# Understanding Plane Geometry and Solid Geometry

Plane geometry is concerned with zero, one, and two dimensional objects like points, lines, and circles; while solid geometry is concerned with three dimensional objects (width, length, depth) like cubes, spheres, and pyramids.

Points are defined to have zero dimensions—they only have position. They are drawn as a period.  A line has one dimension—only length. A line is drawn by marking two points and drawing a “straight line” through them. In plane geometry a line is infinite in either direction. A plane has two dimensions—length and width. It is drawn as a rectangle with infinite boundaries. A piece of paper can be used as part of a plane, but it must be kept in mind it has infinite boundaries in line with the piece of paper without any depth.

The polygons of plane geometry include triangles, rectangles, squares, pentagons, hexagons, heptagons, octagons, nonagons, decagons, hendecagons, and dodecagons.

A segment is a part of a line. It is the space on a line determined by two specific points that is between the two points. Triangles are three connecting segments. Squares are four connecting segments of equal length. Rectangles are two connecting pairs of segments of equal length.

Pentagons are five connecting segments. They are called regular if all five sides are equal and irregular if all five connecting sides are not equal. Hexagons are six connecting segments. Heptagons are seven connecting segments. Octagons are eight connecting segments. Nonagons are nine connecting segments.

Decagons are ten connecting segments. A hendecagon is eleven connecting segments. A dodecagon is twelve connecting segments. Any polygon can be regular or irregular. It is regular if all sides are equal and irregular if all sides are not equal.

One of the most important concepts in plane geometry is the plane, already defined above. Planes can be determined by two lines, three points, or one point and one line.

They are infinite in size. Triangles are congruent (same size and shape) if they pass the side-side-side, side-angle-side, angle-side-angle, or angle-angle-side tests. The terminology means the sides or angles must be equal in length (sides) or number of degrees (angles).

Solid geometry is concerned with polyhedra, spheres, and other three dimensional solids. Some of the three dimensional shapes include cubes, rectangular solids, prisms, cylinders, spheres, cones, and pyramids. The difference between plane geometry and solid geometry is easy to see using squares and cubes. A cube is six connecting squares of equal length.

The length, width, and depth of a cube are all one of the sides of the six connecting squares, so the volume is S^3 if S is one of the sides of the squares. A sphere is the set of points that are the same distance from one point, called the center. The volume of a sphere is 4/3 pi times the radius cubed. The surface area of a sphere is 4 times pi times the radius squared. Rectangular solids can also be called rectangular prisms or cuboids. Rectangular solids are similar to cubes, but the length, width, and height can be different.

A cube is just a special kind of rectangular solid that has all its sides equal. The volume of a rectangular solid is length times width times height=Lwh if L is the length, w is the width, and h is the height. The surface area can be found using the formula for the area of each of the six sides. The area of both the top and bottom is Lw, front and back area is Lh, and the area of the two sides is wh. Since there are two sides for each area, the surface area is 2Lw+2Lh+2wh.

A prism has bases that are two congruent parallel polygons. The number of sides of a prism is the number of sides of the base that is a polygon. For example, if the base of the prism is a triangle, there are three sides of equal length between the congruent parallel bases. Or if the base is a pentagon, there are five sides of equal length between the congruent parallel bases.

The equal sides are the height of the prisms. The volume of a prism is the area of the base times the height. To find the surface area of a prism add the perimeter of the base times the height to two times the area of the base.

A cylinder is two congruent circles connected by a curved surface. The volume is the area of the circle times the height or pi times the radius squared times the height. The distance between the two circles is the height. The net or “insides” of a cylinder is two circles and a rectangle. The length of the rectangle is two times pi times the radius and the width is the height. The area of the rectangle is therefore two times pi times the radius times the height.

The area of the circles is pi times the radius squared. The surface area is the sum of the area of the two circles plus the area of the one rectangle. The surface area is therefore two times pi times the radius squared added to two times pi times the radius times the height.