Classifying Two-Dimensional Shapes by Properties

In geometry, we can group or classify two-dimensional (2D) shapes based on their properties, such as the lengths of their sides and the measures of their angles. Understanding these properties helps us see how different shapes are related in a hierarchy.

Classifying Triangles

Triangles are 3-sided polygons. We can classify them in two ways: by their side lengths and by their angle measures.

By Side Lengths:

• Equilateral Triangle: All $3$ sides are equal in length.
• Isosceles Triangle: At least $2$ sides are equal in length.
• Scalene Triangle: No sides are equal in length.

By Angle Measures:

• Acute Triangle: All $3$ angles are less than $90^\circ$.
• Right Triangle: Exactly $1$ angle is exactly $90^\circ$ (a right angle).
• Obtuse Triangle: Exactly $1$ angle is greater than $90^\circ$.

Shapes can have multiple names! For example, a triangle can be both right and isosceles if it has one $90^\circ$ angle and two sides of the exact same length.

Classifying Quadrilaterals

Quadrilaterals are 4-sided polygons. They have a special hierarchy, meaning some shapes automatically fit into the categories of other shapes based on their rules.

• Trapezoid: A quadrilateral with at least one pair of parallel sides.
• Parallelogram: A quadrilateral with two pairs of parallel sides. Opposite sides are equal in length.
• Rectangle: A parallelogram with $4$ right angles ($90^\circ$).
• Rhombus: A parallelogram with $4$ equal sides.
• Square: A parallelogram with $4$ right angles AND $4$ equal sides.

Understanding Subcategories

Because shapes are grouped by their rules, broad categories can contain specific subcategories. If a shape has all the properties of another shape category, it belongs to that category too!

• Is a square always a rectangle? Yes! The rule for a rectangle is having $4$ right angles. A square always has $4$ right angles, so every square is also a rectangle.
• Is a square always a rhombus? Yes! The rule for a rhombus is having $4$ equal sides. A square has $4$ equal sides, so every square is also a rhombus.
• Is a rectangle always a square? No. A rectangle has $4$ right angles, but its sides do not have to be equal.

When you classify a shape with $4$ right angles and $4$ equal sides, its most specific name is a square, but because of its properties, it is also a quadrilateral, a parallelogram, a rectangle, and a rhombus!