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Polyhedron

a solid bounded by polygons called faces

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Face (of a Polyhedron)

polygons that bound a polyhedron

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Edge (of a Polyhedron)

Line Segment formed by the intersection of two faces

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Vertex (of a Polyhedron)

point where three or more edges meet

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Base

used to name a prism or pyramid

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Regular Polyhedron

all faces are congruent regular polygons

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Convex Polyhedron

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

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Platonic Solids

the five regular polyhedral

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Tetrahedron

4 faces (Platonic Solid)

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Cube

6 faces (Platonic Solid)

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Octahedron

8 faces (Platonic Solid)

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Dodecahedron

12 faces (Platonic Solid)

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Icosahedron

20 faces (Platonic Solid)

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Cross Section

intersection of the plan and the solid

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Prism

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

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Lateral Faces

parallelograms formed by connecting the corresponding vertices of the bases

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Lateral Edges

segments connecting the vertices of the lateral faces

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Surface Area (of a Polyhedron)

the sum of the areas of its faces

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Lateral Area (of a Polyhedron)

the sum of the areas of its lateral faces

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Net

two-dimensional representation of the faces

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Right Prism

each lateral edge is perpendicular to both bases

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Oblique Prism

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

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Cylinder

solid with congruent circular bases that lie in parallel planes

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Right Cylinder

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

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Pyramid

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

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Vertex of a Pyramid

the common vertex in a pyramid

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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

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Slant Height (of a Regular Pyramid)

the height of a lateral face of the regular pyramid

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Vertex (of a Cone)

not in the same plane as the base

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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

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Lateral Surface (of a Cone)

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

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Volume (of a Solid)

the number of cubic units contained in its interior

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Sphere

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

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Center (of a Sphere)

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

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Radius (of a Sphere)

a segment from the center to a point on the sphere

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Chord (of a Sphere)

a segment whose endpoints are on the sphere

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Diameter (of a Sphere)

a chord that contains the center

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Great Circle

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

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Hemisphere

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

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Similar Solids

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

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Volume of a Cube

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

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Volume Congruence Postulate

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

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Volume Addition Postulate

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

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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.

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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

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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

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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

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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.

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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

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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

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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

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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

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Surface Area of a Sphere

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

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Volume of a Sphere

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

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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

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