Perimeter: Polygon

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

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Polygon

Use the workspace below. Write equations like P = 24 to solve for the perimeter.

📖 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.

Why is a square's perimeter formula P = 4s?

A square is a special case of a rectangle where width and height are equal (w = h = s). Substituting this into the rectangle formula gives P = 2(s + s) = 4s.

How does a hole affect the perimeter?

Since perimeter measures the entire boundary of a shape, a hole adds to the perimeter. The total perimeter is the outer perimeter plus the inner perimeter (the perimeter of the hole).

How do you find the perimeter of a right-angled triangle if one side is missing?

Since we've already learned the Pythagorean theorem (a² + b² = c²), we can calculate the missing side length first and then sum all three sides together to get the perimeter.

What unit is used for perimeter?

Since perimeter is a one-dimensional length (boundary), it is measured in linear units like meters (m), centimeters (cm), feet (ft), or inches (in). It is never measured in square units, which are reserved for area.

How does doubling a shape's dimensions affect its perimeter and area?

Doubling all dimensions (scale factor of 2) doubles the perimeter (ratio 2:1) because perimeter is linear (1D). However, it quadruples the area (ratio 4:1) because area is two-dimensional (2D) and scales quadratically (2² = 4).

Perimeter: Polygon | SealMath