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Geometry

Geometry is the branch of mathematics that that deals with points, lines, planes, and solid bodies and examines their properties, measurements and mutual relationships in spaces.

Shapes

Shapes are said to have dimension. One dimensional shapes are called lines and two dimensional shapes are flat shapes like circles and squares. Three dimensional shapes are shapes with depth to them like cubes and spheres.

Polygons are a type of two dimensional shapes. Polygons are closed figures with straight edges of three or more edges or sides. Examples of polygons are triangles, squares, pentagons, and octagons. Triangles have 3 sides, squares have 4, pentagons have 5 and octagons have 8. A figure or shape is not considered a polygon if it has rounded sides or if its sides intersects at any place other than at the ends of each side. Therefore a circle is not a polygon and an hourglass shape is not a polygon. Instead, it is two polygons, two triangles. Sometimes though a circle is considered to be a polygon with an infinite number of sides.

Polygons Non Polygons

A regular polygon is a polygon where all of the sides and angles are the same. Therefore, a square is a regular polygon with four sides and a rectangle is not. The sum of the angles in a polygon of n sides is 180º x (n-2).

The triangle, a type of polygon has 3 special types. An equilateral triangle is a regular polygon. It has all the same sides and the same angles. An isosceles triangle has two equal sides and two equal angles. A scalene triangle has all different sides and angles.

Equilateral Isosceles Scalene

Another type of triangle which can be either isosceles or scalene is a right triangle. A right triangle is a triangle where one of the angles is 90º. A right triangle cannot be an equilateral triangle because all of the angles in an equilateral triangle are 60º. There are also two other designations of triangles, acute and obtuse. Acute triangles are triangles where all three angles are less than 90º. An equilateral triangle is an acute triangle. Obtuse triangles contain one angle that is larger than 90º.

Right Acute Obtuse

Quadrilaterals are shapes with four sides and internal angles adding up to 360º. There are several special types of quadrilaterals. Quadrilaterals with all of their angles equal to 90º are called rectangles. Rectangles with all of their sides equal are called squares. Parallelograms are a kind of quadrilateral with opposite sides equal in length and parallel. Rectangles are considered parallelograms. A rhombus is a parallelogram with all of its sides equal in length. Squares can be considered rhombuses. Trapezoids contain one pair of parallel sides of unequal length.

Rectangle Square Parallelogram Rhombus Trapezoid

Shapes have certain measurable properties. Aside from angles and side length which have already been discussed, there is also area and perimeter. Area is the grayed portion in each of the figures above. The perimeter is the length of all of the sides added. The perimeter of a circle is also called the circumference. The circumference of a circle is equal to p times the diameter of the circle. The equations for the area for the main types of shapes can be seen below.

Three dimensional shapes have measurable properties as well. Instead of area and perimeter, there is volume and surface area. For shapes with no rounded surfaces, only flat sides, the surface area is the sum of each face of the shape. The surface area of rounded shapes is slightly more complicated. The surface areas of a sphere, a cone, and a cylinder can be found in the following table.