ISEE Upper Level

Area, Perimeter & Composite Shapes

Calculating perimeter and area of rectangles, squares, parallelograms, trapezoids, and composite/irregular shapes — excludes circles (see circles) and triangles (see triangles)

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Imagine you are building a playpen for your new pet velociraptor 🦖. You need to know exactly how much fencing to buy so he doesn't escape, and you need to know how much grass he has to run around on. That is exactly what Perimeter and Area are all about!

Perimeter is the outside edge. It's the fence! To find it, you just add up all the outside sides. Area is the inside space. It's the grass! To find it, you multiply the length by the width. You can also think of it like a giant cake 🍰. The perimeter is the border of sprinkles around the edge, and the area is all the delicious frosting spread on top.

The ISEE test loves to ask questions about Perimeter and Area. Sometimes, they will give you a regular rectangle. Other times, they will give you a weird, chunky shape that looks like a set of stairs! When that happens, just use your math sword to chop the weird shape into normal rectangles. Find the area of each piece, and add them together. Remember, the ISEE has no penalty for guessing, so if a shape looks too crazy, take your best guess and keep moving! 🏃‍♂️

Key Formula

For a rectangle:

A r e a = l \times w

and

P er im e t er = 2 l + 2 w

. For a triangle:

A r e a = \frac{1}{2} \times b \times h

Practice Questions

5 practice questions for ISEE Upper Level

Q1 Hard

If the length of a rectangle is increased by

20%

and the width is decreased by

10%

, what is the percent increase in the area of the rectangle?

A 8%

B 10%

C 12%

D 28%

Show Solution

Let the original length be $L$ and the original width be $W$ . The original area is $L \times W$ . A $20%$ increase in length makes the new length $1.2 L$ , and a $10%$ decrease in width makes the new width $0.9 W$ . The new area is $1.2 L \times 0.9 W = 1.08 (L \times W)$ . Since the new area is $1.08$ times the original area, this represents an $8%$ increase.

Answer: A

Q2 Hard

A rectangle has an area of

60

square centimeters. If the length and the width of the rectangle are measured in whole centimeters, what is the greatest possible perimeter of the rectangle?

A 32 cm

B 34 cm

C 64 cm

D 122 cm

Show Solution

To find the greatest possible perimeter for a given area with integer side lengths, we must find the factor pair of $60$ with the greatest difference. The factor pairs of $60$ are (1, 60), (2, 30), (3, 20), (4, 15), (5, 12), and (6, 10). The perimeter is calculated as $2 \times (length + width)$ . Using the pair $1$ and $60$ , the perimeter is $2 \times (1 + 60) = 2 \times 61 = 122$ centimeters. All other factor pairs result in smaller perimeters (for example, $2 \times (6 + 10) = 32$ ). Therefore, the greatest possible perimeter is $122$ cm.

Answer: D

Q3 Hard

A circle has an area of

16 π

square inches. A square has an area of

64

square inches.

Column A

The circumference of the circle

Column B

The perimeter of the square

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

First, find the radius of the circle. The area of a circle is $π r^{2}$ , so $π r^{2} = 16 π$ , which means $r^{2} = 16$ and $r = 4$ . The circumference of the circle (Column A) is $2 π r = 2 π (4) = 8 π$ inches. Since $π$ is approximately $3.14$ , $8 π$ is approximately $25.12$ inches. Next, find the side length of the square. The area is $s^{2} = 64$ , so $s = 8$ . The perimeter of the square (Column B) is $4 s = 4 (8) = 32$ inches. Comparing the two quantities, $32$ is greater than $8 π$ (since $8 \times 4 = 32$ and $π < 4$ ). Therefore, the quantity in Column B is greater.

Answer: B

Q4 Hard

Square

X

has an area of

4 y^{2}

. Rectangle

Y

has an area of

4 y^{2}

and a length of

4 y

. Assume

y > 0

Column A

The perimeter of Square

X

Column B

The perimeter of Rectangle

Y

A The quantity in Column A is greater.

B The quantity in Column B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Show Solution

First, determine the dimensions of Square $X$ . Since its area is $4 y^{2}$ , its side length is $4 y^{2} = 2 y$ . The perimeter of Square $X$ (Column A) is $4 \times 2 y = 8 y$ . Next, determine the dimensions of Rectangle $Y$ . Its area is $4 y^{2}$ and its length is $4 y$ . Since Area = length $\times$ width, $4 y^{2} = 4 y \times width$ , making the width equal to $y$ . The perimeter of Rectangle $Y$ (Column B) is $2 \times (4 y + y) = 2 \times 5 y = 10 y$ . Since $y > 0$ , $10 y$ must be strictly greater than $8 y$ . Therefore, the quantity in Column B is greater.

Answer: B

Q5 Hard

A rectangular garden is

12

meters long and

9

meters wide. It is surrounded by a paved walkway that is

2

meters wide on all sides. What is the area of the walkway?

A 42 square meters

B 84 square meters

C 100 square meters

D 108 square meters

Show Solution

To find the area of the walkway, subtract the area of the garden from the total area of the garden and walkway combined. The garden has an area of $12 \times 9 = 108$ square meters. The walkway adds $2$ meters to each side of the garden, so the total length is $12 + 2 + 2 = 16$ meters, and the total width is $9 + 2 + 2 = 13$ meters. The combined area of the garden and the walkway is $16 \times 13 = 208$ square meters. The area of the walkway alone is $208 - 108 = 100$ square meters.

Answer: C

Tips & Strategies

Draw it out! ✏️ If the ISEE gives you a word problem without a picture, immediately sketch the shape on your scratch paper and label the sides.
Look out for Quantitative Comparison traps! Sometimes the test will ask you to compare the Area in Column A to the Perimeter in Column B. Read carefully so you don't calculate the wrong thing.
Chop up weird shapes! If you see an L-shaped room, draw a line to cut it into two normal rectangles. Find the area of both, then add them up.

Common Mistakes

Watch out for the walkway trap! If a path goes around a garden, it adds to BOTH sides. Don't just add the width once—you have to add it twice to the length and twice to the width.
Don't forget the $\frac{1}{2}$ when finding the area of a triangle! If you just multiply base times height, you are finding the area of a rectangle, which is a very common wrong answer choice.

Frequently Asked Questions

What's the difference between Quantitative Reasoning and Mathematics Achievement on the ISEE?

Quantitative Reasoning has the tricky Column A vs Column B questions and tests how well you think about numbers. Mathematics Achievement is more like a normal school math test where you just solve the problem!

What if I forget a formula during the test?

Don't panic! Try drawing the shape on your scratch paper. You can often estimate the answer by thinking about what makes sense. And remember, the ISEE has no guessing penalty, so always pick your best guess!

Why do the answers say 'sq ft' or 'sq cm'?

Area is measured in flat little squares! So 'sq ft' just means 'square feet'. Perimeter is just a straight line, like a string, so it uses regular feet or centimeters.

Do I need to simplify fractions on the ISEE?

Yes! The answer choices on the ISEE will almost always be in simplest form. If you get $\frac{4}{8}$ , you should look for $\frac{1}{2}$ in the choices.

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