Chapter 12: Extending Surface Area and Volume

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Cavalieri's Principle

If any two solid figures have the same height h and the same cross-sectional area B at every level, then they have the same volume.

How can we know the volume of the cone directly from knowing the volume of a pyramid?

If the pyramid and prism have the same base area B and height h as the cylinder and cone, then by Cavalieri's principle, the volume of the cone must be 1/3 the volume of the cylinder, if the volume of the pyramid is 1/3 the volume of the prism.

THEOREM

If two similar solids have a scale factor of a:b, then the surface areas have a ratio of a^2:b^2, and the volumes have a ratio of a^3:b^3.

Volume of a General Cylinder

The volume V of a cylinder is V=Bh or V=πr^2h, where B is the area of the base (πr^2), h is the height of the cylinder, and r is the radius of the base.

Volume of a Hemisphere

The volume V of a hemisphere is V=(1/2)(4/3)πr^3, or simply (2/3)πr^3; no need to add the area of the great circle.

Volume of a General Prism

The volume V of a prism is V=Bh, where B is the area of a base and h is the height of the prism.

Volume of a Sphere

The volume V of a sphere is V=(4/3)πr^3, where r is the radius of the sphere.

Volume of a cone

The volume of a circular cone is V=(1/3)Bh, or V=(1/3)πr^2h, where B is the area of the base (πr^2), h is the height of the cone, and r is the radius of the base.

Volume of a Pyramid

The volume of a pyramid is V=(1/3)Bh, where B is the area of the base and h is the height of the pyramid.

The given point is called the...

center

isometric views

corner views of three-dimensional geometric solids on two-dimensional paper

lateral faces

faces that are not bases

Properties of Congruent Solids

~Corresponding angles are congruent. ~Corresponding edges are congruent. ~Corresponding faces are congruent. ~Volumes are equal.

Lateral Area of a Regular Pyramid

The lateral area L of a regular pyramid is L=0.5Pℓ, where ℓ is the slant height and P is the perimeter of the base.

Lateral Area of a Cone

The lateral area L of a right circular cone is L=πrℓ, where r is the radius of the base and ℓ is the slant height.

Lateral Area of a Right Cylinder

The lateral area L of a right cylinder is L=2πrh, where r is the radius of a circular base and h is the height.

Surface Area of a Hemisphere

The surface area S of a hemisphere is S=(1/2)4πr^2+πr^2, because a hemisphere also contains a great circle.

Surface Area of a Regular Pyramid

The surface area S of a regular pyramid is S=0.5Pℓ+B, where P is the perimeter of the base, ℓ is the slant height, and B is the area of the base.

Surface Area of a Cone

The surface area S of a right circular cone is L=πrℓ+πr^2, where r is the radius of the base and ℓ is the slant height.

Surface Area of a Right Cylinder

The surface area S of a right cylinder is 2πrh+2πr^2, where r is the radius of a circular base and h is the height.

Surface Area of a General Prism

The surface area S of a right prism is S=L+2B, where L is its lateral area and B is the area of a base.

Surface Area of a Sphere

The surface area S of a sphere is S=4πr², where r is the radius.

What is true about the lateral edges in a solid?

They are both parallel and congruent.

Property of Similar Solids

Two solids are similar if they have the same shape and their corresponding linear measures are proportional.

right cone

a cone with an axis that is also an altitude

oblique cone

a cone with an axis that is not an altitude

Euclidean geometry

a geometrical system in which a plane is a flat surface made up of points that extend infinitely in all directions

non-Euclidean geometry

a geometry in which at least one of the postulates from Euclidean geometry fails

altitude

a perpendicular segment that joins the planes of the bases in any solid figure

regular pyramid

a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles

axis of a cylinder

a segment with endpoints that are centers of the circular bases

The locus of all points in space that are equidistant from a given point is called...

a sphere

composite solid

a three-dimensional figure that is composed of simpler two-dimensional figures

How many great circles does a sphere contain?

infinitely many

lateral edges

intersection of lateral faces

All spheres and cubes are...

similar, because their respective radii and base edges are directly proportional to their volumes, etc.

spherical geometry

the branch of geometry that deals with a system of points, great circles (lines), and spheres (planes)

poles

the endpoints of a diameter of a great circle

slant height

the height of each lateral face of a REGULAR pyramid; represented by the letter ℓ

cross section

the intersection of a solid and a plane

great circle

the intersection of a sphere and a plane that contains the center of the sphere

base edges

the intersection of the lateral faces and bases in a solid figure

Lateral Area of a General Prism

the lateral area of a right prism is the product of the perimeter of the base and the height of the prism; L=Ph

height

the length of the altitude

surface area

the sum of all the areas of all the faces or surfaces that enclose a solid

lateral area

the sum of the areas of the lateral faces

hemispheres

two congruent halves formed by dividing a great circle

similar solids

two solids that have exactly the same shape, but not necessarily the same size

congruent solids

two solids with the same shape, size and similar by a scale factor of 1:1


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