A triangle is a class of geometric shapes. There are many types. Triangles are defined by the angles where the sides intersect and how they relate to other angles in the triangle as well as the length of each of the three sides and how they relate. An equilateral triangle's sides always intersect at the same angle: 60 degrees. An isosceles triangle has two sides of equal length and two intersections with equal angles. A scalene triangle has no equal sides and no equal angles. All triangles have a total of 180 degrees if you add the angles of all three intersections.
A quadrilateral is a geometric class of shapes. All quadrilaterals are flat shapes with four sides with each side connecting at a point. Common quadrilaterals are squares which have sides of equal lengths and right angles, and rectangles that have two differing sets of equal sides intersecting in right angles. Other common quadrilaterals include the rhombus, parallelogram and trapezoid. These shapes vary in how their lines relate to one another and how the sides intersect -- some intersections are acute angles (less than 90 degrees) and others are obtuse (greater than 90 degrees).
A polygon is a broader class of geometric shape. It is any two-dimensional shape comprised of three or more sides where each side connects at a point. Polygons are often defined by their number of sides. An octagon, for example, describes its eight sides. Convex polygons are defined by each subsequent intersection turns in the same direction whereas the intersections of concave polygons can turn inward or outward. Complex polygons can span across the center area to connect its lines, such as a star.
Three-dimensional shapes are based on two-dimensional shapes. A three-dimensional circle is a sphere. A circle connected to a point is a cone. With straight-edged shapes, or polygons, four or more shapes can connect to make a contiguous, three-dimensional shape from a pyramid to a cube to complex geodesic structures.