Perimeter of Polygons - SSAT Middle Level Quantitative

1. The perimeter of a square is numerically equal to its area. What is the side length of the square?
   Answer: 4 units This problem tests your ability to set up equations when two different formulas for the same shape are equal to each other. When you see "numerically equal," you need to write an equation setting the two expressions equal. For a square with side length $s$, the perimeter is $4s$ and the area is $s^2$. Since these are numerically equal, you can write: $4s = s^2$ To solve this equation, rearrange it to standard form: $s^2 - 4s = 0$. Factor out $s$: $s(s - 4) = 0$. This gives you two solutions: $s = 0$ or $s = 4$. Since a square can't have zero side length, $s = 4$. Let's verify: A square with side length 4 has perimeter $4 \times 4 = 16$ and area $4^2 = 16$. They're equal, confirming our answer. Looking at the wrong choices: Choice B (8 units) would give a perimeter of 32 and area of 64 - not equal. Choice C (12 units) would give a perimeter of 48 and area of 144 - not equal. Choice D (16 units) would give a perimeter of 64 and area of 256 - not equal. These wrong answers might tempt you if you confuse which formula is which or make calculation errors. The correct answer is A) 4 units. Strategy tip: When two geometric formulas are set equal, always set up the equation algebraically first, then solve. Don't try to guess-and-check with the answer choices - you might make arithmetic errors that lead to wrong conclusions.
2. A rectangular picture frame has outer dimensions of 18 inches by 14 inches. The frame itself is 2 inches wide all around. What is the perimeter of the inner rectangle (the picture opening)?
   Answer: 48 inches When you encounter a problem about frames or borders, you're working with nested rectangles where you need to carefully track the relationship between outer and inner dimensions. The outer frame measures 18 inches by 14 inches, with a 2-inch-wide frame all around. To find the inner rectangle's dimensions, you subtract the frame width from both sides of each dimension. Since the frame is 2 inches wide on each side, you subtract $2 + 2 = 4$ inches from each outer dimension. Inner length: $18 - 4 = 14$ inches Inner width: $14 - 4 = 10$ inches The perimeter of the inner rectangle is: $2(14 + 10) = 2(24) = 48$ inches. Looking at the wrong answers: Choice B (52 inches) likely comes from subtracting only 2 inches from each dimension instead of 4, giving dimensions of 16 by 12. Choice C (56 inches) might result from subtracting 1 inch from each side (17 by 13 dimensions) or making an arithmetic error. Choice D (60 inches) could come from forgetting to subtract the frame width entirely and using dimensions like 16 by 14. The key trap here is forgetting that a frame that's "2 inches wide all around" reduces each dimension by 4 inches total—2 inches from each side. Always visualize the frame surrounding the picture on all four sides, requiring you to account for the frame width twice in each direction.
3. Two regular polygons have the same side length of 7 cm. If one is a pentagon and the other is a heptagon (7 sides), what is the difference between their perimeters?
   Answer: 14 cm When you encounter polygon problems, remember that perimeter simply means the total distance around the shape, which equals the number of sides times the length of each side. To find the difference in perimeters, you need to calculate each polygon's perimeter separately. A pentagon has 5 sides, so its perimeter is $5 \times 7 = 35$ cm. A heptagon has 7 sides, so its perimeter is $7 \times 7 = 49$ cm. The difference is $49 - 35 = 14$ cm. Looking at the wrong answers: Choice B (21 cm) represents three times the side length, which doesn't correspond to any meaningful calculation here. Choice C (28 cm) equals four times the side length—you might get this if you incorrectly calculated the difference in the number of sides (7 - 5 = 2) and then multiplied by some other factor. Choice D (35 cm) is actually the pentagon's complete perimeter, not the difference between the two perimeters. This is a common trap where students calculate one value correctly but forget to complete the final step. The correct answer is A (14 cm) because it represents the actual difference: 49 - 35 = 14 cm. For polygon perimeter problems, always use the formula: perimeter = number of sides × side length. When comparing polygons, calculate each perimeter completely before finding the difference. Don't try to take shortcuts with the difference in side counts, as this often leads to trap answers.
4. The perimeter of a regular hexagon is 72 inches. What is the perimeter of a square whose side length equals the side length of the hexagon?
   Answer: 48 inches This question tests your understanding of perimeter and how it relates to different polygons with the same side length. To find the hexagon's side length, remember that a regular hexagon has 6 equal sides. If the perimeter is 72 inches, then each side is $72 ÷ 6 = 12$ inches. Now you need the perimeter of a square with the same 12-inch side length. Since a square has 4 equal sides, the perimeter is $12 × 4 = 48$ inches. Looking at the wrong answers: Choice B (54 inches) might result from multiplying the hexagon's side length by 4.5, which has no geometric meaning here. Choice C (60 inches) could come from mistakenly thinking a square has 5 sides, leading to $12 × 5 = 60$. Choice D (66 inches) doesn't follow from any clear mathematical relationship and might be a distractor based on the original perimeter of 72. The correct answer is A: 48 inches. When you encounter polygon perimeter problems, always start by identifying how many sides each shape has. For regular polygons, divide the total perimeter by the number of sides to find individual side length. Then multiply that side length by the number of sides in your target shape. Watch out for answer choices that use incorrect numbers of sides or meaningless mathematical operations – they're designed to catch students who rush through the geometric relationships.
5. A rectangular swimming pool is surrounded by a uniform concrete walkway. The pool measures 20 feet by 30 feet, and the walkway extends 4 feet from each side of the pool. What is the perimeter of the outer edge of the walkway?
   Answer: 132 feet The walkway extends 4 feet from every side, so add 4 feet to both ends of each dimension. Outer length: $30 + 4 + 4 = 38$ feet. Outer width: $20 + 4 + 4 = 28$ feet. Perimeter: $2(38 + 28) = 2 \times 66 = 132$ feet. An answer of 100 feet is the perimeter of the pool itself, $2(30 + 20)$, with no walkway. Adding the 4 feet to only one end of each dimension gives $2(34 + 24) = 116$ feet, which is too small. An answer of 136 feet is too large; it would need outer sides adding to 68 feet, for example by adding 10 feet to the width instead of 8. For borders, sketch the outer rectangle and find its length and width before using the perimeter formula.
6. The diagram shows a compound figure made of two rectangles. Rectangle A measures 8 inches by 5 inches, and Rectangle B measures 6 inches by 3 inches. The rectangles share a common side of length 3 inches. What is the perimeter of the compound figure?
   Answer: 38 inches Trace the outer boundary of the figure. Starting at the top left: the top of Rectangle A is 8 inches, the step down to Rectangle B is $5 - 3 = 2$ inches, the top of Rectangle B is 6 inches, its right side is 3 inches, the bottom edge is $8 + 6 = 14$ inches, and the left side is 5 inches. The perimeter is $8 + 2 + 6 + 3 + 14 + 5 = 38$ inches. As a check, the two separate perimeters are $2(8 + 5) = 26$ and $2(6 + 3) = 18$, for $44$ in all. The shared 3-inch side is counted once in each, so subtract $2 \times 3 = 6$: $44 - 6 = 38$ inches. 32 inches comes from subtracting 12 instead of 6, removing the shared side four times. 26 inches is the perimeter of Rectangle A alone, and 18 inches is the perimeter of Rectangle B alone. Strategy tip: For compound figures, trace the outside edge and add only the segments you walk along.
7. A rectangular garden has a length that is 8 feet more than twice its width. If the width is $w$ feet, what is the perimeter of the garden in terms of $w$?
   Answer: $6w + 16$ feet When you encounter word problems involving rectangles and perimeter, you need to translate the given relationships into algebraic expressions, then apply the perimeter formula. Let's break down what we know: the width is $w$ feet, and the length is "8 feet more than twice the width." This means the length equals $2w + 8$ feet. The perimeter of a rectangle is found using the formula: Perimeter = $2 \times \text{length} + 2 \times \text{width}$. Substituting our expressions: Perimeter = $2(2w + 8) + 2w = 4w + 16 + 2w = 6w + 16$ feet. Looking at the wrong answers: Choice B ($4w + 16$) represents a common error where students forget to double the width—they correctly doubled the length expression but only counted the width once. Choice C ($3w + 8$) occurs when students mistakenly think perimeter means adding length and width just once, without doubling either dimension. Choice D ($2w + 8$) is simply the expression for the length alone, not the perimeter—this happens when students get confused about what the question is asking for. The correct answer is A: $6w + 16$ feet. Study tip: In rectangle perimeter problems, always write out the formula $P = 2l + 2w$ first, then substitute your expressions for length and width. This prevents the common mistake of forgetting to double both dimensions. Also, double-check that your final answer makes sense by testing with a simple value for the variable.
8. An equilateral triangle has the same perimeter as a regular hexagon with side length 9 centimeters. What is the side length of the equilateral triangle?
   Answer: 18 centimeters When you encounter problems comparing perimeters of different polygons, you need to set up an equation where the perimeters are equal, then solve for the unknown measurement. First, find the perimeter of the regular hexagon. Since a regular hexagon has 6 equal sides, each measuring 9 centimeters, its perimeter is $6 \times 9 = 54$ centimeters. Now you know the equilateral triangle must also have a perimeter of 54 centimeters. Since an equilateral triangle has 3 equal sides, if each side has length $s$, then $3s = 54$. Solving for $s$: $s = 54 ÷ 3 = 18$ centimeters. Looking at the wrong answers: Choice B (15 centimeters) would give a triangle perimeter of $3 \times 15 = 45$ centimeters, which is too small. Choice C (12 centimeters) would give $3 \times 12 = 36$ centimeters, also too small. Choice D (9 centimeters) represents the hexagon's side length, not the triangle's—this is a common trap where students might think equal perimeters mean equal side lengths, but that's only true for polygons with the same number of sides. The correct answer is A: 18 centimeters. Study tip: For perimeter comparison problems, always calculate the known perimeter first, then set up an equation for the unknown shape. Remember that polygons with different numbers of sides will have different individual side lengths even when their perimeters are equal.
9. A rhombus has the same perimeter as a rectangle with length 18 meters and width 6 meters. What is the side length of the rhombus?
   Answer: 12 meters When you encounter problems comparing perimeters of different shapes, remember that perimeter is simply the total distance around the outside of any shape, regardless of its specific form. First, let's find the rectangle's perimeter. A rectangle has two lengths and two widths, so: $P = 2(18) + 2(6) = 36 + 12 = 48$ meters. Since the rhombus has the same perimeter as the rectangle, the rhombus also has a perimeter of 48 meters. A rhombus is a special quadrilateral where all four sides are equal in length. If we call each side length $s$, then: $4s = 48$, which means $s = 12$ meters. Looking at the wrong answers: Choice B (15 meters) would give a perimeter of 60 meters, which is too large. This might tempt students who incorrectly add 18 + 6 + 15 = 39 and think they're close. Choice C (18 meters) would create a perimeter of 72 meters; students might choose this by focusing only on the rectangle's length and ignoring the perimeter calculation entirely. Choice D (24 meters) results in a 96-meter perimeter, exactly double what we need—this could catch students who forget to divide by 4 after finding the correct perimeter. The correct answer is A (12 meters). Strategy tip: When comparing perimeters between different shapes, always calculate the known shape's perimeter completely first, then use the properties of the unknown shape to work backwards. Remember that rhombuses have four equal sides, so divide the total perimeter by 4.
10. A parallelogram has sides of length 15 cm and 22 cm. What is the perimeter of the parallelogram?
    Answer: 74 cm When you see a parallelogram problem, remember that opposite sides are equal in length. This means a parallelogram has two pairs of equal sides, not four different sides. Since this parallelogram has sides of 15 cm and 22 cm, you actually have two sides that are 15 cm long and two sides that are 22 cm long. To find the perimeter, you add up all four sides: $15 + 15 + 22 + 22 = 74$ cm. Looking at the answer choices, A) 74 cm is correct because it accounts for all four sides of the parallelogram. B) 52 cm represents a common error where someone might have calculated $15 + 22 + 15 = 52$, forgetting to include the fourth side. This happens when students mistakenly think they only need three sides. C) 44 cm comes from simply doubling one of the sides: $22 \times 2 = 44$. This error occurs when students forget that a parallelogram has two different side lengths, not just one repeated four times. D) 37 cm is the result of just adding the two given measurements once: $15 + 22 = 37$. This mistake happens when students don't realize they need to account for the fact that each measurement represents two sides of the parallelogram. Remember this pattern: for any parallelogram with sides $a$ and $b$, the perimeter is always $2a + 2b$. When you see two side lengths given for a parallelogram, double each one and add them together.
11. A triangle has sides of length 15 cm, 20 cm, and 25 cm. If each side is increased by the same amount $x$, what is the new perimeter in terms of $x$?
    Answer: $(60 + 3x)$ cm When you encounter problems involving changes to geometric figures, focus on how those changes affect the measurements you're asked to find. This question tests your understanding of perimeter and algebraic expressions. The original triangle has sides of 15 cm, 20 cm, and 25 cm, giving it a perimeter of $15 + 20 + 25 = 60$ cm. When each side increases by the same amount $x$, the new sides become $(15 + x)$ cm, $(20 + x)$ cm, and $(25 + x)$ cm. The new perimeter is therefore $(15 + x) + (20 + x) + (25 + x) = 15 + 20 + 25 + 3x = 60 + 3x$ cm. Choice A, $(60 + 3x)$ cm, correctly represents this calculation. Choice B, $(60 + x)$ cm, makes the error of adding $x$ only once instead of three times—forgetting that $x$ gets added to each of the three sides. Choice C, $(45 + 3x)$ cm, correctly adds $3x$ but miscalculates the original perimeter as 45 instead of 60. Choice D, $(45 + x)$ cm, combines both errors: the wrong original perimeter and adding $x$ only once. Remember that when the same value is added to multiple measurements in a perimeter problem, that value gets multiplied by the number of sides. Always double-check your arithmetic on the original measurements, as test makers often include answer choices with common calculation errors.
12. A right triangle has legs of length 9 meters and 12 meters. What is the perimeter of the triangle?
    Answer: 36 meters When you encounter a right triangle problem asking for perimeter, you need to find all three side lengths, then add them together. You're given the two legs, but you'll need to calculate the hypotenuse using the Pythagorean theorem. With legs of 9 meters and 12 meters, you can find the hypotenuse using $a^2 + b^2 = c^2$. Substituting: $9^2 + 12^2 = c^2$, which gives you $81 + 144 = 225$, so $c^2 = 225$ and $c = 15$ meters. The perimeter is therefore $9 + 12 + 15 = 36$ meters. Looking at the wrong answers: Choice B (33 meters) might result from miscalculating the hypotenuse as 12 instead of 15, possibly from computational errors with the Pythagorean theorem. Choice C (30 meters) could come from incorrectly assuming the hypotenuse is 9 meters, perhaps by confusing which sides are legs versus hypotenuse. Choice D (27 meters) represents adding only multiples of the given legs without properly calculating the hypotenuse at all. The correct answer is A) 36 meters. Study tip: Memorize common Pythagorean triples like 3-4-5, 5-12-13, and 9-12-15 (which is 3 times the 3-4-5 triangle). Recognizing these patterns will save you calculation time on the SSAT. When you see legs of 9 and 12, you should immediately think "this is a 3-4-5 triangle scaled up by 3," making the hypotenuse 15.
13. A kite has two pairs of adjacent sides. One pair consists of sides that are each 12 inches long, and the other pair consists of sides that are each 8 inches long. What is the perimeter of the kite?
    Answer: 40 inches When you encounter a question about the perimeter of a kite, remember that a kite is a special quadrilateral with a unique property: it has two pairs of adjacent sides that are equal in length. To find the perimeter, you need to add up all four sides. Since the kite has two pairs of equal adjacent sides, you can think of this as: Perimeter = 2(length of first pair) + 2(length of second pair). In this problem, one pair of adjacent sides is 12 inches each, and the other pair is 8 inches each. So the perimeter is: $2(12) + 2(8) = 24 + 16 = 40$ inches. Looking at the wrong answers: Answer B (32 inches) might result from incorrectly calculating $12 + 8 + 12$ or making an arithmetic error. Answer C (28 inches) could come from adding $12 + 8 + 8$ or miscounting the sides. Answer D (24 inches) is what you'd get if you only counted two sides instead of all four, perhaps by calculating $2(12)$ and forgetting the other pair entirely. The correct answer is A) 40 inches. Study tip: For any polygon perimeter question, always count all the sides systematically. With kites specifically, remember the "2 + 2" pattern: two sides of one length plus two sides of another length. This prevents you from accidentally undercounting or making arithmetic mistakes when you're working quickly.
14. The perimeter of a rectangle is 56 meters. If the length is 4 meters more than twice the width, what is the width of the rectangle?
    Answer: 8 meters When you encounter a rectangle problem with constraints on dimensions, you're dealing with a system of equations. Set up variables and translate the word problem into mathematical relationships. Let's call the width $w$ and the length $l$. The problem gives us two key pieces of information: the perimeter is 56 meters, and the length is 4 meters more than twice the width. From the perimeter formula, we get: $2l + 2w = 56$, which simplifies to $l + w = 28$. From the length relationship: $l = 2w + 4$. Substituting the second equation into the first: $(2w + 4) + w = 28$. This becomes $3w + 4 = 28$, so $3w = 24$, giving us $w = 8$ meters. Let's verify: if width is 8, then length is $2(8) + 4 = 20$. The perimeter is $2(20) + 2(8) = 56$ ✓ Choice A (8 meters) is correct. Choice B (10 meters) would give a length of 24 and perimeter of 68 - this comes from incorrectly setting up the length equation as $l = 2w + 2$. Choice C (12 meters) results from solving $l + w = 28$ and $l = 2w$ (forgetting the "+4" part), giving length 24 and perimeter 72. Choice D (16 meters) might come from confusing which dimension should be larger or misreading the constraint. Always define your variables clearly, write out both constraint equations, and substitute carefully. Double-check by plugging your answer back into the original conditions.
15. Use the figure. The shaded region shows a rectangular courtyard with a circular fountain removed from its center. If the courtyard is 24 feet by 16 feet and the fountain has a radius of 4 feet, what is the perimeter of the shaded region?
    Answer: $(80 + 8\pi)$ feet The perimeter of the shaded region includes both the outer rectangular boundary and the inner circular boundary. The rectangle's perimeter is $2(24 + 16) = 80$ feet. The circle's circumference is $2\pi r = 2\pi(4) = 8\pi$ feet. Total perimeter is $(80 + 8\pi)$ feet. Choice B uses $4\pi$ instead of $8\pi$. Choice C ignores the circular boundary. Choice D uses wrong rectangular perimeter. Choice E includes only the circular part.
16. A regular pentagon and a regular octagon have the same side length. If the pentagon has a perimeter of 45 centimeters, what is the perimeter of the octagon?
    Answer: 72 centimeters When you encounter problems about regular polygons with the same side length, focus on the relationship between the number of sides and perimeter. A regular polygon has all sides equal, so perimeter equals the number of sides times the side length. Let's find the side length of the pentagon first. Since the pentagon has 5 equal sides and a perimeter of 45 centimeters, each side length is $45 ÷ 5 = 9$ centimeters. Now we can find the octagon's perimeter. The octagon has 8 sides, each with the same 9-centimeter length as the pentagon. Therefore, the octagon's perimeter is $8 × 9 = 72$ centimeters. Looking at the wrong answers: Choice B (64 centimeters) might result from incorrectly thinking the side length is 8 centimeters, then multiplying $8 × 8 = 64$. Choice C (56 centimeters) could come from using 7 as the side length instead of 9, giving $8 × 7 = 56$. Choice D (48 centimeters) might occur if you mistakenly calculated the pentagon's side length as 6 centimeters (perhaps dividing 45 by something other than 5), then computed $8 × 6 = 48$. The correct answer is A) 72 centimeters. Strategy tip: For regular polygon problems, always identify what information you have and what you need. When polygons share the same side length, use one polygon's known perimeter to find that side length, then apply it to calculate the other polygon's perimeter. Remember: perimeter = number of sides × side length.
17. A trapezoid has parallel sides of length 12 cm and 18 cm, and non-parallel sides of length 8 cm and 10 cm. What is the perimeter of the trapezoid?
    Answer: 48 cm When you encounter perimeter problems involving polygons like trapezoids, remember that perimeter is simply the sum of all sides, regardless of the shape's other properties or classifications. A trapezoid is defined by having exactly one pair of parallel sides, but for perimeter calculations, you treat it like any polygon - just add up every side length. Here, you have four sides: the two parallel sides measuring 12 cm and 18 cm, plus the two non-parallel sides measuring 8 cm and 10 cm. The perimeter calculation is straightforward: $12 + 18 + 8 + 10 = 48$ cm, making choice A correct. Let's examine why the other answers are wrong. Choice B (46 cm) likely results from a simple arithmetic error, perhaps miscalculating $12 + 18 = 30$ instead of $30$, then adding $8 + 10 = 18$ to get $48$, but making an error somewhere in the process. Choice C (38 cm) might come from accidentally using only three sides instead of all four, or from misreading one of the measurements. Choice D (30 cm) represents adding only the parallel sides ($12 + 18 = 30$), which reflects a fundamental misunderstanding of what perimeter means. The key strategy here is recognizing that perimeter problems are usually straightforward addition, regardless of the polygon type. Don't let geometric terminology like "trapezoid," "parallel sides," or "non-parallel sides" distract you from the basic task: add all the side lengths together. Always double-check that you've included every side in your calculation.
18. Based on the figure shown, what is the perimeter of the hexagonal garden plot? (Note: figure is not drawn to scale)
    Answer: 84 feet Adding all six sides: $15 + 12 + 18 + 14 + 13 + 12 = 84$ feet. Choice B results from omitting one of the 12-foot sides. Choice C results from calculation error. Choice D results from omitting the 18-foot side. Choice E results from omitting multiple sides.
19. An isosceles triangle has two sides of length 13 inches each and a perimeter of 35 inches. What is the length of the third side?
    Answer: 9 inches When you encounter an isosceles triangle problem, remember that an isosceles triangle has exactly two sides of equal length. The key is using the perimeter formula: perimeter equals the sum of all three sides. Given information: two sides are 13 inches each, and the total perimeter is 35 inches. To find the third side, set up the equation: $13 + 13 + \text{third side} = 35$. This simplifies to $26 + \text{third side} = 35$, so the third side equals $35 - 26 = 9$ inches. Let's examine why each answer choice is right or wrong: A) 9 inches is correct, as shown by our calculation above. B) 11 inches would give a perimeter of $13 + 13 + 11 = 37$ inches, which exceeds the given perimeter of 35 inches. C) 13 inches might seem tempting since you already know two sides are 13 inches, but this would create an equilateral triangle (all sides equal) with perimeter $13 + 13 + 13 = 39$ inches, not 35 inches. D) 15 inches would result in a perimeter of $13 + 13 + 15 = 41$ inches, which is far too large. Study tip: Always double-check your answer by substituting back into the original conditions. Here, verify that $13 + 13 + 9 = 35$ inches, and confirm the triangle is indeed isosceles (exactly two equal sides). This verification step catches calculation errors and ensures you haven't misread the problem.
20. A regular decagon (10-sided polygon) has a perimeter of 140 inches. A square is constructed so that its perimeter equals the perimeter of the decagon. What is the side length of the square?
    Answer: 35 inches When you encounter problems involving regular polygons and perimeter comparisons, remember that a regular polygon has all sides equal in length, which makes finding individual side lengths straightforward. Start with the regular decagon. Since it has 10 equal sides and a perimeter of 140 inches, each side length is $140 ÷ 10 = 14$ inches. The square's perimeter must equal the decagon's perimeter, so the square also has a perimeter of 140 inches. For a square with four equal sides, if the perimeter is 140 inches, then each side length is $140 ÷ 4 = 35$ inches. This confirms that A) 35 inches is correct. Looking at the wrong answers: B) 28 inches likely comes from incorrectly calculating $140 ÷ 5 = 28$, perhaps confusing the number of sides. C) 25 inches might result from the error $140 ÷ 5.6 ≈ 25$, which has no clear mathematical basis in this problem. D) 20 inches could come from calculating $140 ÷ 7 = 20$, again using an incorrect divisor. The key strategy here is to work systematically: first find what you need from the given shape (the decagon's perimeter), then apply that information to find what the question asks for (the square's side length). Always double-check that you're dividing the perimeter by the correct number of sides for each polygon. Regular polygon problems often test whether you can keep track of different shapes' properties simultaneously.

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