ISEE Lower 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 Lower Level

Q1 Easy

A rectangular wall is 12 feet long and 9 feet high. It has a square window with a side length of 4 feet. What is the area of the wall that is not covered by the window?

88 ft^{2}

92 ft^{2}

100 ft^{2}

108 ft^{2}

Show Solution

One way the answer can be found is by: 1) finding the area of the whole rectangular wall: $A = 12 \times 9 = 108 ft^{2}$ ; 2) finding the area of the square window: $A = 4 \times 4 = 16 ft^{2}$ ; 3) subtracting the area of the window from the total area of the wall: $108 - 16 = 92 ft^{2}$ .

Answer: B

Q2 Easy

A playroom floor consists of a rectangular section measuring 8 feet by 6 feet, and an attached square section measuring 4 feet on each side. What is the total area of the playroom floor?

48 ft^{2}

56 ft^{2}

64 ft^{2}

80 ft^{2}

Show Solution

One way the answer can be found is by: 1) finding the area of the rectangular section: $A = 8 \times 6 = 48 ft^{2}$ ; 2) finding the area of the attached square section: $A = 4 \times 4 = 16 ft^{2}$ ; 3) adding the three areas together to find the total: $48 + 16 + 16 = 80 ft^{2}$ .

Answer: D

Q3 Easy

The perimeter of an equilateral triangle is

15 x

. What is the length of one side?

3 x

5 x

12 x

45 x

Show Solution

An equilateral triangle has 3 sides of equal length. To find the length of one side, divide the total perimeter by 3. $15 x \div 3 = 5 x$ .

Answer: B

Q4 Easy

If the area of a rectangle is

36 m^{2}

, which equation can be used to determine the length of that rectangle? (

A = l w

, where

A

= Area,

l

= length, and

w

= width.)

l = 36 - w

l = 36 + w

l = \frac{w}{36}

l = \frac{36}{w}

Show Solution

The formula for the area of a rectangle is $A = l w$ . Since the area is $36$ , substitute that into the equation to get $36 = l w$ . To solve for the length ( $l$ ), divide both sides by the width ( $w$ ), resulting in $l = \frac{36}{w}$ .

Answer: D

Q5 Easy

What is the perimeter of a rectangle that has a length of 14 centimeters and a width of 6 centimeters? (

P = 2 l + 2 w

)

A 20 centimeters

B 40 centimeters

C 84 centimeters

D 168 centimeters

Show Solution

The perimeter of a rectangle is the total distance around the outside, found using the formula $P = 2 l + 2 w$ . Substitute the given length and width into the formula: $P = 2 (14) + 2 (6)$ . Multiply to get $28 + 12$ , which equals 40 centimeters.

Answer: B

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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