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UNIT EQ: HOW DO YOU CALCULATE THE SURFACE AREA AND VOLUME OF A 3-DIMENSIONAL FIGURE Surface Area & Volume

ESSENTIAL QUESTION: WHAT ARE THE PARTS OF A POLYHEDRON? Space Figures

Nets  A net is a 2-dimensional pattern that you can fold to form a 3-dimensional figure.  Which one?

Draw & Label a net for this box:

Polyhedron

Euler’s Formula (Oiler)  F =  V =  E =

Cross Sections:

Examples

Cross Sections:  Vertical Plane?  Horizontal Plane?  Diagonal Plane?

Different Cross Sections?  How many different polygonal cross-sections can you create?

Openers:

ESSENTIAL QUESTION: HOW DO YOU FIND THE SURFACE AREA AND VOLUME OF PRISMS AND CYLINDERS? Surface Area of Prisms & Cylinders

Vocabulary  PRISM – A polyhedron with exactly two parallel and congruent faces, called bases. The other faces are lateral faces.  An altitude is a segment that is perpendicular to both bases. The height is the length of the altitude.

Right vs. Oblique  What shape are the Lateral Faces of each?

Surface Area  The lateral area is the sum of the areas of the lateral faces.  The surface area is the sum of the areas of the lateral faces and the bases.

Use the net to find Surface Area:

Formulas for Surface Area:

Find the Lateral and Surface Areas:

Cylinders  Cylinders have two congruent and parallel bases that are circles.

Lateral & Surface Areas:

Example:

Opener

ESSENTIAL QUESTION: HOW DO YOU FIND THE SURFACE AREA AND VOLUME OF PYRAMIDS AND CONES? Surface Area of Pyramids & Cones

Vocabulary  Base vs Lateral edges  Height vs. SLANT Height

Lateral Area & Surface Area

 Find the Surface Area of a Square Pyramid with base edges 5 cm and height of 3 cm.

Cones  Height vs. Slant Height  Lateral Area: (½ pl)  Surface Area: (LA + B)

Opener  Find Lateral & Surface Areas:

ESSENTIAL QUESTION: HOW DO YOU FIND THE SURFACE AREA AND VOLUME OF PRISMS AND CYLINDERS? Volume of Prisms & Cylinders

Volume

Cavalieri’s Principle

Volume of Prisms  Rectangular Prisms: Area of Base x Height  Cavalieri allows formula to extend to ANY Prism!

Volume of a Prism

Cylinders  If V = Bh, then what is the area of the base of a generic Cylinder?

Examples:

Volume of Composite Figures:  Volume Addition Postulate

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ESSENTIAL QUESTION: HOW DO YOU FIND THE SURFACE AREA AND VOLUME OF PYRAMIDS AND CONES? Volume of Pyramids & Cones

Volume of a Pyramid?  Prism: A = phV = Bh  Pyramid:A = ½ phV =

Cones are just like Pyramids!

Examples

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ESSENTIAL QUESTION: HOW DO YOU FIND THE SURFACE AREA AND VOLUME OF SPHERES? Spheres

Sphere  A sphere is the set of all points in space equidistant from a point called the center.  A sphere has a radius from the center to the sphere, and a diameter is any segment through the center with both endpoints on the sphere.  A sphere can be cut into hemispheres, Whose cross-section is a great circle.

Sphere Surface Area  Area Formula?

Examples:

Sphere Volume:

Volume of a Sphere

Volume to Surface Area!  If the Volume of a Sphere is 5,000in 3 then what is the Sphere’s Surface Area?

L – A – V  If two solids are SIMILAR and the ratio of similarity is ¾ then what would be the ratios for the following:  L engths:  Surface A reas:  V olumes:

ESSENTIAL QUESTION: HOW DO YOU FIND SURFACE AREA AND VOLUME OF SIMILAR 3-D FIGURES COMPARE? Similar Solids

 Similar Solids have the same shape and proportional dimensions. The ratio between corresponding edges is the similarity ratio.

Are the following solids Similar?  If so, give the similarity ratio:

Similar?  Similarity Ratio:  Surface Areas:  Surface Area Ratio:  Volumes:  Volume Ratio:

Two Cubes  Two cubes have volumes 729 cm 3 and 1331 cm 3.  Find their ratio of Similarity and their Surface Area ratio.

Example: