Volume and Surface Area of Solids

• The word Geometry comes from two Greek words: ‘geo’, which means earth and ‘metria’, which means measurement.
• The Greek mathematician Euclid (c. 325-300 BC) made a significant contribution to the study of geometry.
• the set of all points is called space.
• A two-dimensional shape possesses linear measures like length and width but has no depth in it.
• a three-dimensional shape possesses length, width and depth.
• The region bounded by a three-dimensional shape is spatial and the quantity associated to this space is called volume.
• the quantity associated to the region occupied by the surface of a three-dimensional shape is called surface area.
• The surface area of a three-dimensional figure is the total area of the planar regions which comprise the surface of the figure. These planar regions can have polygonal shapes.
• A polygonal region is the union of a finite number of triangular regions, in a plane, such that if two of these intersect, their intersection is either a point or a segment.
• The Area Postulate

  To every polygonal region there corresponds a unique positive real number.
• The area of a polygonal region is the number assigned to it by the Area Postulate. The area of a region R is denoted by aR read as “area of R”.
• the area of a region (polygonal region) does not depend on the place where the region is located in space. Rather, it is solely dependent on the size and shape of the region
• The Congruence Postulate

  If two polygons are congruent, then the regions determined by them have the same area
• The Area Addition Postulate

  If two polygonal regions intersect, and that they intersect only in edges and vertices, then the area of their union is the sum of their areas.
• The Unit Postulate for Area

  The area of a square region is the square of the length of its edge.
• A polyhedron is a three-dimensional geometric figure that is made up of a number of polygonal surfaces which are joined along their sides and that encloses a space.
• The Unit Postulate for Volume

  The volume of a rectangular parallelepiped is the product of the altitude and the area of the base.
• The Volume Postulate

  For every polyhedron, there is a real number that gives the number of unit cubes, and parts of unit cubes, which fill the region enclosed by the polyhedron.
• The polygons that made up the polyhedron are called the faces
• the sides of these polygons are called the edges
• the points where the edges meet or intersect are called the vertices
• A polyhedron is a prism if and only if it has two congruent and parallel faces, and its other faces are parallelograms.
• Right prism - the lateral edges are perpendicular to a base
• Oblique prism - the lateral edges are at an angle with a base other than 90◦
• for oblique prisms, the shape of the lateral faces can be any parallelogram. However, for right prisms, they are rectangles
• A right prism is said to be regular if and only if its bases are regular polygons.
• Volume of a Prism

  Bh
• Lateral Area L of a Prism

  ph
• Surface Area of a Prism

  L + 2B
• The volume of a cube with edge s is the cube of s, or V = s3 .
• A polyhedron is a pyramid if and only if all the faces except one have a common vertex, called the vertex of the pyramid.
• A pyramid is regular if and only if its base is a regular polygon and its lateral edges are equal in length.
• An oblique pyramid has non-congruent lateral edges.
• The slant height of a regular pyramid is the perpendicular distance from the vertex to any base edge.
• the lateral faces of a regular pyramid are either isosceles or equilateral triangles
• the segment that gives the slant height of a regular pyramid bisects the base edge it intersects with.
• Volume of a Pyramid

  $1/3 Bh$
• Lateral Area of a Prism

  1/2pl
• Surface Area of a Pyramid

  L + B
• Area of a regular hexagon of side s

  $(3 \sqrt3/2 ) s^2$ or 1/2 Pa