Chapter 12: Extending Surface Area and Volume
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