Perimeter & Area - Accuplacer Qas

1. A rhombus has a perimeter of 48 cm and one diagonal of length 18 cm. If the diagonals of a rhombus are perpendicular bisectors of each other, what is the area of the rhombus?
   Answer: 108 square centimeters When you encounter a rhombus problem, remember that all sides are equal and the diagonals are perpendicular bisectors of each other. This creates four congruent right triangles inside the rhombus. Since the perimeter is 48 cm, each side length is $48 ÷ 4 = 12$ cm. You know one diagonal is 18 cm, so each half of this diagonal is 9 cm. To find the area, you need the length of the other diagonal. Using the Pythagorean theorem on one of the right triangles formed by half-diagonals and a side: $9^2 + b^2 = 12^2$, where $b$ is half the unknown diagonal. Solving: $81 + b^2 = 144$, so $b^2 = 63$ and $b = 3\sqrt{7}$ cm. The full second diagonal is $6\sqrt{7}$ cm. The area of a rhombus equals $\frac{1}{2} × d_1 × d_2 = \frac{1}{2} × 18 × 6\sqrt{7} = 54\sqrt{7} ≈ 108$ square centimeters. Choice A (216) likely comes from multiplying the diagonals without the $\frac{1}{2}$ factor. Choice B (144) might result from incorrectly using the side length squared as area. Choice C (162) could come from using an incorrect second diagonal length, perhaps assuming it's also 18 cm and calculating $\frac{1}{2} × 18 × 18 = 162$. The correct answer is D (108 square centimeters). For rhombus problems, always remember: find both diagonal lengths using the Pythagorean theorem if needed, then use the formula $Area = \frac{1}{2} × d_1 × d_2$.
2. The area of a square plot of land is 81 square meters. A second square plot has a perimeter that is double the perimeter of the first plot. What is the area of the second square plot in square meters?
   Answer: 324 First, find the dimensions of the first square. If the area is 81, the side length is $\sqrt{81} = 9$ meters. The perimeter of the first square is $4 \times 9 = 36$ meters. The perimeter of the second square is double this, so $2 \times 36 = 72$ meters. The side length of the second square is $72 \div 4 = 18$ meters. The area of the second square is $18^2 = 324$ square meters.
3. A rectangular poster has a length of $4\frac{1}{2}$ feet and a width of $2\frac{1}{3}$ feet. What is the area of the poster in square feet?
   Answer: $10\frac{1}{2}$ To find the area, multiply the length by the width. First, convert the mixed numbers to improper fractions: $4\frac{1}{2} = \frac{9}{2}$ and $2\frac{1}{3} = \frac{7}{3}$. Now multiply the fractions: $A = \frac{9}{2} \times \frac{7}{3} = \frac{63}{6}$. Simplify the fraction by dividing the numerator and denominator by 3: $\frac{21}{2}$. Convert back to a mixed number: $10\frac{1}{2}$ square feet.
4. A parallelogram has adjacent sides of lengths 12 cm and 8 cm, with an included angle of 60°. What is the area of the parallelogram?
   Answer: $48\sqrt{3}$ square centimeters Area of a parallelogram = base × height = ab sin θ, where a and b are adjacent sides and θ is the included angle. Area = 12 × 8 × sin(60°) = 96 × (√3/2) = 48√3 cm². Choice A results from using only one side length. Choice C results from using cos(60°) instead of sin(60°). Choice D results from not applying the sine function correctly.
5. The length of each side of a rectangular field is tripled. The area of the new, larger field is how many times the area of the original field?
   Answer: 9 Let the original length and width be $l$ and $w$, respectively. The original area is $A_{original} = l \times w$. The new length is $3l$ and the new width is $3w$. The new area is $A_{new} = (3l) \times (3w) = 9(l \times w) = 9 \times A_{original}$. Therefore, the new area is 9 times the original area.
6. A triangular garden plot has a side that measures 12 meters, and the height corresponding to this side is 8 meters. The other two sides of the plot measure 10 meters and 14 meters. What is the area of the garden plot in square meters?
   Answer: 48 The area of a triangle is calculated by the formula $A = \frac{1}{2} \times \text{base} \times \text{height}$. The problem provides a base of 12 meters and its corresponding height of 8 meters. The lengths of the other two sides (10 and 14 meters) are extra information not needed to find the area. The area is $\frac{1}{2} \times 12 \times 8 = 6 \times 8 = 48$ square meters.
7. The circumference of a circular pond is $50\pi$ feet. What is the area of the pond in square feet?
   Answer: $625\pi$ The formula for the circumference of a circle is $C = 2\pi r$. We are given $C = 50\pi$, so $50\pi = 2\pi r$. Dividing both sides by $2\pi$ gives the radius, $r = 25$ feet. The formula for the area of a circle is $A = \pi r^2$. Substituting the radius, we get $A = \pi (25)^2 = 625\pi$ square feet.
8. A window is shaped like a rectangle with a semicircle on top. The rectangular part is 4 feet wide and 5 feet high. The diameter of the semicircle is the 4-foot width of the rectangle. What is the approximate perimeter of the entire window?
   Answer: $14 + 2\pi$ feet The perimeter of the window consists of three sides of the rectangle and the curved part of the semicircle. The sides are the two 5-foot heights and the 4-foot bottom width, for a total of $5 + 5 + 4 = 14$ feet. The top of the rectangle is not part of the window's perimeter. The diameter of the semicircle is 4 feet, so its radius is 2 feet. The circumference of a full circle is $C = 2\pi r$, so the length of the semicircle arc is $\frac{1}{2}(2\pi r) = \pi r = 2\pi$ feet. The total perimeter is the sum of the straight sides and the arc: $14 + 2\pi$ feet.
9. A sector of a circle has a central angle of 120° and a radius of 9 cm. What is the perimeter of this sector?
   Answer: $6\pi + 18$ cm When you encounter sector problems, remember that a sector's perimeter consists of two parts: the arc length and the two radii that form the "sides" of the sector. To find the arc length, use the formula: arc length = $\frac{\theta}{360°} \times 2\pi r$, where $\theta$ is the central angle in degrees. With a 120° angle and radius of 9 cm, the arc length is $\frac{120°}{360°} \times 2\pi \times 9 = \frac{1}{3} \times 18\pi = 6\pi$ cm. The perimeter also includes the two radii that extend from the center to each end of the arc. Since each radius is 9 cm, you add $9 + 9 = 18$ cm to the arc length. Therefore, the total perimeter is $6\pi + 18$ cm, making D correct. Let's examine why the other answers are wrong. Choice A ($18\pi + 18$) uses the full circumference ($2\pi r = 18\pi$) instead of just the arc portion. Choice B ($9\pi + 18$) incorrectly calculates the arc as $\frac{1}{2}$ of the circumference rather than $\frac{1}{3}$. Choice C ($12\pi + 18$) uses $\frac{2}{3}$ of the circumference, which would correspond to a 240° angle, not 120°. Study tip: Always break sector perimeter into two components—arc length plus two radii. Double-check your fraction: a 120° sector represents $\frac{120°}{360°} = \frac{1}{3}$ of the full circle, so the arc should be $\frac{1}{3}$ of the circumference.
10. A circular garden has a radius of 10 meters. A path of uniform width is to be built around it. If the area of the path is $44\pi$ square meters, what is the width of the path in meters?
    Answer: 2 Let $w$ be the width of the path. The radius of the garden is $r_{inner} = 10$. The radius of the garden plus the path is $r_{outer} = 10+w$. The area of the path is the difference between the outer area and the inner area: $A_{path} = \pi(r_{outer})^2 - \pi(r_{inner})^2$. We are given $A_{path} = 44\pi$, so $44\pi = \pi(10+w)^2 - \pi(10)^2$. Divide by $\pi$: $44 = (10+w)^2 - 100$. Add 100 to both sides: $144 = (10+w)^2$. Take the square root: $12 = 10+w$. Solving for $w$ gives $w=2$ meters.
11. The perimeter of a rectangle is 50 feet. The length of the rectangle is 5 feet greater than its width. What is the area of the rectangle in square feet?
    Answer: 150 Let $w$ be the width and $l$ be the length. We are given $l = w + 5$. The perimeter formula is $P = 2(l + w)$. Substituting the given values, we get $50 = 2((w + 5) + w)$, which simplifies to $50 = 2(2w + 5)$, then $25 = 2w + 5$. Solving for $w$ gives $2w = 20$, so $w = 10$ feet. The length is $l = 10 + 5 = 15$ feet. The area is $A = l \times w = 15 \times 10 = 150$ square feet.
12. A rectangular garden has a length that is 3 feet more than twice its width. If the perimeter of the garden is 42 feet, what is the area of the garden in square feet?
    Answer: 108 square feet Let w = width. Then length = 2w + 3. Perimeter = 2(w) + 2(2w + 3) = 2w + 4w + 6 = 6w + 6 = 42. Solving: 6w = 36, so w = 6 feet. Length = 2(6) + 3 = 15 feet. Area = 6 × 15 = 108 square feet. Choice B results from using the wrong perimeter formula. Choice C comes from incorrectly setting up the length relationship. Choice D results from calculation errors in the algebra.
13. A circular track has a radius of 50 meters. A rectangular warm-up area is constructed inside the track such that the rectangle's length is 60 meters and its width is 40 meters. What is the difference between the area of the circular track and the area of the rectangular warm-up area? (Use $\pi \approx 3.14$)
    Answer: 5,450 square meters Area of circular track = πr² = 3.14 × 50² = 3.14 × 2,500 = 7,850 m². Area of rectangular area = 60 × 40 = 2,400 m². Difference = 7,850 - 2,400 = 5,450 m². Choice A results from calculation errors with the circle area. Choice C gives only the circular area without subtracting. Choice D results from adding the areas instead of finding the difference.
14. A rectangular swimming pool that is 30 feet long and 15 feet wide is surrounded by a concrete walkway of a uniform width of 3 feet. What is the area of the walkway in square feet?
    Answer: 306 First, calculate the area of the pool: $30 \times 15 = 450$ square feet. The walkway adds 3 feet to each side, so the new total length is $30 + 3 + 3 = 36$ feet and the new total width is $15 + 3 + 3 = 21$ feet. The total area of the pool and walkway is $36 \times 21 = 756$ square feet. To find the area of the walkway, subtract the pool's area from the total area: $756 - 450 = 306$ square feet.
15. A rectangular painting measures 10 inches by 16 inches. It is placed in a wooden frame that has a uniform width of 2 inches. What is the area of the frame only, in square inches?
    Answer: 120 First, find the area of the painting itself: $10 \times 16 = 160$ square inches. The frame adds 2 inches to each side, so the total width becomes $10 + 2 + 2 = 14$ inches, and the total length becomes $16 + 2 + 2 = 20$ inches. The total area of the painting plus the frame is $14 \times 20 = 280$ square inches. To find the area of the frame only, subtract the painting's area from the total area: $280 - 160 = 120$ square inches.
16. A regular hexagon is inscribed in a circle with radius 10 cm. What is the perimeter of the hexagon?
    Answer: 60 centimeters In a regular hexagon inscribed in a circle, each side length equals the radius of the circle. This is because connecting the center to two adjacent vertices forms an equilateral triangle. Since the radius is 10 cm, each side is 10 cm. Perimeter = 6 × 10 = 60 cm. Choice A results from using 4 sides instead of 6. Choice B results from using 5 sides. Choice D results from incorrectly calculating side length as radius × √3.
17. A rectangular room has a length of 5 yards and a width of 9 feet. What is the area of the room in square feet?
    Answer: 135 To calculate the area in square feet, both dimensions must be in feet. We are given that 1 yard = 3 feet. Convert the length from yards to feet: $5 \text{ yards} \times 3 \text{ feet/yard} = 15$ feet. The width is already given as 9 feet. Now, calculate the area: $A = \text{length} \times \text{width} = 15 \text{ feet} \times 9 \text{ feet} = 135$ square feet.
18. The area of a rectangle is 96 square centimeters. If its width is 8 centimeters, what is its perimeter in centimeters?
    Answer: 40 The formula for the area of a rectangle is $A = l \times w$. Given $A = 96$ and $w = 8$, we can find the length: $96 = l \times 8$, so $l = 96 \div 8 = 12$ cm. The formula for the perimeter is $P = 2(l + w)$. Substituting the length and width, we get $P = 2(12 + 8) = 2(20) = 40$ cm.
19. A rectangular garden is 25 meters long and 10 meters wide. Fencing for the garden costs $12 per meter. A 2-meter wide gate will be installed, which does not require fencing material. What is the total cost for the fencing material required?
    Answer: $816 First, calculate the perimeter of the garden: $P = 2(25 + 10) = 2(35) = 70$ meters. The total length of material needed is the perimeter minus the width of the gate: $70 - 2 = 68$ meters. The total cost is the length of the fencing material multiplied by the cost per meter: $68 \text{ m} \times \$12/\text{m} = \$816$.
20. A parallelogram has a base of 15 cm and a corresponding height of 6 cm. The length of the adjacent, non-parallel side is 8 cm. What is the area of the parallelogram in square centimeters?
    Answer: 90 The area of a parallelogram is given by the formula $A = \text{base} \times \text{height}$. The length of the adjacent side (8 cm) is extra information not needed to calculate the area. Using the given base and height, the area is $15 \text{ cm} \times 6 \text{ cm} = 90$ square centimeters.

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