What’s the perimeter and area of that polygon?
You’ve probably seen the diagram on a worksheet, a homework sheet, or a quick sketch on a whiteboard. It’s a shape that looks a little like a jigsaw piece, with straight sides that don’t all line up the same way. You’re standing there, calculator in hand, wondering: “How do I actually get the perimeter and area?”
Let’s cut through the confusion. In the next few sections I’ll walk you through the exact steps, show you the tricks that make the math feel less like a chore, and point out the slip‑ups that trip most people up. By the end, you’ll be able to tackle that polygon (and any other shape that comes your way) with confidence And that's really what it comes down to..
What Is the Polygon?
A polygon is just a closed figure made of straight sides. Think of a triangle, a square, or a hexagon—any shape that starts and ends at the same point without curves. The one in question is an irregular hexagon: six sides, but the lengths and angles aren’t all the same.
Because it’s irregular, you can’t just use a one‑size‑fits‑all formula like Area = side² or Perimeter = 6 × side. You’ll need to break it into pieces you can measure or calculate Easy to understand, harder to ignore..
Why It Matters / Why People Care
When teachers hand out that diagram, they usually want you to practice two core skills:
- Adding up side lengths – that’s the perimeter.
- Finding the space inside – that’s the area.
Both of these are building blocks for geometry, algebra, and real‑world problems (think tiling a floor or cutting out a shape from cardboard). Which means if you skip the details, you’ll miss out on how each side contributes to the whole. And if you mis‑measure one piece, the whole answer can be off.
How It Works
1. Label the vertices
Give each corner a letter (A, B, C, D, E, F). It sounds trivial, but labeling keeps you from mixing up side lengths later.
A ───── B
│ │
F ───── C
│ │
E ───── D
Your diagram might look a little different, but the idea stays the same Surprisingly effective..
2. Measure each side
Use a ruler or the given measurements. Write them down next to the corresponding side:
- AB = 4 cm
- BC = 5 cm
- CD = 3 cm
- DE = 6 cm
- EF = 2 cm
- FA = 4 cm
(If the sides are given in feet, inches, or any other unit, keep the units consistent.)
3. Add them up for the perimeter
Just sum the six numbers:
Perimeter = 4 + 5 + 3 + 6 + 2 + 4 = 24 cm
That’s it. No angles needed for the perimeter Most people skip this — try not to. But it adds up..
4. Split the polygon into triangles
Area is trickier. So the easiest way is to cut the shape into triangles whose areas you can calculate. Since it’s a hexagon, you can draw diagonals from one vertex to the others that aren’t adjacent Which is the point..
To give you an idea, draw diagonals from A to D and A to E. You’ll end up with three triangles:
- Triangle ADE
- Triangle AED (actually the same as ADE, but we’ll treat each segment separately)
- Triangle ABC
(Your actual split might differ based on the shape.)
5. Find each triangle’s area
Use the ½ × base × height formula or the Heron formula if you only know side lengths.
Triangle ADE
If you know the base (AD) and the height from A to the line DE, compute:
Area = ½ × AD × height
If you only have side lengths (AD = 7 cm, DE = 6 cm, AE = 5 cm), use Heron:
- s = (7+6+5)/2 = 9
- Area = √[s(s-7)(s-6)(s-5)] = √[9×2×3×4] ≈ 14.7 cm²
Triangle ABC
Do the same. Suppose AB = 4 cm, BC = 5 cm, AC = 6 cm:
- s = (4+5+6)/2 = 7.5
- Area = √[7.5×3.5×2.5×1.5] ≈ 11.8 cm²
6. Sum the triangle areas
Add up the areas of all the triangles you split the polygon into:
Total Area = 14.7 + 11.8 + … = 39.5 cm² (rounded)
That’s the area of the whole hexagon.
Common Mistakes / What Most People Get Wrong
- Mixing up units – If one side is in inches and another in centimeters, the perimeter will be nonsense.
- Counting a side twice – In an irregular shape, it’s easy to forget a diagonal or double‑count a side when drawing triangles.
- Using the wrong triangle formula – Forgetting that Heron requires all three sides, or mis‑applying the base‑height method when the height isn’t obvious.
- Overlooking a diagonal – Skipping a diagonal can leave a “gap” in your area calculation.
- Rounding too early – If you round intermediate values, the final perimeter or area can drift. Keep decimals until the end.
Practical Tips / What Actually Works
- Draw a clean diagram: Even a rough sketch helps you see which diagonals to draw.
- Label everything: Numbers next to each side and angle make the math less mental gymnastics.
- Check your arithmetic: A single mis‑added side can throw off the perimeter by a full unit.
- Use a calculator for Heron: Square roots can be messy; a quick calculator saves time.
- Practice with a ruler: If you’re measuring from a printed sheet, a good ruler or a digital measuring tool keeps the data accurate.
FAQ
Q1: Can I use the shoelace formula instead of splitting into triangles?
A1: Yes, if you have the coordinates of each vertex, the shoelace formula gives the area directly. It’s handy for irregular polygons but requires coordinate data And that's really what it comes down to..
Q2: What if one side length is missing?
A2: You can often solve for it using the law of cosines or by setting up an equation based on the known area or perimeter. It turns into a bit of algebra.
Q3: Does the shape need to be convex?
A3: No, the method works for concave polygons too, but you’ll need to split the shape into triangles that don’t overlap.
Q4: How do I find the height of a triangle when I only have side lengths?
A4: Use Heron’s formula to find the area first, then solve for the height: height = (2 × area) / base Easy to understand, harder to ignore..
Q5: Is there a shortcut for regular polygons?
A5: Absolutely. For a regular n‑gon with side s, perimeter = n × s and area = (n × s²) / (4 × tan(π/n)). But that only works when all sides and angles are equal.
Finding the perimeter and area of a polygon is a step-by-step dance between measurement, geometry, and a bit of algebra. Label, measure, split, calculate, and sum. Keep your units straight, double‑check your triangle formulas, and don’t rush the arithmetic. Practically speaking, once you’ve mastered this routine, any polygon—regular or not—will be just another shape to conquer. Happy calculating!
