Compare Geometric Figures - HSPT Quantitative

1. Rectangular prism (a) has dimensions 2×3×6, rectangular prism (b) has dimensions 1×4×9, and cube (c) has side length 3. Which comparison of surface areas is correct?
   Answer: Prism (b) has the largest surface area, followed by prism (a), then cube (c) When you encounter surface area problems involving different 3D shapes, you need to apply the specific formula for each shape and calculate carefully. For rectangular prisms, surface area equals $2(lw + lh + wh)$ where l, w, and h are the dimensions. For cubes, it's $6s^2$ where s is the side length. Let's calculate each surface area: Prism (a) with dimensions 2×3×6: $SA = 2(2×3 + 2×6 + 3×6) = 2(6 + 12 + 18) = 2(36) = 72$ square units Prism (b) with dimensions 1×4×9: $SA = 2(1×4 + 1×9 + 4×9) = 2(4 + 9 + 36) = 2(49) = 98$ square units Cube (c) with side length 3: $SA = 6(3^2) = 6(9) = 54$ square units From largest to smallest: prism (b) = 98, prism (a) = 72, cube (c) = 54. Choice A incorrectly places the cube first, likely assuming cubes always have larger surface areas than prisms. Choice B incorrectly ranks prism (a) as largest, possibly due to calculation errors with the middle-sized dimensions. Choice D is wrong because the surface areas span 44 square units (98 - 54), well beyond the 20-unit range suggested. Choice C correctly identifies the ranking: prism (b), then prism (a), then cube (c). Study tip: Don't assume shape determines size—always calculate. Rectangular prisms with one very small dimension (like the 1×4×9) often have surprisingly large surface areas due to their elongated faces.
2. Regular hexagon (a) has side length 2, equilateral triangle (b) has side length 6, and regular pentagon (c) has side length 3. Which has the greatest perimeter?
   Answer: Equilateral triangle (b) has the greatest perimeter by exactly 3 units Find each perimeter by multiplying the side length by the number of sides. Regular hexagon (a): 6 × 2 = 12 units. Equilateral triangle (b): 3 × 6 = 18 units. Regular pentagon (c): 5 × 3 = 15 units. The triangle has the greatest perimeter. To find how much greater, compare it with the next-largest perimeter, the pentagon: 18 − 15 = 3 units. Comparing the triangle with the hexagon gives 6 units, but the hexagon is the smallest shape, not the runner-up. The hexagon cannot have the greatest perimeter, since 12 is the smallest of the three. The pentagon is second, not first. The perimeters span from 12 to 18, which is 6 units, so they are not all within 3 units of each other. The equilateral triangle has the greatest perimeter by exactly 3 units.
3. Trapezoid (a) has parallel sides of length 4 and 8 with height 3, parallelogram (b) has base 6 and height 3, and triangle (c) has base 12 and height 3. Compare their areas.
   Answer: All three figures have exactly the same area When comparing areas of different geometric figures, you need to calculate each area using the appropriate formula and then compare the numerical results. Let's calculate each area systematically: Trapezoid (a): Use the formula $A = \frac{1}{2}(b_1 + b_2)h$ where $b_1$ and $b_2$ are the parallel sides and $h$ is the height. $A = \frac{1}{2}(4 + 8)(3) = \frac{1}{2}(12)(3) = 18$ Parallelogram (b): Use the formula $A = bh$ where $b$ is the base and $h$ is the height. $A = 6 \times 3 = 18$ Triangle (c): Use the formula $A = \frac{1}{2}bh$ where $b$ is the base and $h$ is the height. $A = \frac{1}{2}(12)(3) = 18$ All three figures have an area of 18 square units, making answer choice A correct. Answer choice B incorrectly suggests the triangle has the largest area at 18, followed by the trapezoid, then parallelogram. Answer choice C wrongly claims the parallelogram is largest, followed by triangle, then trapezoid. Answer choice D mistakenly states the trapezoid is largest, followed by triangle, then parallelogram. All these options fail to recognize that the areas are identical. Study tip: When comparing areas, always calculate each one completely before making comparisons. Don't assume that larger-looking dimensions automatically mean larger areas—the specific formulas and how the dimensions interact determine the final result.
4. Cylinder (a) has radius 2 and height 6, cylinder (b) has radius 3 and height 4, and cylinder (c) has radius 1 and height 12. Compare their volumes.
   Answer: Cylinder (b) has the largest volume, followed by (a), then (c) When you encounter cylinder volume problems, remember that the formula is $V = \pi r^2 h$. The key insight is that radius is squared, making it more influential than height in determining volume. Let's calculate each volume systematically: Cylinder (a): $V = \pi (2)^2 (6) = \pi \cdot 4 \cdot 6 = 24\pi$ Cylinder (b): $V = \pi (3)^2 (4) = \pi \cdot 9 \cdot 4 = 36\pi$ Cylinder (c): $V = \pi (1)^2 (12) = \pi \cdot 1 \cdot 12 = 12\pi$ Ranking from largest to smallest: cylinder (b) at $36\pi$, cylinder (a) at $24\pi$, and cylinder (c) at $12\pi$. Choice A correctly identifies this order: (b) largest, followed by (a), then (c). Choice B incorrectly places cylinder (a) first. While it has decent height, its smaller radius compared to cylinder (b) means less volume overall. Choice C wrongly suggests cylinder (c) is largest. Despite having the greatest height (12), its tiny radius of 1 severely limits its volume since radius gets squared in the formula. Choice D claims all volumes are within $10\pi$ of each other, but the difference between the largest ($36\pi$) and smallest ($12\pi$) is $24\pi$, far exceeding $10\pi$. Remember: when comparing cylinder volumes, pay special attention to the radius since it's squared. A cylinder with a larger radius often beats one with greater height, as demonstrated here where cylinder (b)'s radius of 3 trumps cylinder (c)'s height advantage.
5. Circle (a) has radius 3, circle (b) has diameter 8, and circle (c) has circumference $12\pi$. Which statement about their areas is correct?
   Answer: Circle (c) has the largest area, followed by (b), then (a) When comparing circle areas, you need to find each circle's radius first, since area formula is $A = \pi r^2$. For circle (a): radius = 3, so area = $\pi(3)^2 = 9\pi$ For circle (b): diameter = 8, so radius = 4, and area = $\pi(4)^2 = 16\pi$ For circle (c): circumference = $12\pi$. Using $C = 2\pi r$, we get $12\pi = 2\pi r$, so $r = 6$. Therefore area = $\pi(6)^2 = 36\pi$ Comparing the areas: circle (c) has $36\pi$, circle (b) has $16\pi$, and circle (a) has $9\pi$. So circle (c) is largest, followed by (b), then (a). Choice A incorrectly assumes circle (a) is largest—this happens when students confuse the given measurements without properly converting to compare areas. Circle (a) has the smallest radius when all are calculated. Choice B correctly identifies that circle (b) is larger than circle (a), but misses that circle (c) is actually the largest. Students might make this error by not fully working through the circumference-to-radius conversion for circle (c). Choice D is wrong because the circles clearly have different radii (3, 4, and 6), so their areas must be different. The correct answer is C. Strategy tip: Always convert all measurements to the same form (radius) before comparing circle areas. Watch for mixed units—diameter, radius, and circumference—in the same problem, as this is a common HSPT trap.
6. Isosceles triangle (a) has two sides of length 5 and base 6, isosceles triangle (b) has two sides of length 8 and base 6, and isosceles triangle (c) has two sides of length 10 and base 6. Which statement about their heights is true?
   Answer: Triangle (c) has the greatest height, and all heights differ by more than 2 units When you encounter isosceles triangles with the same base but different equal sides, you need to find how the height changes as those equal sides get longer. To find the height of an isosceles triangle, draw a line from the apex perpendicular to the base. This creates two right triangles, each with hypotenuse equal to one of the equal sides and base equal to half the original base (3 units here). Using the Pythagorean theorem: $h^2 + 3^2 = (\text{equal side})^2$ For triangle (a): $h^2 + 9 = 25$, so $h = 4$ For triangle (b): $h^2 + 9 = 64$, so $h = \sqrt{55} \approx 7.42$ For triangle (c): $h^2 + 9 = 100$, so $h = \sqrt{91} \approx 9.54$ Triangle (c) has the greatest height at about 9.54 units. The difference between the largest and smallest heights is approximately 5.54 units, which is indeed more than 2 units. This confirms answer A. Answer B is wrong because triangle (c), not (b), has the greatest height, and the heights don't stay within 3 units of each other. Answer C incorrectly claims the heights form an arithmetic sequence—the differences between consecutive heights aren't equal (about 3.42 and 2.12). Answer D reflects a common misconception that equal bases mean equal heights, ignoring how the equal sides affect the triangle's shape. Remember: longer equal sides in an isosceles triangle create a "taller" triangle, increasing the height even when the base stays constant.

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