Perimeter: Polygon

Learn and practice calculating the perimeter of polygons, composite polygons, and solving algebraic perimeter equations.

Solved: 0

Polygon

📖 Perimeter Study Guide

4. Polygons and Sum of Sides

A polygon is any closed two-dimensional shape formed by straight line segments. The perimeter of a polygon is the total length of its boundary. To calculate the perimeter, we sum the lengths of all its sides:
s1s2s3s4s5
💡 Why do we use subscripts/indices (s1,s2,…s_1, s_2, \dots)?
In mathematics, when we have a group of similar items (like the sides of a polygon), we could label them with different letters like a,b,c,d,…a, b, c, d, \dots. But what if a polygon has 50 sides? The alphabet only has 26 letters!

To solve this, we use subscripts (indices). We choose one base letter (like ss for side) and attach a small index number (s1,s2,s3,…s_1, s_2, s_3, \dots):
• The base letter tells us what kind of item it is (all of them are sides).
• The index acts as a counter that identifies each specific side in our set.

This allows mathematicians to write clean, general formulas like P=s1+s2+⋯+snP = s_1 + s_2 + \dots + s_n that work for a polygon of any number of sides (nn) without running out of letters!

General Triangle as a Special Case

A triangle is a polygon with 3 sides. The triangle perimeter formula is just a special case of the general polygon perimeter formula when n=3n = 3:

Polygons with Holes

As we've already learned, if a polygon has holes inside, walking the entire boundary includes the outer boundary PLUS the inner boundaries of all the holes. The sides include all sides of both the outer and inner boundaries. For example, for a polygon with two holes, the perimeter is calculated by summing the outer boundary and the boundaries of both holes:
s1s2s3s4h1h2h3h4h5h6h712
P=Pouter+Phole 1+Phole 2P = P_{\text{outer}} + P_{\text{hole 1}} + P_{\text{hole 2}}
P=(s1+s2+s3+s4)+(h1+h2+h3)+(h4+h5+h6+h7)P = (s_1 + s_2 + s_3 + s_4) + (h_1 + h_2 + h_3) + (h_4 + h_5 + h_6 + h_7)
Learning Topics

Frequently Asked Questions

What is perimeter?

Perimeter is the total boundary of a two-dimensional shape. It includes both the outer boundary and any inner boundaries (like the edges of holes inside the shape).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is the sum of all its four sides: P = 2w + 2h or P = 2(w + h), where w is the width and h is the height.

Perimeter: Polygon | SealMath