Right Prisms & Cylinders, Right
Pyramids & Cones, Platonic
Solids, Composite Figures
• Right Solids
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Cylinders
Interior Angle of a Polygon
Area of a Polygon
Right Prisms
• Pyramids & Cones
– Pyramids
– Cones
• Platonic Solids
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Why Only 5?
Tetrahedron
Cube (Hexahedron)
Octahedron
Dodecahedron
Icosahedron
• Composite Figures
• Faces & Vertices & Edges,
Oh My!
Two congruent ends connected by
rectangle(s).
One end is the “Base”.
The “height” connects the two ends.
Volume is Base times height.
Surface Area is 2 times Base + the
rectangles connecting the ends.
Two congruent circles connected by one rectangle.
Area of the base is πr2.
Volume = πr2*h.
Lateral Surface Area is one rectangle. The area of the rectangle is
circumference of the base times height. LSA = 2πr*h.
SA = 2(πr2) + 2πr*h.
h
2πr
πr2
1) Find volume given h=10 and
r=5. Round to .001.
2) Find r given h=13 and V= 52π.
3) Find SA given h=3 and V=12π.
4) Find SA given that d=14 and
h=10.
5) Find h given that d=8 and
V=220. Round to the 10th.
The long way which creates understanding…
135⁰
45⁰
67.5⁰
67.5⁰
360⁰ = 45⁰
180⁰-45⁰
= 67.5⁰
82
Regular polygons are made of
congruent isosceles triangles.
Around the center of the polygon
there is 360⁰.
Divide 360⁰ by the number of sides
and you have the apex angle of the
isosceles triangle.
Subtract the apex angle from 180⁰,
divide by 2 and you have the base
angles of the isosceles triangle.
The interior angle is twice the base
angle.
The shortcut!
135⁰
180⁰(n-2)
n
180⁰(8-2)
=
135⁰
8
14
12
a
s
A=nΔ=n(½sa)
A=6(½*14*12)=504
Regular polygons are made of
congruent isosceles triangles.
The apothem of the polygon is the
altitude of the isosceles triangle.
The side of the polygon is the base of
the isosceles triangle.
Area of the polygon is n times the
area of the isosceles triangle.
Area of the polygon is also ½ the
perimeter times the apothem.
A=n(½sa)=½(ns)a=½Pa
Two congruent n-gons connected by n rectangles.
If the polygons are regular then
Area of the base is nΔ=n(½ s*a)= ½ Pa.
Volume = (nΔ)*h=(½Pa)*h.
Lateral Surface Area is n rectangles. The area of one rectangle is
side of the polygon times height. LSA = n(s*h).
SA = 2(½Pa) + n(s*h).
s
a
n(s*h)
nΔ
= ½ Pa
h
s*h
1) Find V & SA of a triangular prism
given h (of prism)=5, s=12, a=6√3.
2) Find s of a hexagonal prism given
h=13 and V= 1950√3.
3) Find P of a hexagonal prism given
h=10, a=7√3 and V= 2940√3.
4) Find SA of a octagonal prism given
that h=14 and s=10.
5) Find V & SA of a pentagonal prism
given h=10 and s=14. Round to .001.
All polygons are regular.
h
πr2
2 h l One end connected to a point directly over
LA=
V= 1/13/3πr(2πr)
* *
the center of the end.
The end is the “Base”.
l
The “height” connects the center to the point.
Volume is 1/3 Base times height.
Lateral Area connects the base to the point.
Surface Area is Base + lateral surface area.
1 s2 h
LA=V=1/2/P
3 *l *
h
s2
l
The base can be any polygon. The lateral area, “sides”, are triangles.
The lateral area is the sum of the areas of all the triangles.
The side of the polygon is the base of the triangle.
The altitude of the triangle is the slant height of the pyramid.
Surface Area = Base + Lateral Surface Area. If the polygon is regular than you can use
the surface area formula on the reference sheet.
The height, apothem, and slant height form a right triangle.
Volume = 1/3Base*height
LA= n(½ s*l)=½(ns)*l =½P*l
SA=½P*l + B
l2=h2 + a2
V=1/3(½Pa)*h
60⁰
90⁰
108⁰
1) What is the upper limit 1) The sum must be less
of the sum of the
than 360⁰ or the
angles of a vertex of a
“corner” will be flat.
polyhedra?
2) The interior angle of a
2) Find the interior angles
regular triangle is 60⁰,
of the regular: triangle,
square is 90⁰, pentagon
square, pentagon,
is 108⁰, hexagon is
hexagon, septagon,
120⁰, septagon is
octagon, & nonagon.
128.57⁰, octagon is
135⁰, & nonagon is
140⁰.
120⁰
128.57⁰
135⁰
3) What is happening?
4) What is the minimum
number of faces that will
meet at the vertex of a
polyhedra?
5) What congruent regular
polygon can meet at a
vertex? (i.e. What are the
regular polygons that can
be the faces of a regular
polyhedra?)
3) As the number of sides
increases, the interior angle
gets closer to 180⁰.
4) There must be at 3 faces or the
figure will not be 3D.
5) Triangle, Square, & Pentagon.
90⁰*3<360 ⁰
108⁰*3<360 ⁰
60⁰*3<360 ⁰
60⁰*4<360 ⁰
60⁰*5<360 ⁰
• Describe the five platonic solids by face and
number of polygons that meet at a vertex.
Justify why each platonic solid is possible
using the previous reasoning.
1) What is the number of faces,
vertices, & edges of a
pentagonal prism?
2) What is the number of faces,
vertices, & edges of a triangular
prism?
3) What is the number of faces,
vertices, & edges of a
pentagonal pyramid?
4) What is the number of faces,
vertices, & edges of a
tetrahedron?
5) What are the patterns?
1) In a pentagonal prism,
there are 7 faces, 10
vertices & 15 edges.
2) In a triangular prism,
there are 5 faces, 6
vertices & 9 edges.
3) In a pentagonal pyramid,
there are 6 faces, 6
vertices & 10 edges.
4) In a tetrahedron, there
are 4 faces, 4 vertices & 6
edges.
What can you conclude about the sum of the exterior
angles of a polygon?