Classifying Two-Dimensional Figures

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we look at the classification of two-dimensional figures based on their properties. From that, we'll have a better understanding of the relationship between various figures.

The Properties Game

Ever play the game where you can only answer ''yes'' or ''no'' to questions as people try to guess which person in history you are? Playing a variation of this game will help classify two-dimensional figures. For example, ''does this figure have four sides?'' might have an answer of yes. Thus, we could rule out triangles.

In this lesson we will explore how various properties of figures help to classify the figures. These properties include the number of sides, the size of the angles, parallel sides and equal sides. Ready for the game?

Quadrilaterals

We start with a kite. A kite has four sides, four interior angles and the sides are interconnected while enclosing an area (there are no gaps). Figures with these properties are called quadrilaterals . The prefix ''quad'' means four.


Kite
Kite


Another quadrilateral is the rhombus.


Rhombus
Rhombus


How is the rhombus the same as the kite?

  • They both have four sides
  • They both have four interior angles
  • They both have interconnected sides

In short, both the rhombus and the kite are quadrilaterals.

How is the rhombus different from the kite?

In a rhombus, all four sides are equal. A geometrical math term for equal is congruent.

Here's something even more interesting: all rhombi are kites, but only some kites (the ones with four congruent sides) are rhombi.

When all the interior angles congruent of the rhombus are congruent i.e. with four equal angles totaling 360o means each is equal to 90o, the rhombus has a special name. We have just described a square!


Square
Square


So far, we have a hierarchy of two-dimensional figures from quadrilateral to kite to rhombus to square. This math hierarchy is a classification in terms of inclusiveness. For example, all the rhombi and squares are included in the figures called ''kites.''

The square is pretty amazing. For example, the quadrilateral best approximating the circle is the square. The square may be approached from another path. Start with a quadrilateral having at least one pair of parallel sides. This figure is called a trapezoid.


Trapezoid
Trapezoid


There are trapezoids whose non-parallel sides are equal. These are called isosceles trapezoids.


Isosceles Trapezoid
Isosceles_Trapezoid


Another special type of trapezoid has one of the interior angles equal to 90o. This is called a right trapezoid.


Right Trapezoid
Right_Trapezoid


Okay, getting back to the idea of parallel sides. Consider a trapezoid with two pairs of parallel sides. This figure is called a parallelogram.


Parallelogram
Parallelogram


Interestingly enough, if the parallelogram has four equal sides, it is the same as a rhombus. Thus, all rhombi are also parallelograms.

And if the four interior angles in a parallelogram are congruent, we have a rectangle.


Rectangle
Rectangle


Finally, if the rectangle's four sides are congruent, this figure is once again our friend the square.

Thus, the hierarchy has another path: quadrilateral to trapezoid to parallelogram to rectangle to square.

But wait, there are other figures to classify. Figures with more than four sides like pentagons (5 sides), hexagons (6 sides), … are classified as regular if all the sides are congruent and irregular otherwise. We also have three-sided figures called triangles. All of these figures are polygons; meaning many sides.

What about the ellipse and the circle? An ellipse has two radii. A circle is a special ellipse where the two radii are the same. A circle and an ellipse are not polygons unless you theoretically let the number of sides in a regular polygon go to infinity. In this lesson, we stick to a finite (countable) number of sides. Which brings us back to triangles.

Triangles

Our classification game with triangles starts by counting the number of equal sides. With no sides equal, it's a scalene triangle. With two sides equal, the triangle is isosceles. And if all three sides are equal, the triangle is equilateral.

Now if a triangle has all three sides equal, then the angles are also congruent. Meaning, the angles in an equilateral triangles are all equal to 60o. The square is the quadrilateral with congruent sides and congruent angles. The equilateral triangle is a three-sided figure with congruent sides and congruent angles. Thus, the equilateral triangle is the ''square'' of three-sided figures.

What if one of the interior angles of a triangle is 90o? Then, we have a right triangle. And if one of the angles if greater than 90o, the triangle is obtuse.

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