Two-Dimensional Figures
A polygon is a closed figure made of straight segments — no gaps, no curves. Polygons get their names from how many sides they have: three sides make a triangle, four a quadrilateral, five a pentagon, six a hexagon, and eight an octagon. Once you can name a figure, geometry asks two more things of you: classify its shape and measure it.
The two measurements that matter most are perimeter and area. Perimeter is the distance around the figure — add up the sides. Area is the space inside — for a rectangle, multiply length by width. Students mix these up constantly, so this lesson pins down which is which and when to use each formula.
Naming and classifying polygons
Count the sides — that gives the name. A polygon is regular when all of its sides are congruent and all of its angles are congruent. A square is a regular quadrilateral; a stop sign is a regular octagon. If even one side or angle is different, the polygon is not regular.
Polygons are also classified as convex or concave. In a convex polygon, every segment connecting two vertices stays inside the figure. In a concave polygon, at least one vertex is pushed inward — a dent — so a segment between two vertices can pass outside the figure. A quick check: if any vertex points into the figure, it's concave. The concave figure below has that tell-tale inward dent.
Perimeter: the distance around
Perimeter adds up every side length. For a rectangle with length and width , that shortcut is , because the sides come in two matching pairs.
For a regular polygon the shortcut is even simpler: all sides match, so , where is the length of one side. A regular hexagon with side has perimeter . This also runs in reverse — if you know the perimeter of a regular polygon, divide by the number of sides to get one side.
Perimeter is measured in plain length units: units, feet, centimeters.
Area: the space inside
The area of a rectangle is — length times width. Area counts how many unit squares fit inside the figure, so it is measured in square units: square feet, square centimeters.
Here is the trap to avoid: adding all four sides of a rectangle gives the perimeter, not the area. If a problem asks for area, you must multiply. Read the question twice — "fencing around" means perimeter, "carpet to cover" means area.
Worked examples
Example 1: perimeter and area of a rectangle
A rectangle has length and width . Find its perimeter and its area.
Answer: units and square units
Example 2: perimeter of a regular polygon
A regular pentagon has sides of length cm. Find its perimeter.
Answer: cm
Example 3: working backward from the perimeter
A regular octagon has a perimeter of inches. Find the length of one side.
Answer: inches
Try one yourself
Common questions
What is the difference between perimeter and area?
Perimeter is the distance around the figure — add the sides, answer in length units. Area is the space inside — for a rectangle, multiply length by width, answer in square units. If the answer should be in square units, you needed area.
How can I tell if a polygon is concave?
Look for a dent: a vertex that points into the figure instead of out. Formally, a polygon is concave if some segment between two of its vertices passes outside the figure. If no such segment exists, the polygon is convex.
Does regular just mean the sides are equal?
It means both: all sides congruent and all angles congruent. A rhombus has four equal sides but unequal angles, so it is not regular. A square, with equal sides and four right angles, is.
Is a circle a polygon?
No. A polygon must be made entirely of straight segments, and a circle has no straight parts. Circles get their own formulas for distance around (circumference) and area.
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