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# Geometry SURFACE AREA AND VOLUME

## A Cavalieri’s Principle Theorem

A polyhedron is a three-dimensional geometric object characterized by flat polygonal faces, straight edges, and vertices where these edges intersect. The term "polyhedron" is derived from the Greek words "poly," meaning many, and "hedra," meaning faces, emphasizing its defining feature of having multiple flat surfaces. The faces themselves are polygons, and the edges are straight line segments connecting the vertices. The vertices are the sharp corners where three or more edges meet. Buckle up!  We’re moving into 3-d volume calculations.  Before moving on to this topic you should have a very solid understanding of the different types of polygons, and how to determine their 2-D areas.

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I NEED HELP WITH FINDING 3-D VOLUMES

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In geometry, a "net" refers to a two-dimensional, flat representation of a three-dimensional object, particularly a polyhedron. It is essentially a flat plane that, when folded along its edges and assembled, creates the three-dimensional shape of the corresponding polyhedron. The net displays the individual faces of the polyhedron and the connecting edges, providing a visual guide for constructing the solid figure. The concept of nets is especially helpful in aiding individuals to visualize how flat shapes combine to form complex, three-dimensional objects.

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A geometric prism is a three-dimensional solid with two identical, parallel polygonal bases and rectangular sides.  These sides are called lateral faces and they connect corresponding vertices of these bases. The bases can be any polygon, such as a square, triangle, pentagon, etc.,.  The connecting faces are perpendicular to the bases. Prisms are classified based on the shape of their bases.  For example, hexagonal prisms have hexagonal bases. The surface area of a prism is twice the area of the base plus the perimeter of the base times the height of the prism. The height of the prism is the perpendicular distance between the bases.  The volume of a geometric prism is calculated by multiplying the area of one of its bases by its height.

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A cylinder is similar to a prism in that it has two bases.  However, the two parallel identical bases are circular.  Instead of separate lateral faces, a cylinder  has one curved lateral surface that essentially unrolls into a rectangle. The length of the face is the circumference of the circle it surrounds, and its height in the linear distance between bases. To determine the surface area of a cylinder, you must consider the contributions from its two circular bases and the curved lateral surface.  It is calculating by adding twice the area of the bases to the total area of the lateral side.  Its volume is the surface area of one base times the height of the cylinder.

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A geometric pyramid has only one polygonal base and triangular faces that converge at a common point called the apex. The base can be any polygon, and the triangular faces extend from each vertex of the base to the apex. The height of the pyramid is the perpendicular distance from the base to the apex. Geometric pyramids are classified based on the shape of their base, with common types including square pyramids, rectangular pyramids, and triangular pyramids. The surface area is determined by adding the area of the base to the sum of the areas of the triangular faces. In other polyhedral, the volume was determined by multiplying the area of the base times the height.  However, the volume of a pyramid cannot be calculated in this way because its shape is not uniform along its height.  The volume of a pyramid is 1/3 the volume of its corresponding prism.  For example, the volume of a triangular pyramid is 1/3 the volume of a triangular prism.

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A geometric cone is analogous to a pyramid in that it has only one base.  However, the base is circular and instead of triangular faces, there is single curved surface connected to the apex. The height of the cone is the perpendicular distance from the apex to the plane of the circular base, and the length of edge of the curved surface is equal to the circumference of the circle it surrounds.  Last, the slant height of a cone is the distance from the outer edge of the prism to the apex. A cone’s total surface area is a sum of the circular base area and the lateral surface area. The volume of a cone is 1/3 the volume of a cylinder.

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A sphere is a three-dimensional geometric shape that is perfectly round and symmetrical. It has no sides and is defined as the set of all points in space in all directions that are equidistant from a common center. The distance from the center to any point on the surface of the sphere is called the radius.  The surface area formula is the product of 4 times pi times the radius squared.  The volume of a sphere is 4/3 pi times the radius cubed.

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Cavalieri's Principle is a geometric concept that deals with the comparison of volumes in two-dimensional and three-dimensional spaces. It states that if two objects (usually solids) have the same cross-sectional area at every level parallel to a chosen base, then they have the same volume.

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