HSPT Quantitative Quiz: Compare Geometric Figures

What this quiz covers

This quiz focuses on Compare Geometric Figures, giving you a quick way to practice the rules, question types, and explanations that matter most for HSPT Quantitative.

How to use this quiz

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.

Question 1

1. Regular hexagon (a) has the greatest perimeter by more than 4 units
2. Equilateral triangle (b) has the greatest perimeter by exactly 6 units (correct answer)
3. Regular pentagon (c) has the greatest perimeter by less than 3 units
4. All three polygons have perimeters within 3 units of each other

Explanation: When you encounter perimeter problems with regular polygons, remember that perimeter equals the number of sides times the side length. This tests your ability to work with different polygon types and compare results accurately. Let's calculate each perimeter systematically. The regular hexagon has 6 sides of length 2, so its perimeter is $6 \times 2 = 12$ units. The equilateral triangle has 3 sides of length 6, giving a perimeter of $3 \times 6 = 18$ units. The regular pentagon has 5 sides of length 3, so its perimeter is $5 \times 3 = 15$ units. The equilateral triangle has the greatest perimeter at 18 units. Comparing this to the others: 18 - 12 = 6 units greater than the hexagon, and 18 - 15 = 3 units greater than the pentagon. Since the triangle exceeds both by at least 3 units, and specifically exceeds the hexagon by exactly 6 units, answer B is correct. Now for the incorrect choices: A is wrong because the hexagon actually has the smallest perimeter, not the greatest. C is incorrect because the pentagon has the middle perimeter value, not the greatest. D fails because the polygons don't have perimeters within 3 units of each other—the triangle's perimeter exceeds the hexagon's by 6 units. For polygon perimeter problems, always write down the formula (number of sides × side length) and calculate each value before comparing. Don't try to estimate or reason through these mentally, as the specific numerical differences often determine the correct answer choice.

Question 2

1. Triangle (c) has the greatest height, and all heights differ by more than 2 units (correct answer)
2. Triangle (b) has the greatest height, but all heights are within 3 units of each other
3. Triangle (a) has the smallest height, and the heights form an arithmetic sequence
4. All three triangles have the same height since they have the same base length

Explanation: 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.

Question 3

1. Cylinder (b) has the largest volume, followed by (a), then (c) (correct answer)
2. Cylinder (a) has the largest volume, followed by (c), then (b)
3. Cylinder (c) has the largest volume, followed by (b), then (a)
4. All three cylinders have volumes within $10\pi$ cubic units of each other

Explanation: 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.

Question 4

1. Cube (c) has the largest surface area, followed by prism (a), then prism (b)
2. Prism (a) has the largest surface area, followed by cube (c), then prism (b)
3. Prism (b) has the largest surface area, followed by prism (a), then cube (c) (correct answer)
4. All three solids have surface areas within 20 square units of each other

Explanation: 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.

Question 5

1. All three figures have exactly the same area (correct answer)
2. Triangle (c) has the largest area, followed by trapezoid (a), then parallelogram (b)
3. Parallelogram (b) has the largest area, followed by triangle (c), then trapezoid (a)
4. Trapezoid (a) has the largest area, followed by triangle (c), then parallelogram (b)

Explanation: 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.

Question 6

1. Circle (a) has the largest area, followed by (c), then (b)
2. Circle (b) has the largest area, followed by (c), then (a)
3. Circle (c) has the largest area, followed by (b), then (a) (correct answer)
4. All three circles have exactly the same area

Explanation: 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.