G E O M E T R Y
CHAPTER 12
SURFACE AREA &
VOLUME
Notes & Study Guide
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TABLE OF CONTENTS
POLYHEDRA ....................................................................................................... 3
(Section 12.1)
REGULAR & PLATONIC SOLIDS ...................................................................... 4
SURFACE AREA & VOLUME ............................................................................. 5
PRISMS................................................................................................................ 6
(Sections 12.2 & 12.4)
CYLINDERS......................................................................................................... 7
PYRAMIDS .......................................................................................................... 9
(Sections 12.3 & 12.5)
CONES............................................................................................................... 10
SPHERES .......................................................................................................... 12
(Section 12.6)
FORMULA SUMMARIES .................................................................................. 13
ADDITIONAL EXAMPLES ................................................................................ 15
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POLYHEDRA
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A polyhedron is a 3-D figure (solid) that is formed
from polygons. (the plural form is polyhedra)
“poly” means many and “-hedron” means surfaces
PARTS OF POLYHEDRA
Faces  each surface of a polyhedron (the panels)
Edges  segments formed where 2 faces meet (the lines)
Vertices  points formed where 3+ faces meet (the corners)
NO CURVES!
Polyhedra MUST be formed from polygons! Since polygons
can’t have curves, polyhedra cannot have curves either.
CONCAVE vs CONVEX
Polygons were concave if they had a dent. Polyhedra are
concave if they have a dent in them
TYPES OF SOLIDS
In Geometry, there are 5 types of 3-D figures (called solids). Only 2 of them are
considered polyhedra. We will be studying all five.
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REGULAR & PLATONIC SOLIDS
If all of the faces of a polyhedron are congruent, regular
polygons then that polyhedron is called a regular
polyhedron.
Every face is identical and made from identical segments
PLATONIC SOLIDS
Regular polyhedra are rare. There are only five that exist. We call them the
Platonic Solids (named for person who discovered them).
TETRAHEDRON
4 triangles
DODECAHEDRON
12 pentagons
CUBE
6 squares
OCTAHEDRON
8 triangles
ICOSAHEDRON
20 triangles
All polyhedra have a pattern between their faces, edges and vertices…
EULER’S THEOREM
The number of faces (F), vertices (V) and edges (E) of a polyhedron meet the
following pattern…
F+V=E+2
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SURFACE AREA & VOLUME
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SURFACE AREA
SURFACE AREA (SA), is the combined area of the faces
of any polyhedron.
Surface area pertains to the “outer shell” of a solid only
and not the space contained within.
Think of surface area as the “perimeter” of a solid
In general, to find surface area, find the area of the individual faces and add
them all together. (Formulas exist to make it easier)
VOLUME
VOLUME (V), is the total amount of space a polyhedron
occupies (sometimes called its’ capacity).
Think of volume as how much a solid can hold if you
filled it with water
In general, to find volume, you multiply the length, width and height of a solid
together.
HOWEVER! Some solids have changing dimensions, so formulas exist to make
it easier.
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PRISMS
A PRISM is a polyhedron that has 2 parallel, congruent faces (bases) connected
by a set of rectangles (lateral faces).
There are many types of prisms and they get their names from the shape of
their bases (rectangular, triangular, pentagonal, etc.)
RIGHT PRISM – when the lateral faces form right
angles to the bases.
OBLIQUE PRISM – when the lateral faces do not
form right angles to the bases.
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CYLINDERS
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A CYLINDER is a solid with 2 parallel congruent CIRCULAR faces (bases)
connected by a single rectangle (lateral face).
Circular bases cannot change; therefore there are no other types of cylinders like
there are for prisms.
CYLINDERS ARE NOTHING MORE THAN ROUND PRISMS
RIGHT CYLINDER – when the lateral face
forms a right angle to the bases
OBLIQUE CYLINDER – when the lateral face
does not form a right angle to the bases
Oblique solids have slight differences in their heights! Be careful!
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FORMULAS: PRISMS & CYLINDERS
PRISMS
SURFACE AREA
VOLUME
SA = 2B + Ph
V = Bh
B = area of base P = perimeter of base h = height
EXAMPLE: Find the surface area and volume of the prism.
CYLINDERS
SURFACE AREA
VOLUME
SA = 2πr2 + 2πrh
V = πr2h
r = radius h = height
EXAMPLE: Find the surface area and volume of the cylinder.
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PYRAMIDS
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A PYRAMID is a solid that has a polygon base with triangular lateral faces that
meet at a single point (vertex).
There are many types of pyramids and they get their names from the shape of
their base (rectangular, triangular, pentagonal, etc).
BEWARE! PYRAMIDS HAVE TWO HEIGHTS!
HEIGHT (true height) – perpendicular distance
from the vertex to the base (down the interior)
SLANT HEIGHT – distance from the vertex to
the base along one lateral face (on the exterior)
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CONES
A CONE is a solid that has a circular base and whose lateral face is a single
sector that meets at a single point (vertex).
Since circular bases cannot change there are no extra types of cones (like
pyramids or prisms).
CONES ARE NOTHING MORE THAN CIRCULAR PYRAMIDS
BEWARE! PYRAMIDS HAVE TWO HEIGHTS!
Refer to the same definitions from pyramids (page 9).
RIGHT CONE – when the true height, slant
height and the radius form a right triangle.
OBLIQUE CONE – when the true height, slant
height and the radius do not form a right triangle
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FORMULAS: PYRAMIDS & CONES
PYRAMIDS
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SURFACE AREA
VOLUME
SA = B + 1/2Pl
V = 1/3Bh
B = area of base P = perimeter of base h = height l = slant height
EXAMPLE: Find the surface area and volume of the pyramid.
CONES
SURFACE AREA
VOLUME
SA = πr2 + πrl
V = 1/3πr2h
r = radius h = height l = slant height
EXAMPLE: Find the surface area and volume of the cylinder.
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SPHERES
A SPHERE is a solid that consists of the set of points in space that are a given
distance away from a given point (center).
When a plane intersects a sphere, the
intersection is a circle. If that circle
happens to contain the center, then it is
called a great circle.
SPHERES
SURFACE AREA
VOLUME
SA = 4πr2
V = 4/3πr3
EXAMPLE: Find the surface area and volume of the sphere.
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FORMULA SUMMARY
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SUMMARY OF SURFACE AREA
AND VOLUME FORMULAS
SOLID
Surface Area
Volume
Prisms
S = 2B + Ph
V = Bh
Cylinders
S = 2πr2 + 2πrh
V = πr2h
Pyramids
Cones
Spheres
S = 4πr2
B = area of base r = radius
h = height
= slant height P = perimeter of base
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AREA OF POLYGONS
AREA FORMULAS
FOR POLYGONS
Triangle
Rectangle
Square
Parallelogram
Trapezoid
Rhombus
Kite
Circle
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