Measurement – 3D

Right Prisms & Cylinders, Right Pyramids & Cones,

Platonic Solids, Composite Figures

Table of Contents

• Right Solids– Cylinders– Interior Angle of a

Polygon– Area of a Polygon– Right Prisms

• Pyramids & Cones– Pyramids– Cones

• Platonic Solids– Why Only 5?– Tetrahedron– Cube (Hexahedron)– Octahedron– Dodecahedron– Icosahedron

• Composite Figures• Faces & Vertices & Edges,

Oh My!

Right SolidsTwo 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.

CylinderTwo 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.

2πrπr2

h

Cylinder

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.

What is missing?

Interior Angle of a Regular Polygon

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.

360⁰8

= 45⁰180⁰-45⁰2 = 67.5⁰

45⁰

67.5⁰67.5⁰

135⁰

The long way which creates understanding…

Interior Angle of a Regular Polygon

180⁰(n-2)n

= 135⁰

135⁰

The shortcut!

180⁰(8-2)8

Area of a Regular Polygon

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=nΔ=n(½sa)A=6(½*14*12)=504

14

12

A=n(½sa)=½(ns)a=½Pa

a

s

Right PrismTwo 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).

n(s*h)

nΔ = ½ Pa

h s*h

sa

Right Prism

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.

What is missing?All polygons are regular.

Pyramids & ConesOne end connected to a point directly over the center of the end. The end is the “Base”.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.πr2

h

V= 1/3 πr2*h

h

s2

V= 1/3 s2*h

l

LA= 1/3 (2πr)*l

lLA= 1/2 P*l

PyramidsThe 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 + a2V=1/3(½Pa)*h

Coming Soon – Cones!

Platonic Solids

Why are there only five?1) What is the upper limit

of the sum of the angles of a vertex of a polyhedra?

2) Find the interior angles of the regular: triangle, square, pentagon, hexagon, septagon, octagon, & nonagon.

1) The sum must be less than 360⁰ or the “corner” will be flat.

2) The interior angle of a regular triangle is 60⁰, square is 90⁰, pentagon is 108⁰, hexagon is 120⁰, septagon is 128.57⁰, octagon is 135⁰, & nonagon is 140⁰.

60⁰

90⁰

108⁰120⁰

128.57⁰135⁰

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.

60⁰*3<360 ⁰

90⁰*3<360 ⁰108⁰*3<360 ⁰

60⁰*5<360 ⁰

60⁰*4<360 ⁰

Why are there only five? Cont.

• 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.

Why are there only five? Cont.

Faces and Vertices and Edges, Oh My!

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?