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Perimeter of
Polygons

Master the art of calculating distance around shapes: from simple rectangles to challenging composite figures on the SSAT.

SSAT Middle Level

What Is Perimeter?

The perimeter of a polygon is the total distance around the outside of the shape. Imagine walking along each side of a figure and adding up every step: that total distance is the perimeter.

side 1 side 2 side 3 side 4 side 5 Add all sides!

Common Polygons You'll See

Triangle
3 sides
Rectangle
4 sides, opposite sides equal
Pentagon
5 sides
Hexagon
6 sides

Why Perimeter Matters

Perimeter shows up everywhere in real life and on the SSAT. You need perimeter whenever you calculate how much fencing goes around a yard, how much ribbon wraps a frame, or how far you run around a track.

Fencing a Garden
Total fence length = perimeter of the garden plot
Framing a Picture
Length of frame material = perimeter of the picture
Running a Track
One lap distance = perimeter of the track shape

The General Perimeter Formula

For a regular polygon (all sides equal to \( s \)):

Shortcuts: Rectangle & Square

Rectangle l w Square s s
Rectangle
\( P = 2l + 2w \)
Square
\( P = 4s \)

Perimeter vs. Area

Perimeter Area
Measures Distance around Space inside
Units cm, m, in, ft cm², m², in², ft²
Rectangle \( 2l + 2w \) \( l \times w \)
Think of it as Fence around a yard Grass covering the yard

Step-by-Step Approach

1
Identify the shape. Is it a rectangle, triangle, regular polygon, or a composite figure?
2
Label every side. Write down each length. For missing sides, use the shape's properties to find them.
3
Choose the right formula. Use \( P = 2l + 2w \) for rectangles, \( P = 4s \) for squares, or just add all sides.
4
Compute and include units. Add up the values and write your answer with the correct linear unit.

Example 1: Rectangle

Find the perimeter of a rectangle with length \( 12 \) cm and width \( 5 \) cm.

12 cm 5 cm 12 cm 5 cm

Example 2: Triangle

Find the perimeter of a triangle with sides \( 7 \) in, \( 10 \) in, and \( 8 \) in.

10 in 7 in 8 in A B C

Example 3: Regular Hexagon

A regular hexagon has each side measuring \( 6 \) m. Find its perimeter.

6 m 6 m 6 m 6 m 6 m 6 m

Composite Shapes

A composite shape is formed by combining two or more simple shapes. To find the perimeter, trace only the outer edges: do not count any side that is inside the figure.

300 ft 100 ft 140 ft 100 ft 160 ft 200 ft inside edges (don't count!)

Finding Missing Sides

On the SSAT, composite shape problems often leave some side lengths unlabeled. Use the fact that opposite sides of the overall rectangle must match to solve for the unknowns.

15 5 10 ? ? ?
1
Bottom side: total width = 15, right portion = 15 − (15 − unknown). Compare horizontal spans: bottom = \( 15 - (15 - ?) \). Use the notch width: top is 15, notch goes in at some point.
2
Vertical unknown: total height = 10, known vertical = 5, so the drop = \( 10 - 5 = 5 \).
3
Horizontal notch: total width = 15, bottom piece starts where the notch ends. Read the diagram carefully to find each segment.

Example 4: L-Shape

14 cm 5 cm 7 cm 6.5 cm 7 cm 11.5 cm

Common Mistakes

✕ Confusing Perimeter with Area
Perimeter is the distance around, not the space inside. Don't multiply length by width when asked for perimeter.
✕ Forgetting a Side
Count every outer edge. In composite shapes, trace your finger around the outside to make sure you haven't skipped a segment.
✕ Counting Interior Edges
When two shapes join, the shared edge is inside the figure. Never include it in the perimeter.

Perimeter Formulas Summary

Shape Formula Example
Any Polygon \( s_1 + s_2 + \cdots + s_n \) \( 3 + 5 + 7 = 15 \)
Rectangle \( 2l + 2w \) \( 2(8) + 2(3) = 22 \)
Square \( 4s \) \( 4(6) = 24 \)
Regular Polygon \( n \times s \) \( 5 \times 4 = 20 \)
Composite Add outer edges only Trace the outline

Question 1 of 5

A rectangle has a length of \( 9 \) cm and a width of \( 4 \) cm. What is its perimeter?

Question 2 of 5

A regular octagon has each side measuring \( 3.5 \) in. What is its perimeter?

Question 3 of 5

A triangle has sides of \( 6 \) ft, \( 8 \) ft, and \( 10 \) ft. What is its perimeter?

Question 4 of 5

An L-shaped figure is formed by removing a \( 3 \) m by \( 2 \) m rectangle from the corner of a \( 10 \) m by \( 6 \) m rectangle. What is the perimeter of the L-shape?

Question 5 of 5

A square has a perimeter of \( 48 \) cm. What is the length of one side?