Calculate perimeter of polygons and composite shapes. |...

SECTION 1

Historical Context & Motivation

Long ago, people needed to measure land for farming. Ancient Egyptians used ropes to mark perimeter – the distance around a shape. This helped them build pyramids and plan fields after floods. Today, you use it for fences or picture frames.

3000 BC

Egyptian Land Measurement

Farmers measured fields with ropes after Nile floods.

300 BC

Euclid's Geometry

Greek mathematician Euclid wrote rules for shapes and perimeters.

1800s

Modern Surveying

Tools improved to measure perimeters accurately for maps.

Today

SSAT Prep

You calculate perimeters quickly without tools!

Perimeter solves real problems like fencing a playground. You can master it step by step.

SECTION 2

Core Principles & Definitions

A polygon is a shape with straight sides, like a triangle or square. Perimeter is the total length of all outside sides added up. It does not include the inside. Composite shapes combine polygons, so add outer sides only.

Polygon Sides

Straight lines only. No curves.

Perimeter Formula

Add all side lengths: P = a + b + c + …

Composite Rule

Trace outer path. Skip hidden sides.

Units Match

If sides are in feet, perimeter is feet.

✦ Think Like a Fence Builder

Perimeter is the fence around your yard. You add every board on the outside. Ignore gates or inside walls – just the boundary!

SECTION 3

Visual Explanation of Polygon Perimeter

Labeled sides of a rectangle (P=5+5+12+12=34) and triangle (P=9+9+8=26). Arrows show path.

See the orange arrows trace the outside. Add those lengths only. This visual helps you picture any polygon.

SECTION 4

Mathematical Framework

POLYGON PERIMETER

P = a + b + c + …

a, b, c are side lengths. Units stay the same.

REGULAR POLYGON

P = n × s

n = number of equal sides s.

For any polygon, just add the sides given. Regular ones like squares make it faster. Practice with simple numbers first. You got this!

SECTION 5

Detailed Breakdown of Composite Shapes

House-like composite: rectangle + triangle. Pink path shows outer perimeter. Dashed lines are internal – ignore them.

Trace the pink line around the outside. Add only those sides. Internal lines like doors cancel out.

SECTION 6

Worked Example: L-Shaped Composite

Find perimeter of an L-shape with sides 3, 4, 2, 5, 1, 6 units.

Step-by-Step Solution

Step 1 – Sketch and LabelDraw the L. Label outer sides: bottom 6, right 5, top short 1, indent 2, left 4, bottom short 3.

Outer sides: 6, 5, 1, 2, 4, 3

Step 2 – Add Them Up6 + 5 = 11; 11 + 1 = 12; 12 + 2 = 14; 14 + 4 = 18; 18 + 3 = 21.

Step 3 – Check UnitsAll in units, so P = 21 units.

💡 Tip

Always trace the outline like a bug walking around.

SECTION 7

Strengths and Common Mistakes

Do	Don't
Add only outer sides.	Include internal lines.
Trace the boundary first.	Count corners twice.
Check all labels match.	Forget units.

✦ KEY TAKEAWAY

Perimeter is easy like adding game scores – just the edges count. Practice avoids traps!

SECTION 8

Connection to Circles

Polygons	Circles
Straight sides, add lengths.	Curved, C = 2πr or πd.
Many sides ≈ circle.	No straight sides.

Polygons build to circles in harder math. Master sides now for future success!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

What does perimeter measure? A) Area inside B) Total side lengths C) Number of sides D) Diagonal length E) Half the sides

PROBLEM 2 — BASIC CALCULATION

Rectangle sides 4 and 7. Perimeter? A) 11 B) 22 C) 28 D) 14 E) 18

PROBLEM 3 — INTERMEDIATE

Triangle sides 5, 6, 7. Perimeter? A) 18 B) 210 C) 9 D) 36 E) 16

PROBLEM 4 — APPLIED

Pentagon sides 3,3,3,3,5. Perimeter? A) 17 B) 15 C) 20 D) 9 E) 14

PROBLEM 5 — CRITICAL THINKING

L-shape: horizontal 10 (split 4+6), vertical 8 and 3 internal. Outer path? A) 25 B) 29 C) 21 D) 34 E) 16

SUMMARY

Quick Review

Master perimeter by adding outer sides of polygons. For composites, trace the boundary like a fence.

Practice visuals and steps. You can ace SSAT questions now! Keep adding confidently.