ISEE Middle Level Quiz: Area And Perimeter
19 questions · exam conditions
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Area And PerimeterQuestion 1 of 19

Two identical squares, each with an area of 144 square feet, are placed so that they overlap. The overlapping region is a rectangle with an area of 30 square feet. What is the total area of the figure formed by the two overlapping squares?

258 square feet
288 square feet
318 square feet
348 square feet
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: Area And Perimeter

Practice Area And Perimeter in ISEE Middle Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Area And Perimeter, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Two identical squares, each with an area of 144 square feet, are placed so that they overlap. The overlapping region is a rectangle with an area of 30 square feet. What is the total area of the figure formed by the two overlapping squares?

  1. 258 square feet (correct answer)
  2. 288 square feet
  3. 318 square feet
  4. 348 square feet
Explanation: To find the total area of the combined figure, add the areas of the two individual squares and then subtract the area of the overlapping region, because it was counted twice. The area of each square is 144 square feet. The sum of their areas is 144+144=288144 + 144 = 288 square feet. The overlapping area is 30 square feet. The total area of the figure is 288−30=258288 - 30 = 258 square feet.

Question 2

The area of a square is 64 square centimeters. A rectangle has the same perimeter as the square. If the length of the rectangle is 10 centimeters, what is the area of the rectangle in square centimeters?

  1. 60 square centimeters (correct answer)
  2. 64 square centimeters
  3. 80 square centimeters
  4. 100 square centimeters
Explanation: First, find the side length of the square. If the area is 64 cm², the side length is 64=8\sqrt{64} = 8 cm. The perimeter of the square is 4×8=324 \times 8 = 32 cm. The rectangle has the same perimeter, 32 cm. The formula for the perimeter of a rectangle is P=2(L+W)P = 2(L+W). We have 32=2(10+W)32 = 2(10+W). Dividing by 2 gives 16=10+W16 = 10+W, so the width W=6W = 6 cm. The area of the rectangle is L×W=10×6=60L \times W = 10 \times 6 = 60 square centimeters.

Question 3

A rectangular garden measures 10 meters by 15 meters. A walkway of uniform width of 2 meters is built around the outside of the garden. What is the area of the walkway?

  1. 100 square meters
  2. 116 square meters (correct answer)
  3. 150 square meters
  4. 266 square meters
Explanation: The area of the garden is 10×15=15010 \times 15 = 150 square meters. The walkway adds 2 meters to each side of the garden. So, the new length is 15+2+2=1915 + 2 + 2 = 19 meters, and the new width is 10+2+2=1410 + 2 + 2 = 14 meters. The total area of the garden with the walkway is 19×14=26619 \times 14 = 266 square meters. To find the area of the walkway, subtract the garden's area from the total area: 266−150=116266 - 150 = 116 square meters.

Question 4

A rectangular park has a perimeter of 120 meters. Its length is 10 meters longer than its width. What is the area of the park?

  1. 875 square meters (correct answer)
  2. 900 square meters
  3. 950 square meters
  4. 1000 square meters
Explanation: Let the width of the park be ww meters. The length is w+10w+10 meters. The perimeter is given by P=2(L+W)P=2(L+W), so 120=2((w+10)+w)120 = 2((w+10)+w). Simplifying gives 120=2(2w+10)120 = 2(2w+10), which becomes 60=2w+1060 = 2w+10. Subtracting 10 gives 50=2w50 = 2w, so the width w=25w=25 meters. The length is 25+10=3525+10=35 meters. The area of the park is L×W=35×25=875L \times W = 35 \times 25 = 875 square meters.

Question 5

A piece of wire 60 inches long is bent to form a square. The same wire is then re-bent to form a rectangle with a length of 18 inches. What is the positive difference in area between the square and the rectangle?

  1. 9 square inches (correct answer)
  2. 15 square inches
  3. 99 square inches
  4. 225 square inches
Explanation: The length of the wire is the perimeter of both shapes. For the square, the perimeter is 60 inches, so each side is 60÷4=1560 \div 4 = 15 inches. The area of the square is 15×15=22515 \times 15 = 225 square inches. For the rectangle, the perimeter is also 60 inches. Using P=2(L+W)P=2(L+W), we have 60=2(18+W)60 = 2(18+W). This simplifies to 30=18+W30 = 18+W, so the width is 1212 inches. The area of the rectangle is 18×12=21618 \times 12 = 216 square inches. The difference in areas is 225−216=9225 - 216 = 9 square inches.

Question 6

A rectangular piece of cardboard measures 15 inches by 20 inches. A square with a side length of 4 inches is cut from each of the four corners. What is the perimeter of the remaining piece of cardboard?

  1. 38 inches
  2. 54 inches
  3. 62 inches
  4. 70 inches (correct answer)
Explanation: The perimeter of the original rectangle is 2(15+20)=2(35)=702(15+20) = 2(35) = 70 inches. When a square is cut from a corner, two segments of the original perimeter are removed, but two new segments of the same length are added. For example, at one corner, a 4-inch segment from the length and a 4-inch segment from the width are conceptually removed. They are replaced by the two inner sides of the cut-out square, each 4 inches long. The net change to the perimeter at each corner is −4−4+4+4=0-4 - 4 + 4 + 4 = 0. Since this happens at all four corners, the perimeter of the new shape is the same as the original perimeter, which is 70 inches.

Question 7

A square has a side length that is a positive integer. The numerical value of its perimeter is greater than the numerical value of its area. What is the sum of all possible integer side lengths for the square?

  1. 3
  2. 6 (correct answer)
  3. 7
  4. 10
Explanation: Let the side length of the square be ss. The perimeter is 4s4s and the area is s2s^2. The condition is that the perimeter is greater than the area: 4s>s24s > s^2. Since ss is a positive integer, we can divide both sides by ss without changing the inequality direction: 4>s4 > s. The positive integers ss that satisfy this condition are 1, 2, and 3. The sum of these possible side lengths is 1+2+3=61 + 2 + 3 = 6.

Question 8

A triangle has an area of 30 square meters. It is one of two congruent triangles that together form a rectangle. If the perimeter of that rectangle is 34 meters, what is the length of the rectangle's longer side?

  1. 5 meters
  2. 10 meters
  3. 12 meters (correct answer)
  4. 17 meters
Explanation: The area of the rectangle is twice the area of the triangle, so the rectangle's area is 2×30=602 \times 30 = 60 square meters. The perimeter of the rectangle is 34 meters. The formula for perimeter is P=2(L+W)P = 2(L+W), so 34=2(L+W)34 = 2(L+W), which means L+W=17L+W = 17. We need two numbers that have a sum of 17 and a product of 60. The integer pairs that multiply to 60 are (1,60), (2,30), (3,20), (4,15), (5,12), and (6,10). The pair that adds up to 17 is (5, 12). The length of the longer side is 12 meters.

Question 9

A rectangular field must be enclosed with a fence that has 4 strands of wire. The field is 60 yards long and 40 yards wide. If the wire costs $0.50 per yard, what will be the total cost of the wire needed?

  1. $100
  2. $200
  3. $400 (correct answer)
  4. $1200
Explanation: First, calculate the perimeter of the field: P=2(L+W)=2(60+40)=2(100)=200P = 2(L+W) = 2(60+40) = 2(100) = 200 yards. This is the length of one strand of wire. Since the fence requires 4 strands, the total length of wire needed is 4×200=8004 \times 200 = 800 yards. The cost of the wire is $0.50 per yard, so the total cost is 800 \text{ yards} \times \0.50/\text{yard} = $400$.

Question 10

A floor plan is shaped like a large rectangle measuring 20 feet by 30 feet, from which a square corner measuring 8 feet by 8 feet has been removed. What is the perimeter of this new L-shaped floor plan?

  1. 68 feet
  2. 84 feet
  3. 100 feet (correct answer)
  4. 116 feet
Explanation: The perimeter of the original large rectangle is 2(20+30)=2(50)=1002(20+30) = 2(50) = 100 feet. When a square corner is removed, two parts of the original perimeter are taken away, but two new sides are created. For example, a segment of 8 feet is removed from the 20-foot side, and a segment of 8 feet is removed from the 30-foot side. These are replaced by two new 8-foot sides on the inside cut. The change in perimeter is −8−8+8+8=0-8 - 8 + 8 + 8 = 0. Therefore, the perimeter of the L-shaped floor plan is the same as the perimeter of the original rectangle, which is 100 feet.

Question 11

A square-shaped garden has an area of 225 square feet. If a farmer wants to build a fence around the garden, how many feet of fencing will be required?

  1. 15 feet
  2. 30 feet
  3. 60 feet (correct answer)
  4. 225 feet
Explanation: The amount of fencing required is the perimeter of the garden. The area of the square is given as 225 square feet. The formula for the area of a square is A=s2A = s^2, where ss is the side length. So, s2=225s^2 = 225. To find the side length, take the square root of the area: s=225=15s = \sqrt{225} = 15 feet. The perimeter of a square is P=4sP = 4s. So, the required fencing is 4×15=604 \times 15 = 60 feet.

Question 12

The length of a rectangle is twice its width. If the area of the rectangle is 98 square feet, what is its perimeter in feet?

  1. 21 feet
  2. 35 feet
  3. 42 feet (correct answer)
  4. 49 feet
Explanation: Let the width be ww. The length is 2w2w. The area is L×WL \times W, so (2w)(w)=98(2w)(w) = 98. This gives 2w2=982w^2 = 98. Dividing by 2, we get w2=49w^2 = 49, so the width w=7w = 7 feet. The length is 2×7=142 \times 7 = 14 feet. The perimeter is 2(L+W)=2(14+7)=2(21)=422(L+W) = 2(14+7) = 2(21) = 42 feet.

Question 13

The perimeter of Square X is 4 times the perimeter of Square Y. The area of Square X is how many times the area of Square Y?

  1. 4 times
  2. 8 times
  3. 16 times (correct answer)
  4. 64 times
Explanation: Let the side length of Square Y be sys_y. Its perimeter is 4sy4s_y. Let the side length of Square X be sxs_x. Its perimeter is 4sx4s_x. We are given 4sx=4(4sy)4s_x = 4(4s_y), which simplifies to 4sx=16sy4s_x = 16s_y, and further to sx=4sys_x = 4s_y. The area of Square Y is Ay=(sy)2A_y = (s_y)^2. The area of Square X is Ax=(sx)2=(4sy)2=16(sy)2A_x = (s_x)^2 = (4s_y)^2 = 16(s_y)^2. Therefore, the area of Square X is 16 times the area of Square Y.

Question 14

If the length and width of a rectangle are both tripled, which statement accurately describes the change in its perimeter and area?

  1. The perimeter is tripled, and the area is tripled.
  2. The perimeter is tripled, and the area is nine times as large. (correct answer)
  3. The perimeter is six times as large, and the area is tripled.
  4. The perimeter is nine times as large, and the area is nine times as large.
Explanation: Let the original length and width be L and W. The original perimeter is P=2(L+W)P = 2(L+W) and the original area is A=LWA = LW. The new length and width are 3L and 3W. The new perimeter is Pnew=2(3L+3W)=2(3(L+W))=3(2(L+W))=3PP_{new} = 2(3L+3W) = 2(3(L+W)) = 3(2(L+W)) = 3P. The new area is Anew=(3L)(3W)=9(LW)=9AA_{new} = (3L)(3W) = 9(LW) = 9A. So, the perimeter is tripled and the area becomes nine times as large.

Question 15

The length of a rectangular sign is (3x+2)(3x + 2) meters and its width is xx meters. If the perimeter of the sign is 52 meters, what is its area in square meters?

  1. 6 square meters
  2. 20 square meters
  3. 52 square meters
  4. 120 square meters (correct answer)
Explanation: The formula for the perimeter of a rectangle is P=2(L+W)P = 2(L+W). Substitute the expressions for length and width: 52=2((3x+2)+x)52 = 2((3x+2) + x). Simplify the expression inside the parentheses: 52=2(4x+2)52 = 2(4x+2). Distribute the 2: 52=8x+452 = 8x + 4. Subtract 4 from both sides: 48=8x48 = 8x. Divide by 8 to find x=6x=6. Now find the dimensions: the width is x=6x = 6 meters, and the length is 3(6)+2=18+2=203(6) + 2 = 18 + 2 = 20 meters. The area is length times width: A=20×6=120A = 20 \times 6 = 120 square meters.

Question 16

A rectangle's length is 5 inches more than its width. If the perimeter of the rectangle is 54 inches, what is its area in square inches?

  1. 154 square inches
  2. 168 square inches
  3. 176 square inches (correct answer)
  4. 184 square inches
Explanation: Let the width be ww. Then the length is w+5w+5. The perimeter is 2(L+W)2(L+W), so 2((w+5)+w)=542((w+5)+w) = 54. This simplifies to 2(2w+5)=542(2w+5) = 54, so 4w+10=544w+10 = 54. Subtracting 10 from both sides gives 4w=444w = 44, so w=11w = 11 inches. The length is 11+5=1611+5=16 inches. The area is length times width, which is 16×11=17616 \times 11 = 176 square inches.

Question 17

A rectangular floor measuring 10 feet by 12 feet is to be completely covered by square tiles measuring 8 inches on a side. How many tiles are required?

  1. 180
  2. 240
  3. 270 (correct answer)
  4. 360
Explanation: First, convert all measurements to the same unit, such as inches. The floor dimensions are 10 ft×12 in/ft=12010 \text{ ft} \times 12 \text{ in/ft} = 120 inches and 12 ft×12 in/ft=14412 \text{ ft} \times 12 \text{ in/ft} = 144 inches. The area of the floor is 120×144120 \times 144 square inches. The area of one tile is 8×8=648 \times 8 = 64 square inches. The number of tiles needed is the floor area divided by the tile area: (120×144)/64(120 \times 144) / 64. We can simplify this: (120×(2×72))/64=(120×2×8×9)/(8×8)=(120×2×9)/8=(240×9)/8=30×9=270(120 \times (2 \times 72)) / 64 = (120 \times 2 \times 8 \times 9) / (8 \times 8) = (120 \times 2 \times 9) / 8 = (240 \times 9) / 8 = 30 \times 9 = 270.

Question 18

A rectangle has an area of 72 square inches. If its length and width are both integers, what is the greatest possible perimeter of the rectangle in inches?

  1. 34 inches
  2. 44 inches
  3. 74 inches
  4. 146 inches (correct answer)
Explanation: To find the greatest perimeter for a fixed area, the length and width must be as far apart as possible. The integer factor pairs of 72 are (1, 72), (2, 36), (3, 24), (4, 18), (6, 12), and (8, 9). The perimeter is calculated as 2(L+W)2(L+W). For (1, 72), the perimeter is 2(1+72)=2(73)=1462(1+72) = 2(73) = 146 inches. For (8, 9), the perimeter is 2(8+9)=2(17)=342(8+9) = 2(17) = 34 inches. The greatest possible perimeter corresponds to the factor pair (1, 72), which is 146 inches.

Question 19

A rectangular room is 15 feet long and 12 feet wide. What is the area of the room in square yards? (Note: 1 yard = 3 feet)

  1. 20 square yards (correct answer)
  2. 30 square yards
  3. 60 square yards
  4. 180 square yards
Explanation: There are two ways to solve this. First, find the area in square feet: 15 ft×12 ft=180 sq ft15 \text{ ft} \times 12 \text{ ft} = 180 \text{ sq ft}. Since 1 yard = 3 feet, 1 square yard = 3 ft×3 ft=9 sq ft3 \text{ ft} \times 3 \text{ ft} = 9 \text{ sq ft}. Convert the area to square yards by dividing: 180 sq ft÷9 sq ft/sq yd=20 sq yd180 \text{ sq ft} \div 9 \text{ sq ft/sq yd} = 20 \text{ sq yd}. Alternatively, convert the dimensions to yards first: 15 ft÷3 ft/yd=5 yd15 \text{ ft} \div 3 \text{ ft/yd} = 5 \text{ yd} and 12 ft÷3 ft/yd=4 yd12 \text{ ft} \div 3 \text{ ft/yd} = 4 \text{ yd}. Then calculate the area: 5 yd×4 yd=20 sq yd5 \text{ yd} \times 4 \text{ yd} = 20 \text{ sq yd}.