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In Glorious 3-D!. Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height. Polyhedron. - PowerPoint PPT Presentation
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In Glorious 3-D!
Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.
Polyhedron
A solid formed by polygons that enclose a single region of space is called a polyhedron.
Parts of Polyhedrons
• Polygonal region = face• Intersection of 2 faces = edge• Intersection of 3+ edges = vertex
face vertexedge
Surface Area of Prisms, Cylinders
Objectives:
1. To find and use formulas for the lateral and total surface area of prisms, cylinders, pyramids, and cones
Prism
A polyhedron is a prism iff it has two congruent parallel bases and its lateral faces are parallelograms.
Pyramid
A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex.
Pyramid
A regular pyramid is one whose base is a regular polygon.
• The slant height is the height of one of the congruent lateral faces.
Cylinder
A cylinder is a 3-D figure with two congruent and parallel circular bases.
• Radius = radius of base
Nets
Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.
An unfolded pizza box is a net!
Nets
Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.
Exercise 1
There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net?
Surface Area
The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid.
• Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it.
Lateral Surface Area
The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid.
• Think of the lateral surface area as the size of a label that you could put on the figure.
Exercise 2
What solid corresponds to the net below?
How could you find the lateral and total surface area?
Exercise 3
Draw a net for the rectangular prism below.
To find the lateral surface area, you could:
• Add up the areas of the lateral rectangles
A B C D
Exercise 3
Draw a net for the rectangular prism below.
To find the lateral surface area, you could:
• Find the area of the lateral surface as one, big rectangle
Hei
gh
t o
f P
rism
Perimeter of the Base
Exercise 3
Draw a net for the rectangular prism below.
To find the total surface area, you could:
• Find the lateral surface area then add the two bases
Hei
gh
t o
f P
rism
Perimeter of the Base
Exercise 5
Draw a net for the cylinder.
Notice that the lateral surface of a cylinder is also a rectangle. Its height is the height of the cylinder, and the base is the circumference of the base.
Exercise 7
The net can be folded to form a cylinder.
What is the approximate lateral and total surface area of the cylinder?
Exercise 8
Draw a net for the square pyramid below.
To find the lateral surface area:
• Find the area of one triangle, then multiply by 4
12 40 25A
124 40 25S
12 4 40 25S
Exercise 8
Draw a net for the square pyramid below.
To find the lateral surface area:
• Find the area of one triangle, then multiply by 4
12 40 25A
124 40 25S
12 4 40 25S
12 4S s l
12S Pl
Exercise 8
Draw a net for the square pyramid below.
To find the total surface area:
• Just add the area of the base to the lateral area
12 40 25A
124 40 25S
12 4 40 25S
12 4S s l
12S Pl12S Pl B
Surface Area of a Pyramid
Lateral Surface Area of a Pyramid:
• P = perimeter of the base• l = slant height of the
pyramid
Total Surface Area of a Prism:
• B = area of the base
12S Pl 1
2S Pl B