Calculate perimeter of a figure from a diagram. - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Measure Perimeter?

Have you ever walked around a playground? Or put a fence around a garden? People have been measuring the distance around shapes for thousands of years!

The word perimeter comes from two Greek words: peri (meaning "around") and metron (meaning "measure"). So perimeter literally means "measure around." Pretty cool, right?

3000 BC • Ancient Egypt

Egyptian farmers measured the perimeter of their fields along the Nile River to know how much fencing they needed.

300 BC • Greek Mathematicians

A mathematician named Euclid wrote rules about shapes and how to measure them. He helped create the word "perimeter."

100 AD • Roman Builders

Romans used perimeter to plan roads, walls, and buildings all across their empire.

Today • Everyday Life

We use perimeter to build fences, frame pictures, wrap gifts, and even plan running tracks!

On the ISEE test, you will see diagrams of shapes with side lengths labeled. Your job is to find the perimeter. Let's learn how!

SECTION 2

Core Ideas About Perimeter

Before we start calculating, let's learn the big ideas. These four rules will help you solve every perimeter problem!

Perimeter = Total Distance Around

Imagine an ant walking along every edge of a shape. The total distance it walks is the perimeter.

Add Up All the Sides

To find perimeter, add the length of every side together. Don't skip any sides!

Count Every Side

A rectangle has 4 sides. A triangle has 3. A pentagon has 5. Make sure you count them all.

Units Stay the Same

If the sides are measured in feet, your perimeter is in feet. If in centimeters, your answer is in centimeters.

SECTION 3

See It: Perimeter in a Picture

Let's look at a rectangle with labeled sides. Follow the arrows around the shape. That path is the perimeter!

Follow the dashed arrows from START all the way around the rectangle. Add each side: 12 + 8 + 12 + 8 = 40 ft. That's the perimeter!

Notice how the arrows go all the way around the rectangle. We add each side as we go: top (12 ft), right side (8 ft), bottom (12 ft), and left side (8 ft). The total is 40 feet.

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ISEE Test Tip

SECTION 4

The Perimeter Formulas

For any shape, you can always just add all the sides. But for common shapes, there are shortcuts. These formulas save you time!

ANY SHAPE

Perimeter = Side 1 + Side 2 + Side 3 + ... (add all sides)

This works for every shape. Just add up every single side.

RECTANGLE SHORTCUT

Perimeter = length + width + length + width = 2 × length + 2 × width

A rectangle has two pairs of equal sides. The length is the longer side and the width is the shorter side.

SQUARE SHORTCUT

Perimeter = side × 4

A square has four equal sides. Multiply one side by 4 and you're done!

TRIANGLE

Perimeter = side 1 + side 2 + side 3

A triangle has three sides. Add all three together.

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Remember!

SECTION 5

Different Shapes, Same Idea

The ISEE may show you rectangles, squares, triangles, or even L-shaped figures. Let's look at several shapes and their perimeters side by side.

Four different shapes all solved the same way: add every side! The square uses a shortcut (side × 4). The L-shape has 6 sides, so be careful to count them all.

Notice the L-shaped figure at the bottom left. It has 6 sides, not 4! On the ISEE, you might see shapes like this. The trick is to count every side carefully.

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ISEE Test Tip

SECTION 6

Step-by-Step Example

Let's work through a problem together, step by step. Imagine you see this on the ISEE:

A rectangle has a length of 14 cm and a width of 9 cm. What is the perimeter?

Worked example

1
Step 1 — Read the Problem

We have a rectangle. The length is 14 cm and the width is 9 cm. We need the perimeter.
2
Step 2 — Identify All Four Sides

A rectangle has two lengths and two widths. So the four sides are: 14, 9, 14, and 9.

Sides: 14, 9, 14, 9
3
Step 3 — Add All the Sides

Add them in order: 14 + 9 = 23. Then 23 + 14 = 37. Then 37 + 9 = 46.

14 + 9 + 14 + 9 = 46
4
Step 4 — Write the Answer with Units

Don't forget the units! The sides were in centimeters, so the perimeter is in centimeters too.

Perimeter = 46 cm

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Quick Check

SECTION 7

Common Mistakes to Avoid

Even smart students make mistakes on perimeter problems. Here are the most common ones and how to avoid them.

Mistake	Why It Happens	How to Fix It
Forgetting a side	You add only 3 sides of a rectangle instead of 4.	Count the sides first. Put a check mark on each side as you add it.
Mixing up perimeter and area	You multiply length × width instead of adding all sides.	Remember: perimeter = add. Area = multiply. Perimeter is the fence; area is the grass inside.
Missing unlabeled sides	The diagram doesn't label every side, so you skip one.	Use what you know. Opposite sides of a rectangle are equal. Figure out missing sides.
Wrong addition	Simple adding mistakes when no calculator is allowed.	Double-check your addition. Add the numbers a second time to make sure.

Common perimeter mistakes and how to avoid them

SECTION 8

Perimeter of Trickier Figures

Sometimes the ISEE gives you shapes that aren't plain rectangles. You might see L-shapes or figures with missing side labels. Don't worry — you can handle them!

Simple Shapes	Trickier Shapes
All sides are labeled	Some sides may be missing labels
3 or 4 sides	5, 6, or more sides
Just add the numbers you see	Figure out missing sides first, then add
Example: rectangle, triangle	Example: L-shape, T-shape, U-shape

Simple vs. trickier perimeter problems

To find a missing side on an L-shape, look at the sides that run in the same direction. If the full top is 10 and one part of the top is 6, the other part must be 10 − 6 = 4.

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Looking Ahead

SECTION 9

Practice Problems

Time to practice! Remember: no calculator, just like on the real ISEE. Read each problem carefully, and add up all the sides. You've got this!

Question 1: Calculate Perimeter from a Diagram

What is the perimeter of the rectangle shown below?

A. 16 ft
B. 32 ft
C. 40 ft
D. 60 ft

Show answer (B)

The perimeter of a rectangle is the sum of all four sides: 10 + 6 + 10 + 6 = 32 ft.A (16 ft) is incorrect. A student may have added only one length and one width: 10 + 6 = 16, forgetting that a rectangle has four sides.C (40 ft) is incorrect. A student may have multiplied 10 × 4 = 40, treating all four sides as if they were each 10 ft long.D (60 ft) is incorrect. A student may have found the area (10 × 6 = 60) instead of the perimeter.

Question 2: Calculate Perimeter from a Diagram

What is the perimeter of the triangle shown below?

A. 21 cm
B. 27 cm
C. 30 cm
D. 36 cm

Show answer (C)

The perimeter of a triangle is the sum of all three sides: 9 + 9 + 12 = 30 cm.A (21 cm) is incorrect. A student may have added only two different side lengths: 9 + 12 = 21, forgetting to include the second 9 cm side.B (27 cm) is incorrect. A student may have multiplied 9 × 3 = 27, treating all three sides as if they were each 9 cm.D (36 cm) is incorrect. A student may have multiplied 12 × 3 = 36, treating all three sides as if they were each 12 cm.

Question 3: Calculate Perimeter from a Diagram

A rectangle has length 9 m and width 4 m. What is its perimeter?

A. 22 m
B. 26 m
C. 30 m
D. 36 m

Show answer (B)

Correct answer: B. Perimeter = 2(9 + 4) = 26 m.A (22 m): this can come from adding 9 + 9 + 4 and forgetting the final side of length 4.C (30 m): this can come from using 9 + 9 + 4 + 4 + 4, accidentally counting one width twice.D (36 m): this can come from computing area 9 × 4 instead of perimeter.

Question 4: Calculate Perimeter from a Diagram

A square garden has a side length of 15 yards. A fence is built around the entire garden. What is the total length of the fence?

A. 30 yd
B. 45 yd
C. 60 yd
D. 225 yd

Show answer (C)

A square has four equal sides, so the perimeter is 4 × 15 = 60 yd.A (30 yd) is incorrect. A student may have added only two sides: 15 + 15 = 30, forgetting that a square has four sides.B (45 yd) is incorrect. A student may have added only three sides: 15 + 15 + 15 = 45, missing one side of the square.D (225 yd) is incorrect. A student may have found the area (15 × 15 = 225) instead of the perimeter.

Question 5: Calculate Perimeter from a Diagram

A triangle has side lengths 9 in., 11 in., and 12 in. What is its perimeter?

A. 30 in.
B. 31 in.
C. 32 in.
D. 33 in.

Show answer (C)

Correct answer: C. Add all side lengths: 9 + 11 + 12 = 32 in.A (30 in.): this can come from misreading 12 as 10, then 9 + 11 + 10 = 30.B (31 in.): this can come from copying one side as 11 instead of 12, then 9 + 11 + 11 = 31.D (33 in.): this can come from an addition slip that adds 1 to the correct sum of 32.