Area & Perimeter of Polygons: Every Formula You Need, Derived

Almost every shape on a standardized math test is either a polygon or built out of polygons. A "polygon" is just a closed figure with straight sides: triangles, rectangles, pentagons, hexagons. Two questions get asked about them over and over — how far is it around the edge (perimeter), and how much space is inside (area)? Get fluent with both and you remove an entire source of lost points.

This guide does three things. First, it derives each area formula so you never have to blindly memorize. Second, it shows how the concept appears on each test we prep at Northside Tutoring. Third, it gives you worked examples and practice problems calibrated to real test difficulty.

Why Area & Perimeter Matter

Area and perimeter are the most-tested topics in all of standardized geometry because they are composable. A test writer can dress up a simple rectangle as an "L-shaped garden," a "shaded region," or a "running track," and suddenly a 10-second calculation looks like a brand-new problem. Students who know the formulas cold — and who can break a weird figure into familiar pieces — turn those traps back into 10-second calculations.

Perimeter: The Distance Around

Perimeter is the simplest idea in geometry: add up the length of every side. For a rectangle with length l and width w, that's two lengths and two widths:

For any other polygon, there is no shortcut — you simply sum the sides. The one wrinkle the tests exploit: when a side isn't labeled, you usually have to deduce it from the sides that are, especially on L-shaped composite figures.

Triangles & Rectangles

A rectangle's area is length times width. A triangle is exactly half of the rectangle (or parallelogram) that boxes it in — which is why its area formula carries a factor of one-half:

Shape	Area formula
Rectangle / Square	A = b × h
Triangle	A = ½ × b × h

Two cautions. First, the height of a triangle is the perpendicular distance from the base to the opposite vertex — not the length of a slanted side. Second, any of the three sides can be the "base"; pick whichever pairs with a height you actually know.

Parallelograms & Trapezoids

A parallelogram is a "leaned-over" rectangle. If you slice off the triangle hanging off one end and slide it to the other side, you get a rectangle with the same base and height — so the area is identical:

A trapezoid has two parallel sides of different lengths (b₁ and b₂). Its area is the average of those two bases times the height — which makes intuitive sense, since a trapezoid is "between" two rectangles:

Regular Polygons

A regular polygon (all sides and angles equal) can be split into identical triangles meeting at the center. The height of each triangle is the apothem (a) — the distance from the center to the middle of a side. Add up all those triangles and the formula collapses to something elegant:

where P is the perimeter. You'll most often see this with hexagons, where the regular hexagon also conveniently breaks into six equilateral triangles — tying directly back to special right triangles.

Composite Figures — The Real Test Skill

The hardest area questions aren't hard formulas — they're familiar formulas hidden inside an unfamiliar shape. The strategy is always the same: decompose the figure into rectangles and triangles, find each piece, then add (or subtract a hole). For a shaded region, find the big shape's area and subtract the unshaded part.

Where You'll See This — Test by Test

The College Board and ACT both include the basic area formulas on their reference sheets, but knowing which formula to reach for — and how to handle composite figures — is still on you. The SSAT provides no reference sheet at all.

Watch the Lesson

Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.

Worked Example Problems

These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.

Common Mistakes to Avoid

Practice Problems — You Try

Three problems below. Work each before checking the solution.

The Northside Method — How We Teach This 1-on-1

Reading a blog is a great starting point. But there's a meaningful gap between understanding a concept and reflexively applying it under timed conditions. That gap is exactly what our tutors close.

Every Northside student works through a four-step framework:

1. Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
2. Perfect-match coach. We pair them with an elite tutor (we accept only the top 1% of applicants) whose teaching style fits how your student actually learns.
3. Bespoke plan. A roadmap built around your student's target score, target timeline, and current pacing data.
4. Data-driven adjustment. Every session ends with a check on whether the student's accuracy and speed are moving in the right direction.

And if a student meets all eligibility requirements but doesn't hit the defined score improvement? We provide 5 additional hours of cohort learning at no cost. That's the Northside guarantee — built on 25 years of measured outcomes.