Unit 10: 3D Figures

Polygon

Closed 2-D shape whose sides are segments

Polyhedron

Closed 3-D shape whose faces are polygons (no curves)

Cavaliers’s principle (3D)

If 3D figures have the same height and equal areas everywhere along the height then they have the same volume

(You must know the height of the object and the width along the height, or the base)

Rectangular prism (volume equation)

lwh=V

LENGTH X WIDTH = Base —> full equation could be: Base area x height = volume

Volume of prism (equation)

Vprisim = Base area x height

= B x h

Note: Base AREA used for polygons AND height is defined as the distance between the bases, height is ALWAYS perpendicular to the bases

Height of prism

Height of a prism is the distance between the bases, height is always perpendicular to the bases

Other equations (volume)

Volume/base area = height

Volume/height = base area

Base area x height = volume

Some conversions (Ml=? and gallons = ?)

Ml = cm³

Gallons = by definition are cubed

Prism (definition and equation) also: define “height” of a prism

Definition: A 3-D figure with 2 bases that are congruent and parallel and whose other faces are parallelograms

Equation: Base area x height = Volume AKA: B x h = V

Height: of a prison is the distance between the bases, height it always perpendicular to the bases

Conversion between cubic feet and gallons (and how to convert)

1ft³ = 7.48 gallons

Feet³ = 7.48gal/1 foot —> feet cross cancel and the fraction you multiply it back is equal to one

Intersection

Where shapes cross AKA: when a set of points is in all shapes

Cross section

Intersection of a plane and 3D object/figure

Truncate

Means to cut off

Conic sections

Density (and three equations)

Denser objects have more mass for the same volume

D = m/v

m = d x v

V = m/d

Pyramid and equation for volume

Pyramid: 3D figures with triangular lateral faces and polygon bases

Equation: Vpyramid = 1/3 (Base area x height) = Bh/3

Note: 1/3 because it’s one third of a cube

Ellipse

Another word for oval, can be used for talking about cross sections

Max sides of a cross section

The number of faces

Volume of a cone (equation)

Vcone = (πr² x h)/3

—> contains less volume than may appear, it is one third of a cylinder

Sphere (definition and equation)

Sphere: set of points equidistant from a center point

ratio of Vsphere:Vcylinder = 2:3

Vsphere = 4/3 πr³

Cylinder (volume equation and comparison to sphere)

V = πr²h

Compared: sphere is 2/3 of the volume of a cylinder

Trapezoid AREA formula

A = ((b1 + b2)/2)) (h)

Surface Area of a sphere and why?

Surface area of a sphere = 4πr²

Think about it like a baseball: 4 circles if you lay out the pieces

How to use pyramids to determine the volume of a sphere?

The pyramids’ height is the radius of the sphere.

The sum of the volumes of the pyramids is…

1/3(B1r) + 1/3(B2r) + 1/3(B3r) +….

Factoring we get: 1/3r(B1 + B2 + B3+…)

The sum of all the bases areas is the surface area of the sphere, which has the formula 4πr²

We can substitute this into the “factoring we get:” equation to get a NEW equation: 1/3r(4πr²) = 4/3πr³

This is the volume formula for a hemisphere!!

Vsphere = 4/3πr³

What is a hemisphere (description and volume formula)

3D shape that is half of a sphere.

Vhemisphere = (2/3)πr³

If the edges of a rectangular prism (or any 3D shape) double, how does the surface area change? How does the volume change?

Edges: times n

Area: times n²

Volume: times n³

The top 1/3 of a pyramid has what percent of the total volume? If we dilate a pyramid by a scale factor of 1/3, how does the volume change? (KNOW HOW TO PROVE)

The top 1/3 of a pyramid…

(You can use example numbers)

1/27=4% (around)

—> Why? Plug in a pyramid of 1,1,1 versus 3,3,3

If we dilate a pyramid by 1/3…

Edges: n

Area: n²

Volume: n³