Perimeter of Polygons — Calculate perimeter of polygons and composite shapes. - SHSAT Math

SECTION 1

Why People Needed to Measure Boundaries

Thousands of years ago, people needed to know the distance around their land. Farmers in ancient Egypt wanted to build fences around their fields. Builders in ancient Greece needed to know how much stone to cut for the edges of a temple floor. The idea of measuring perimeter (the total length around a shape) was born from these real-world needs.

~3000 BCE • Ancient Egypt

Egyptian farmers measured the boundaries of their fields after the Nile River flooded each year. They used ropes and knots to find the distance around each plot.

~500 BCE • Ancient Greece

Greek mathematicians like Euclid wrote rules for measuring shapes. The word perimeter comes from Greek: peri (around) + metron (measure).

~200 BCE • Archimedes & Complex Shapes

Archimedes figured out how to estimate the perimeter of circles by using polygons with many sides. This was an early example of working with composite (combined) shapes.

Modern Day • SHSAT & Everyday Life

Today, perimeter shows up in building design, fencing, picture framing, and on standardized tests like the SHSAT, where you must quickly find distances around polygons and composite shapes.

So here is the big question this lesson answers: How do you find the total distance around any polygon or composite shape, quickly and accurately? Let's find out.

SECTION 2

Core Principles & Definitions

Before you solve any perimeter problem, you need to understand a few key ideas. A polygon is a flat (two-dimensional) shape made of straight sides that close completely. Triangles, rectangles, pentagons, and hexagons are all polygons. A circle is not a polygon because its edge is curved.

Perimeter

The total distance around the outside of a shape. Add up every side length to find it.

Regular Polygon

A polygon where all sides are equal in length and all angles are equal. Example: a square or an equilateral triangle.

Irregular Polygon

A polygon with sides that are different lengths. You must add each side one by one.

Composite Shape

A figure made by joining two or more simpler shapes together. Some sides become interior (hidden) and are NOT part of the perimeter.

Units Matter

Perimeter is a length, so the answer is always in linear units like cm, m, in, or ft — never in square units (that's area).

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KEY TAKEAWAY

Think of perimeter like walking around the edge of a soccer field. You start at one corner, walk along every sideline and goal line, and come back to where you started. The total distance you walked is the perimeter. For composite shapes, just remember: only count the outside edges — skip any edges hidden on the inside.

SECTION 3

Visual Explanation — Seeing Perimeter

This diagram shows five shapes. Notice that the square and hexagon are regular polygons (all sides equal), while the triangle and rectangle may or may not be regular. The green L-shape is a composite shape — you only count the outer edges.

Look at the diagram above. For simple polygons (the square, rectangle, triangle, and hexagon), you just add up every side. For the L-shaped composite figure in green, you trace your finger along the outside edge and add only those lengths. Any edge that sits inside where the two rectangles meet does not count toward the perimeter.

SECTION 4

The Math Behind Perimeter

The core formula for perimeter is beautifully simple. But there are shortcut versions for common shapes that save you time on the SHSAT. Let's go through each one.

GENERAL POLYGON

P = s₁ + s₂ + s₃ + … + sₙ

P = perimeter. s₁, s₂, … sₙ are the lengths of each side. n = number of sides. Just add them all up!

REGULAR POLYGON SHORTCUT

P = n × s

n = number of sides, s = length of one side. Because all sides are equal, multiply instead of adding each one.

RECTANGLE

P = 2 × (l + w)

l = length, w = width. A rectangle has two pairs of equal sides, so this formula adds length + width and then doubles.

SQUARE

P = 4 × s

s = side length. A square is a regular polygon with 4 equal sides.

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SHSAT Tip

On the SHSAT, composite shape problems often give you some side lengths but not all. Use the given info to figure out the missing sides. For example, if an L-shape is made from rectangles, the missing side is usually the difference or sum of two other sides.

SECTION 5

Composite Shapes — Finding Hidden Sides

A composite shape is made by sticking two or more simple shapes together (or by cutting a piece out of a simple shape). On the SHSAT, these are the trickiest perimeter problems. The key skill is finding missing side lengths using the dimensions you are given.

This L-shaped figure has 6 outer sides. Two of those sides (the inner horizontal and inner vertical) are not labeled directly. Use subtraction to find them: missing side = larger parallel side − smaller parallel side.

The trick for every composite shape is the same. First, walk around the outside edge and count how many sides you see. Then label each side. If a side length is missing, look at the parallel sides that run in the same direction. The missing length is usually the difference or the sum of sides you already know.

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Common Mistake Alert

Don't accidentally count interior edges! If you imagine the L-shape as two rectangles glued together, the edge where they meet is inside the shape. That shared edge is not part of the perimeter.

SECTION 6

Worked Example — Step by Step

Let's work through a full SHSAT-style problem together. Read each step carefully.

Finding the Perimeter of a T-Shaped Figure

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Step 1 — Read the Problem

A T-shaped figure is made from two rectangles. The top rectangle is 14 cm wide and 4 cm tall. The bottom rectangle is 6 cm wide and 10 cm tall. The bottom rectangle is centered on the top rectangle. Find the perimeter of the whole T-shape.
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Step 2 — Sketch and Label the Outside Edges

Starting at the top-left corner and going clockwise, the T-shape has 8 outer sides. The top side = 14 cm. Going down the right side of the top bar = 4 cm. Then a horizontal step inward to the right edge of the stem. Then down the right side of the stem = 10 cm. Across the bottom of the stem = 6 cm. Up the left side of the stem = 10 cm. A horizontal step inward to the left edge of the top bar. Up the left side of the top bar = 4 cm.
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Step 3 — Find the Missing Horizontal Steps

The top bar is 14 cm wide, and the stem is 6 cm wide and centered. That means the stem sticks out equally on each side. The overhang on each side is (14 − 6) ÷ 2 = 8 ÷ 2 = 4 cm. So each horizontal step-in is 4 cm.

Each horizontal step = 4 cm
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Step 4 — Add All Outside Edges

Now list every outer side: 14 + 4 + 4 + 10 + 6 + 10 + 4 + 4.
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Step 5 — Calculate

14 + 4 = 18. 18 + 4 = 22. 22 + 10 = 32. 32 + 6 = 38. 38 + 10 = 48. 48 + 4 = 52. 52 + 4 = 56.

Perimeter = 56 cm

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Check Your Work

After you add up all the sides, a good check is to count the number of sides. A T-shape has 8 outer sides. If you only added 6 or 7 numbers, you missed a side! Go back and trace around the shape again.

SECTION 7

Shortcuts, Strengths & Common Pitfalls

Perimeter problems are some of the most straightforward on the SHSAT — if you avoid the traps. Here is a comparison of strategies and the mistakes that trip students up.

Strategy / Shortcut	When to Use It	Watch Out For
P = n × s (regular polygon shortcut)	All sides are the same length (square, equilateral triangle, regular hexagon).	Don't use this for rectangles! Length ≠ width.
P = 2(l + w) (rectangle formula)	Rectangles and squares (since a square is a special rectangle).	Remember to multiply by 2. A common mistake is to add l + w and stop.
Trace the outline	Any composite or irregular shape.	Don't accidentally trace an interior edge. Only count the outside.
Use subtraction for missing sides	Composite shapes where some lengths are not given.	Be sure you know which sides are parallel. Draw arrows to match them.

Strategies for perimeter problems

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KEY TAKEAWAY

Think of a composite shape like a jigsaw puzzle. When two pieces are pushed together, the edges where they touch disappear. Only the outer edges — the ones you can still see — make up the perimeter. If you can trace the outline with your pencil without lifting it, you've found every side of the perimeter.

SECTION 8

Connection to Area and Advanced Topics

Perimeter and area are cousins, but they measure very different things. Perimeter measures the distance around a shape (a length), while area measures the space inside a shape (in square units). The SHSAT loves testing whether you can tell the difference.

Feature	Perimeter	Area
What it measures	Distance around the outside	Space enclosed inside
Units	Linear (cm, m, in, ft)	Square (cm², m², in², ft²)
Rectangle formula	P = 2(l + w)	A = l × w
Can two shapes have the same one but different other?	Yes! A 1 × 8 rectangle and a 3 × 5 rectangle both have P = 18, but different areas.	Yes! A 2 × 6 rectangle and a 3 × 4 rectangle both have A = 12, but different perimeters.

Perimeter vs. Area

In higher-level math, perimeter ideas extend to 3D shapes — the total length of all edges of a rectangular prism, for example. You will also see circumference (the perimeter of a circle, C = 2πr) on the SHSAT and in high school geometry. Mastering polygon perimeter now gives you a strong foundation for all of that.

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Work each one on paper before checking the answer.

Question 1: CONCEPTUAL

A student says: "I found the perimeter of a rectangle by multiplying the length times the width." Is the student correct? Explain why or why not.

Show answer

The student is not correct. Multiplying length × width gives the area (the space inside), not the perimeter (the distance around). To find the perimeter of a rectangle, use P = 2 × (l + w). For example, a 3 × 5 rectangle has area = 15 square units but perimeter = 2 × (3 + 5) = 16 units.

Question 2: BASIC CALCULATION

A regular pentagon (5 equal sides) has a side length of 9 cm. What is its perimeter?

Show answer

Use the regular polygon shortcut: P = n × s. Here n = 5 sides and s = 9 cm. P = 5 × 9 = 45 cm. Since all five sides are the same length, multiplication is faster than adding 9 + 9 + 9 + 9 + 9, but either method gives the same answer.

Question 3: INTERMEDIATE

A rectangle has a perimeter of 38 cm. Its length is 12 cm. What is its width?

Show answer

Start with the formula P = 2(l + w). Substitute: 38 = 2(12 + w). Divide both sides by 2: 19 = 12 + w. Subtract 12 from both sides: w = 7. The width is 7 cm. You can check: 2 × (12 + 7) = 2 × 19 = 38 cm ✓. This type of problem tests whether you can work the perimeter formula backwards.

Question 4: APPLIED

A school playground is shaped like an L. The overall figure is 30 m along the top and 20 m on the left side. A rectangular notch is cut from the bottom-right corner that is 12 m wide and 8 m tall. What is the perimeter of the playground?

Show answer

First, find the missing sides. The bottom edge of the figure = 30 − 12 = 18 m. The right side going down from the top = 20 − 8 = 12 m. Now trace all 6 outer sides: Top = 30 m, right side down = 12 m, step in (horizontal) = 12 m, notch down = 8 m, bottom = 18 m, left side up = 20 m. Add them: 30 + 12 + 12 + 8 + 18 + 20 = 100 m. Always count your sides — an L-shape has exactly 6 outer edges.

Question 5: CRITICAL THINKING

Two rectangles each have a perimeter of 24 cm. Rectangle A has a length of 10 cm. Rectangle B has a length of 8 cm. Which rectangle has the greater area? By how much?

Show answer

Find each width first. Rectangle A: 24 = 2(10 + w), so 12 = 10 + w, w = 2 cm. Area A = 10 × 2 = 20 cm². Rectangle B: 24 = 2(8 + w), so 12 = 8 + w, w = 4 cm. Area B = 8 × 4 = 32 cm². Rectangle B has the greater area by 32 − 20 = 12 cm². This shows that two shapes can have the same perimeter but very different areas. The more "square-like" the rectangle, the larger the area.

SUMMARY

Lesson Summary

Summary

Perimeter is the total distance around the outside of a shape. For any polygon, you find the perimeter by adding up every side length. Regular polygons have all sides equal, so you can use the shortcut P = n × s. For a rectangle, use P = 2(l + w).

For composite shapes, trace only the outer edges and skip any interior (hidden) sides. Find missing side lengths by using subtraction or addition with the dimensions you already know. Remember that perimeter is measured in linear units (cm, m, ft) — never square units. Master these skills and you'll handle any SHSAT perimeter question with confidence!