Chapter 13: Solid Shapes and
their Volume & Surface Area
Section 13.1: Polyhedra and other Solid Shapes
Basic Definitions
• A polyhedron is a closed, connected shape in space whose outer
surfaces consist of polygons
• A face of a polyhedron is one of the polygons that makes up the outer
surface
• An edge is a line segment where two faces meet
• A vertex is a corner point where multiple faces join together
• Polyhedra are categorized by the numbers of faces, edges, and
vertices, along with the types of polygons that are faces.
Examples of Polyhedra
Cube
Pyramid
Icosidodecahedron
Example 1
• Find the number of and describe the faces of the following
octahedron, and then find the number of edges and vertices.
Example 2
• Find the number of and describe the faces of the following
icosidodecahedron, and then find the number of edges and vertices.
Non-Examples
• Spheres and cylinders are not polyhedral because their surfaces are
not made of polygons.
Special Types of Polyhedra
• A prism consists of two copies of a polygon lying in parallel planes
with faces connecting the corresponding edges of the polygons
• Bases: the two original polygons
• Right prism: the top base lies directly above the
bottom base without any twisting
• Oblique prism: top face is shifted instead of
being directly above the bottom
• Named according to its base (rectangular prism)
Prism Examples
More Special Polyhedra
• A pyramid consists of a base that is a polygon,
a point called the apex that lies on a different
plane, and triangles that connect the apex to
the base’s edges
• Right pyramid: apex lies directly above the
center of the base
• Oblique pyramid: apex is not above the center
Pyramid Examples
A very complicated example
• Adding a pyramid to each pentagon of an icosidodecahedron creates
a new polyhedron with 80 triangular faces called a pentakis
icosidodecahedron.
See Activity 13B
Similar Solid Shapes
• A cylinder consists of 2 copies of a closed curve (circle, oval, etc) lying
in parallel planes with a 2-dimensional surface wrapped around to
connect the 2 curves
• Right and oblique cylinders are defined similarly to those of prisms
Other Similar Solid Shapes
• A cone consists of a closed curve, a point in a different plane, and a
surface joining the point to the curve
Platonic Solids
• A Platonic Solid is a polyhedron with each face being a regular
polygon of the same number of sides, and the same number of faces
meet at every vertex.
• Only 5 such solids:
• Tetrahedron: 4 equilateral triangles as faces, 3 triangles meet at
each vertex
• Cube: 6 square faces, 3 meet at each vertex
• Octahedron: 8 equilateral triangles as faces, 4 meet at each vertex
• Dodecahedron: 12 regular pentagons as faces, 3 at each vertex
• Icosahedron: 20 equilateral triangles as faces, 5 at each vertex
Platonic Solids
Scattergories
die
Pyrite crystal
Section 13.2: Patterns and
Surface Area
Making Polyhedra from 2-dimensional surfaces
• Many polyhedral can be constructed by folding and joining
two-dimensional patterns (called nets) of polygons.
• Helpful for calculating surface area of a 3-D shape, i.e. the
total area of its faces, because you can add the areas of each
polygon in the pattern (as seen on the homework)
How to create a dodecahedron calendar
• http://folk.uib.no/nmioa/kalender/
Cross Sections
• Given a solid shape, a cross-section of that shape is formed by slicing
it with a plane.
• The cross-sections of polyhedral are polygons.
• The direction and location of the plane can result in several different
cross-sections
• Examples of cross-sections of the cube:
https://www.youtube.com/watch?v=Rc8X1_1901Q
Section 13.3: Volumes of
Solid Shapes
Definitions and Principles
• Def: The volume of a solid shape is the number of unit cubes that it
takes to fill the shape without gap or overlap
• Volume Principles:
 Moving Principle: If a solid shape is moved rigidly without
stretching or shrinking it, the volume stays the same
 Additive Principle: If a finite number of solid shapes are combined
without overlap, then the total volume is the sum of volumes of
the individual shapes
 Cavalieri’s Principle: The volume of a shape and a shape made by
shearing (shifting horizontal slices) the original shape are the same
Volumes of Prisms and Cylinders
• Def: The height of a prism or cylinder is the perpendicular distance
between the planes containing the bases
Volumes of Prisms and Cylinders
• Formula: For a prism or cylinder, the volume is given by
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 ∙ ℎ𝑒𝑖𝑔ℎ𝑡
• The formula doesn’t depend on whether the shape is right or oblique.
Volumes of Particular Prisms and Cylinders
• Ex 1: The volume of a rectangular box with length 𝑙, width 𝑤, and
height ℎ is
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙 ∙ 𝑤 ∙ ℎ
• Ex 2: The volume of a circular cylinder with the radius of the base
being 𝑟 and height ℎ is
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋 ∙ 𝑟 2 ∙ ℎ
Volumes of Pyramids and Cones
• Def: The height of a pyramid or cone is the perpendicular length
between the apex and the base.
Volumes of Pyramids and Cones
• Formula: For a pyramid or cone, the volume is given by
1
𝑉𝑜𝑙𝑢𝑚𝑒 = ∙ (𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒) ∙ ℎ𝑒𝑖𝑔ℎ𝑡
3
• Again, the formula works whether the shape is right or oblique
Volume Example
• Ex 3: Calculate the volume of the
following octahedron.
Volume of a Sphere
• Formula: The volume of a sphere with radius 𝑟 is given by
4
𝑉𝑜𝑙𝑢𝑚𝑒 = ∙ 𝜋 ∙ 𝑟 3
3
• See Activity 13O for explanation of why this works.
Volume vs. Surface Area
• As with area and perimeter, increasing surface area generally
increases volume, but not always.
• With a fixed surface area, the cube has the largest volume of any
rectangular prism (not of any polyhedron) and the sphere has the
largest volume of any 3-dimensional object.
See examples problem in Activity 13N
Section 13.4: Volumes of
Submerged Objects
Volume of Submerged Objects
• The volume of an 3-dimensional object can be calculated by
determining the amount of displaced liquid when the object is
submerged.
• Ex: If a container has 500 mL of water in it, and the water level rises
to 600 mL after a toy is submerged, how many 𝑐𝑚3 is the volume of
the toy?
Volume of Objects that Float
• Archimedes’s Principle: An object that floats displaces the amount of
water that weighs as much as the object
See example problems in Activity 13Q