ISEE Middle 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

4 practice questions for ISEE Middle Level

Q1 Medium

A circle is inscribed within a square. The perimeter of the square is 40 cm. What is the area of the shaded region outside the circle but inside the square?

40 - 10 π

^{2}

40 - 25 π

^{2}

100 - 10 π

^{2}

100 - 25 π

^{2}

Show Solution

First, find the side length of the square. Since the perimeter is 40 cm, each side is $40 \div 4 = 10$ cm. The area of the square is $10 \times 10 = 100$ cm $^{2}$ . Because the circle is inscribed in the square, its diameter is equal to the side length of the square (10 cm), which makes its radius $r = 5$ cm. The area of the circle is $π (5)^{2} = 25 π$ cm $^{2}$ . The area of the shaded region is the area of the square minus the area of the circle, which is $100 - 25 π$ cm $^{2}$ .

Answer: D

Q2 Medium

A 12 cm by 8 cm rectangle has a 3 cm by 3 cm square cut out of one of its corners.

Column A

The perimeter of the original 12 cm by 8 cm rectangle

Column B

The perimeter of the new shape after the 3 cm by 3 cm square is removed

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

Column A: The perimeter of the original 12 cm by 8 cm rectangle is $2 (12 + 8) = 2 (20) = 40$ cm.

Column B: When a 3 cm by 3 cm square is removed from a corner, the original corner's two 3 cm segments are removed from the perimeter, but two new 3 cm segments are created on the inside of the shape. Therefore, the total perimeter remains unchanged: $40 - 3 - 3 + 3 + 3 = 40$ cm. The two quantities are equal.

Answer: C

Q3 Medium

A composite figure is shaped like a house, consisting of a triangle sitting on top of a rectangle. The rectangular base has a length of 10 cm and a height of 8 cm. If the total area of the composite figure is 110 cm

^{2}

, what is the height of the triangular roof?

Note: Area of a triangle =

\frac{1}{2} bh

A 3 cm

B 5 cm

C 6 cm

D 12 cm

Show Solution

First, find the area of the rectangular part of the figure: $10 \times 8 = 80$ cm $^{2}$ . Subtract this from the total area to find the area of the triangular part: $110 - 80 = 30$ cm $^{2}$ . The base of the triangle is the same as the top length of the rectangle, which is 10 cm. Using the area formula for a triangle, solve for the height $h$ : $30 = \frac{1}{2} (10) h$ . Simplify to $30 = 5 h$ , and divide by 5 to get $h = 6$ cm.

Answer: C

Q4 Medium

A rectangular painting measures 15 inches by 20 inches. It is surrounded by a wooden frame that is uniformly 2 inches wide on all sides. What is the area of the wooden frame?

A 74 in

^{2}

B 156 in

^{2}

C 300 in

^{2}

D 456 in

^{2}

Show Solution

First, calculate the area of the painting itself: $15 \times 20 = 300$ in $^{2}$ . Next, determine the outer dimensions of the frame. Since the frame is 2 inches wide on all sides, you must add 4 inches (2 inches for each side) to both the length and the width. The outer dimensions are $15 + 4 = 19$ inches and $20 + 4 = 24$ inches. The total area of the painting and frame together is $19 \times 24 = 456$ in $^{2}$ . Finally, subtract the area of the painting from the total area to find the area of the frame: $456 - 300 = 156$ in $^{2}$ .

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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