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Polyhedron

a solid bounded by polygons called faces

Face (of a Polyhedron)

polygons that bound a polyhedron

Edge (of a Polyhedron)

Line Segment formed by the intersection of two faces

Vertex (of a Polyhedron)

point where three or more edges meet

Base

used to name a prism or pyramid

Regular Polyhedron

all faces are congruent regular polygons

Convex Polyhedron

any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron

Platonic Solids

the five regular polyhedral

Tetrahedron

4 faces (Platonic Solid)

Cube

6 faces (Platonic Solid)

Octahedron

8 faces (Platonic Solid)

Dodecahedron

12 faces (Platonic Solid)

Icosahedron

20 faces (Platonic Solid)

Cross Section

intersection of the plan and the solid

Prism

polyhedron with two congruent faces, called bases that lie in parallel planes

Lateral Faces

parallelograms formed by connecting the corresponding vertices of the bases

Lateral Edges

segments connecting the vertices of the lateral faces

Surface Area (of a Polyhedron)

the sum of the areas of its faces

Lateral Area (of a Polyhedron)

the sum of the areas of its lateral faces

Net

two-dimensional representation of the faces

Right Prism

each lateral edge is perpendicular to both bases

Oblique Prism

a prism with lateral edges that are not perpendicular to the bases

Cylinder

solid with congruent circular bases that lie in parallel planes

Right Cylinder

segment joining the centers of the bases is perpendicular to the bases

Pyramid

a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex

Vertex of a Pyramid

the common vertex in a pyramid

Regular Pyramid

has a regular polygon for a base and the segment joining the vertex and the center of the base is perpendicular to the base

Slant Height (of a Regular Pyramid)

the height of a lateral face of the regular pyramid

Cone

has a circular base

Vertex (of a Cone)

not in the same plane as the base

Right Cone

the segment joining the vertex and the center of the base is perpendicular to the base, and the slant height is the distance between the vertex and a point on the base edge

Lateral Surface (of a Cone)

consists of all segments that connect the vertex with points on the base edge

Volume (of a Solid)

the number of cubic units contained in its interior

Sphere

the set of all points in space equidistant from a given point

Center (of a Sphere)

point the set of all points of a Sphere are equidistant from

Radius (of a Sphere)

a segment from the center to a point on the sphere

Chord (of a Sphere)

a segment whose endpoints are on the sphere

Diameter (of a Sphere)

a chord that contains the center

Great Circle

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

Hemisphere

half of a sphere, formed when a great circle separates a sphere into two congruent halves

Similar Solids

two solids of the same type with equal ratios of corresponding linear measures, such as height or radii

Volume of a Cube

the volume of a cube is the cube of the length of its sides, or V=s³

Volume Congruence Postulate

If two polyhedral are congruent, then they have the same volume

Volume Addition Postulate

The volume of a solid is the sum of the volumes of all its non-overlapping parts

Euler's Theorem

F+V=E+2

Surface Area of a Right Prism

The surface are S of a right prism is S=2B+Ph= aP+Ph, where a is the apothem of the base, B is the area of a base, P is the perimeter of a base, and h is the height.

Surface Area of a Right Cylinder

The surface area S of a right cylinder is S=2B+Ch=2πr²+2 πrh, where B is the area of a base, C is the circumference of a base, r is the radius of a base, and h is the height

Surface Area of a Regular Pyramid

The surface area S of a regular pyramid is S=B+1/2Pl, where B is the area of the base, P is the perimeter of the base, and l is the slant height

Surface Area of a Right Cone

The surface area S of a right cone is S=B+1/2Cl = πr² = πrl, where B is the area of the base, r is the radius of the base, and l is the slant height

Volume of a Prism

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

Volume of a Cylinder

The volume V of a cylinder is V=Bh=πr²h, where B is the area of a base, h is the height, and r is the radius of a base

Cavalieri's Principle

If two solids have the same height and the same cross-sectional area at every level, then they have the same volume

Volume of a Pyramid

The volume V of a pyramid is V=⅓Bh, where B is the area of the base and h is the height

Volume of a Cone

The volume V of a cone is V=⅓Bh = ⅓πrh, where B is the area of the base, h is the height, and r is the radius of the base

Surface Area of a Sphere

The surface area S of a sphere with radius r is S=4πr²

Volume of a Sphere

The volume V of a sphere with radius r is V=4/3πr³

Similar Solids Theorem

If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of a²:b², and corresponding volumes have a ration of a³:b³