Angles, Triangles and Polygons

Geometry is the study of shapes — the lines, angles, and figures that fit together to build everything from a triangle on a roof to the hexagons on a honeycomb. In this lesson you will sort angles by how wide they open, name triangles by their sides and angles, and recognize polygons by counting how many straight sides they have.

Angles — the Turn Between Two Rays

• An angle is formed by two rays meeting at a single point called the vertex.
• The size of an angle is measured in degrees, written with a small circle: 90 degrees is shown as 90 deg.
• Acute angle: less than 90 degrees — sharper than a square corner.
• Right angle: exactly 90 degrees — looks like the corner of a square or a book.
• Obtuse angle: between 90 and 180 degrees — wider than a square corner but not yet a flat line.
• Straight angle: exactly 180 degrees — the two rays form a straight line.
• Reflex angle: more than 180 degrees — the angle on the OUTSIDE of a turn that has gone past straight.

Triangles — Three Sides, Three Angles

• A triangle is a polygon with three sides and three angles. The three interior angles always add up to 180 degrees.
• Equilateral triangle: all three sides equal AND all three angles equal (60 degrees each).
• Isosceles triangle: exactly two sides equal — the angles opposite those sides are also equal.
• Scalene triangle: all three sides different lengths and all three angles different sizes.
• Right triangle: one of the angles is exactly 90 degrees.
• Obtuse triangle: one of the angles is greater than 90 degrees.
• Acute triangle: ALL three angles are less than 90 degrees.

Polygons — Closed Shapes with Straight Sides

• A polygon is a flat closed figure made of straight line segments — no curves and no gaps.
• Triangle: 3 sides, 3 vertices.
• Quadrilateral: 4 sides, 4 vertices (square, rectangle, rhombus, trapezoid, parallelogram).
• Pentagon: 5 sides, 5 vertices.
• Hexagon: 6 sides, 6 vertices (think of a honeycomb cell).
• Heptagon: 7 sides. Octagon: 8 sides (think of a stop sign).

Regular vs Irregular Polygons

A regular polygon has all sides the same length AND all angles the same size. A square is a regular quadrilateral; an equilateral triangle is a regular triangle. An irregular polygon has at least one side or one angle different from the others — for example, a rectangle is an irregular quadrilateral because its sides are not all equal.