentagonal bases)
    • Pyramids
      • Composed of one base connected to a single vertex by triangular faces (square pyramid, hexagonal pyramid)
      • Identified by the shape of their base (triangular pyramid has a triangular base)
  • Non-polyhedra
    • Cylinders
      • Formed by two congruent, parallel circular bases connected by a curved surface (soup can, pipe)
    • Cones
      • Consist of one circular base connected to a single vertex by a curved surface (ice cream cone, party hat)
    • Spheres
      • Characterized by a curved surface equidistant from a central point (basketball, globe)

Characteristics of geometric solids

  • Prisms
    • Bases are congruent and parallel polygons
    • Lateral faces are rectangles that connect the bases
    • Lateral edges are parallel line segments formed by the intersection of lateral faces
    • Examples: cube (6 square faces), rectangular prism (4 rectangular faces and 2 square or rectangular bases), triangular prism (3 rectangular faces and 2 triangular bases)
  • Cylinders
    • Bases are congruent and parallel circles
    • Lateral surface is a curved surface that connects the bases
    • Axis is an imaginary line segment that connects the centers of the bases
    • Examples: soup can (circular bases and curved lateral surface), pipe (hollow cylinder with circular bases)
  • Pyramids
    • One base of any polygon shape (triangle, square, pentagon)
    • Lateral faces are triangles that connect the base to a single vertex called the apex
    • Examples: square pyramid (square base and 4 triangular lateral faces), hexagonal pyramid (hexagonal base and 6 triangular lateral faces)
  • Cones
    • One circular base
    • Lateral surface is a curved surface that connects the base to a single vertex called the apex
    • Axis is an imaginary line segment that connects the center of the base to the apex
    • Examples: ice cream cone (circular base and curved lateral surface), party hat (circular base and pointed apex)
  • Spheres
    • No faces, edges, or vertices
    • Surface is a curved surface that is equidistant from a central point
    • Examples: basketball (spherical shape), globe (model of Earth as a sphere)

Sketching 3D figures from properties

  • Use the given properties to determine the type of three-dimensional figure
    1. Identify the number and shape of bases (2 circular bases suggest a cylinder)
    2. Examine the shape of lateral faces or surfaces (triangular lateral faces indicate a pyramid)
    3. Check for parallel or perpendicular faces or edges (parallel bases are characteristic of prisms and cylinders)
  • Sketch the figure
    1. Draw the bases according to the given shape and dimensions (2 congruent pentagons for a pentagonal prism)
    2. Connect the bases with lateral faces or surfaces (rectangular lateral faces for prisms, curved surface for cylinders)
    3. Ensure the sketch matches the given properties (height, base dimensions, number of lateral faces)
  • Label the figure with relevant information
    • Dimensions (height, base lengths)
    • Angles (right angles for perpendicular faces)
    • Parallel or perpendicular markings (parallel lines for lateral edges of prisms)

Cross-sections of 3D figures

  • Cross-section: the intersection of a plane with a three-dimensional figure
  • Prisms
    • Parallel to bases: cross-section is congruent to the base (cutting a triangular prism parallel to its bases results in a triangular cross-section)
    • Perpendicular to bases: cross-section is a rectangle or parallelogram (cutting a rectangular prism perpendicular to its bases results in a rectangular cross-section)
    • Oblique to bases: cross-section is a polygon with the same number of sides as the base (cutting a hexagonal prism obliquely results in a hexagonal cross-section)
  • Cylinders
    • Parallel to bases: cross-section is a circle (cutting a cylinder parallel to its bases results in a circular cross-section)
    • Perpendicular to bases: cross-section is a rectangle (cutting a cylinder perpendicular to its bases results in a rectangular cross-section)
    • Oblique to bases: cross-section is an ellipse (cutting a cylinder obliquely results in an elliptical cross-section)
  • Pyramids
    • Parallel to base: cross-section is a polygon similar to the base (cutting a square pyramid parallel to its base results in a smaller square cross-section)
    • Perpendicular to base and through apex: cross-section is a triangle (cutting a pyramid perpendicular to its base and through its apex results in a triangular cross-section)
    • Perpendicular to base and not through apex: cross-section is a polygon with the same number of sides as the base (cutting a pentagonal pyramid perpendicular to its base and not through its apex results in a pentagonal cross-section)
  • Cones
    • Parallel to base: cross-section is a circle (cutting a cone parallel to its base results in a circular cross-section)
    • Perpendicular to base and through apex: cross-section is a triangle (cutting a cone perpendicular to its base and through its apex results in a triangular cross-section)
    • Perpendicular to base and not through apex: cross-section is a circle (cutting a cone perpendicular to its base and not through its apex results in a circular cross-section)
    • Oblique to base: cross-section is an ellipse (cutting a cone obliquely results in an elliptical cross-section)
  • Spheres
    • Any plane through the center: cross-section is a circle (cutting a sphere through its center always results in a circular cross-section)
    • Any plane not through the center: cross-section is a circle (cutting a sphere off-center still results in a circular cross-section, although smaller than a cross-section through the center)

Key Terms to Review (24)

Isometric Drawings: Isometric drawings are a type of three-dimensional representation that allows objects to be depicted in a two-dimensional format while maintaining their proportions. This method uses a specific angle, typically 30 degrees from the horizontal, to visualize height, width, and depth simultaneously, giving a clear perspective of the object's shape and size without distortion. They are essential for understanding the properties of three-dimensional figures by providing a more realistic view compared to standard flat drawings.
Reflective Symmetry: Reflective symmetry is a type of symmetry where one half of an object is a mirror image of the other half. This property can be observed in various shapes and figures, including two-dimensional and three-dimensional objects, where a line or plane divides the object into two identical halves. Reflective symmetry plays a crucial role in understanding the properties of geometric figures, allowing for a deeper appreciation of their structural balance and aesthetic appeal.
Vertices: Vertices are the points where two or more edges meet in a geometric figure. In the context of three-dimensional figures, vertices play a crucial role in defining the shape and structure of solid objects, such as cubes, pyramids, and spheres. Each vertex is essential for determining the properties and characteristics of these figures, influencing aspects like volume, surface area, and overall symmetry.
Perspective Drawings: Perspective drawings are a technique used in art and design to represent three-dimensional objects on a two-dimensional surface, creating an illusion of depth and space. This method relies on the principles of geometry to depict how objects appear smaller as they recede into the distance, utilizing vanishing points and horizon lines to guide the viewer’s perception. The effectiveness of perspective drawings is crucial for accurately representing three-dimensional figures, enhancing the understanding of their properties and spatial relationships.
Rotational Symmetry: Rotational symmetry is a characteristic of a shape that looks the same after being rotated by a certain angle around a central point. This concept is crucial in understanding how shapes and objects can be transformed and how they maintain their identity through various movements and orientations.
Vertical Cross-Section: A vertical cross-section is a two-dimensional representation obtained by slicing through a three-dimensional figure along a vertical plane. This technique is used to reveal the internal structure and shape of the figure, making it easier to analyze its properties and dimensions. Understanding vertical cross-sections allows for a deeper insight into the spatial relationships and geometric characteristics of three-dimensional objects.
Edges: Edges are the line segments where two faces of a three-dimensional figure meet. They play a crucial role in defining the shape and structure of these figures, connecting vertices and outlining the boundaries between different faces. Understanding edges helps in visualizing and analyzing the properties of three-dimensional figures, including how they relate to other geometric concepts such as vertices and faces.
Faces: In geometry, faces are the flat surfaces that make up the boundaries of a three-dimensional figure. Each face is a polygon, and the number of faces varies depending on the type of solid figure, such as a cube, pyramid, or prism. Understanding faces is essential as they contribute to the overall shape, surface area, and volume of three-dimensional objects.
Height: Height is a measurement that indicates the vertical distance from the base of a geometric figure to its topmost point. This concept is essential in understanding the dimensions and properties of various three-dimensional figures, as well as calculating areas, volumes, and surface areas. It serves as a critical measurement in defining the size and shape of figures such as prisms, cylinders, pyramids, and cones.
Horizontal cross-section: A horizontal cross-section is a two-dimensional slice or view of a three-dimensional figure taken parallel to the base or horizontal plane of that figure. This concept helps in understanding the internal structure of solid shapes by providing a way to analyze their properties and dimensions at a specific height.
Radius: The radius is the distance from the center of a circle or sphere to any point on its boundary. This key measurement is essential in understanding the properties and formulas related to circles, spheres, and other three-dimensional figures, as it directly influences calculations for circumference, area, volume, and surface area.
Sa = 2πr(h + r): The formula $$sa = 2\pi r(h + r)$$ represents the surface area of a cylinder, where 'sa' stands for surface area, 'r' is the radius of the base, and 'h' is the height of the cylinder. This equation helps in understanding how to calculate the total surface area, which includes both the curved surface area and the areas of the two circular bases. Understanding this formula is essential for analyzing three-dimensional figures, their properties, and how they interact with their environments.
Diameter: The diameter is a straight line segment that passes through the center of a circle and connects two points on its circumference. It serves as a crucial measurement in understanding various properties of circles and three-dimensional shapes, connecting concepts like area, volume, and relationships within geometric figures.
Surface Area: Surface area is the total area that the surface of a three-dimensional figure occupies. It provides a way to measure the extent of the outer layer of a solid object, making it essential for understanding its properties and real-world applications, such as in packaging and construction.
Cross-section: A cross-section is a two-dimensional shape obtained by cutting through a three-dimensional figure. This concept is important as it helps visualize the internal structure and properties of three-dimensional objects, such as how volume and surface area can be calculated or understood. Cross-sections can reveal various shapes depending on the angle and location of the cut, which plays a key role in understanding the geometry of prisms, cylinders, pyramids, and cones.
Euler's Formula: Euler's Formula states that for any convex polyhedron, the relationship between the number of vertices (V), edges (E), and faces (F) can be expressed as $$V - E + F = 2$$. This formula provides a fundamental connection among these three properties, illustrating how they are interrelated in three-dimensional shapes. It is crucial for understanding the structure and characteristics of various polyhedra and plays a significant role in topology.
V = lwh: The formula $$v = lwh$$ represents the volume of a rectangular prism, where $$v$$ is the volume, $$l$$ is the length, $$w$$ is the width, and $$h$$ is the height. This equation shows how to calculate the space occupied by a three-dimensional figure, specifically focusing on the properties of rectangular prisms. Understanding this formula helps in visualizing and quantifying how three-dimensional objects interact with the space around them.
Volume: Volume is a measure of the amount of space occupied by a three-dimensional figure. It quantifies how much substance, such as liquid or gas, can fit inside a given solid shape, making it essential for understanding the properties and dimensions of various geometric figures in three-dimensional space.
Pyramid: A pyramid is a three-dimensional geometric figure that has a polygonal base and triangular faces that converge at a single point called the apex. The most common type of pyramid is the square pyramid, which has a square base and four triangular faces. The properties of pyramids include their volume, surface area, and the relationships between their vertices, edges, and faces, making them important shapes in geometry.
Cone: A cone is a three-dimensional geometric figure that has a circular base and a single vertex that tapers to a point, called the apex. This shape is unique because it combines both flat and curved surfaces, which allows it to hold volume and have specific surface area properties. The cone is commonly found in everyday objects like ice cream cones and traffic cones, making it relatable and easy to visualize in various contexts.
Sphere: A sphere is a perfectly round three-dimensional geometric figure where every point on its surface is equidistant from its center. This unique property makes it a fundamental shape in geometry, often studied for its volume, surface area, and applications in various fields such as physics and engineering. Spheres can be found in nature and human-made objects, making them significant in understanding spatial relationships.
Prism: A prism is a three-dimensional geometric figure that consists of two parallel, congruent bases connected by rectangular lateral faces. The shape of the bases determines the type of prism, such as triangular, rectangular, or hexagonal. Prisms are important in understanding three-dimensional figures and their properties, as they exemplify concepts of volume, surface area, and symmetry.
Pythagorean Theorem in 3D: The Pythagorean Theorem in 3D extends the classic 2D theorem to three dimensions, expressing the relationship between the lengths of the edges of a right rectangular prism or a 3D coordinate system. It states that for a right triangle formed by three points in space, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other three sides, which can be expressed as $$c^2 = a^2 + b^2 + d^2$$, where $c$ is the length of the hypotenuse and $a$, $b$, and $d$ are the lengths along each axis. This concept is vital for understanding distances and relationships among 3D geometric figures.
Cylinder: A cylinder is a three-dimensional geometric figure with two parallel circular bases connected by a curved surface at a fixed distance from the center of the bases. The properties of a cylinder include its height, radius, and volume, which are critical for understanding how this shape interacts with space. As a common solid in geometry, cylinders can be found in various real-world applications, making it essential to grasp their characteristics and calculations.