GED Math Quiz: Geometry 3d

What this quiz covers

This quiz focuses on Geometry 3d, giving you a quick way to practice the rules, question types, and explanations that matter most for GED Math.

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. $48$
2. $144$ (correct answer)
3. $216$
4. $288$

Explanation: Sphere volume: $\tfrac{4}{3}\pi(6)^3=288\pi$. Cylinder volume: $\pi(1)^2(2)=2\pi$. Number: $288\pi/2\pi=144$. A forgets the $\tfrac{4}{3}$ factor. C uses sphere volume formula $\tfrac{4}{3}\pi r^3$ but divides wrongly. D doubles the answer.

Question 2

1. $754$
2. $888$ (correct answer)
3. $1,021$
4. $1,155$

Explanation: Cylinder volume: $\pi(4)^2(15)=240\pi$. Hemisphere volume: $\frac{1}{2}\cdot\frac{4}{3}\pi(4)^3=\frac{128\pi}{3}$. Total: $240\pi+\frac{128\pi}{3}=\frac{848\pi}{3}\approx 888$. Choice A uses only the cylinder. Choice C adds a full sphere instead of a hemisphere. Choice D adds a cone with height 4 instead of a hemisphere.

Question 3

1. The volume is unchanged.
2. The volume is multiplied by $2$. (correct answer)
3. The volume is multiplied by $4$.
4. The volume is multiplied by $\tfrac{1}{2}$.

Explanation: Original volume $V=\tfrac{1}{3}\pi r^2 h$. New volume: $\tfrac{1}{3}\pi(2r)^2(h/2)=\tfrac{1}{3}\pi\cdot 4r^2\cdot h/2=2V$. A assumes effects cancel (forgetting radius is squared). C forgets to halve the height. D reverses the logic.

Question 4

1. $56.52$
2. $169.56$ (correct answer)
3. $226.08$
4. $678.24$

Explanation: Full volume: $\pi(3)^2(8)=72\pi\approx 226.08$. Three-quarters: $0.75(226.08)=169.56$. A takes $\tfrac{1}{4}$ instead of $\tfrac{3}{4}$. C gives full tank volume. D uses diameter 6 as radius.

Question 5

1. $477$ (correct answer)
2. $523$
3. $816$
4. $1,000$

Explanation: Cube volume: $1000$. Sphere radius is half the edge = $5$. Sphere volume: $\tfrac{4}{3}\pi(5)^3\approx 523.6$. Empty space: $1000-523.6\approx 476.4\approx 477$. B gives the sphere volume. C uses radius 10. D gives cube volume only.

Question 6

1. $60\pi$
2. $65\pi$ (correct answer)
3. $90\pi$
4. $25\pi$

Explanation: Slant height: $\ell=\sqrt{5^2+12^2}=\sqrt{169}=13$. Lateral area: $\pi r\ell=\pi(5)(13)=65\pi$. A uses height 12 as slant. C uses diameter. D is base area only.

Question 7

1. $376.8$
2. $427.04$ (correct answer)
3. $477.28$
4. $527.52$

Explanation: Radius = $4$ cm. Lateral area: $2\pi rh=2\pi(4)(15)=120\pi\approx 376.8$. Top circle: $\pi(4)^2=16\pi\approx 50.24$. Total: $376.8+50.24=427.04$. A omits the top. C includes both top and bottom. D uses diameter as radius for one term.

Question 8

1. $6$
2. $9$ (correct answer)
3. $12$
4. $18$

Explanation: Hemisphere volume: $\tfrac{1}{2}\cdot\tfrac{4}{3}\pi(6)^3=\tfrac{2}{3}\pi(216)=144\pi$. Set equal to cylinder volume: $\pi(4)^2 h=16\pi h=144\pi$, so $h=9$. A uses ratio $6/...$incorrectly. C uses full sphere formula. D forgets to square the cylinder radius.

Question 9

1. $120\sqrt{3}$
2. $240\sqrt{3}$ (correct answer)
3. $480\sqrt{3}$
4. $960\sqrt{3}$

Explanation: Base area: $\tfrac{3\sqrt{3}}{2}(4)^2=24\sqrt{3}$. Volume: $24\sqrt{3}\cdot 10=240\sqrt{3}$. A uses half the base area. C doubles the answer (perhaps confused with surface area). D uses $3\sqrt{3}s^2$ (forgot the $\tfrac{1}{2}$).

Question 10

1. $48\pi$ cubic feet (correct answer)
2. $192\pi$ cubic feet
3. $144\pi$ cubic feet
4. $64\pi$ cubic feet

Explanation: First, find the total volume: $V = \pi r^2 h = \pi(4)^2(12) = 192\pi$ cubic feet. Currently filled to 75%, so current volume is $0.75 \times 192\pi = 144\pi$ cubic feet. Water needed: $192\pi - 144\pi = 48\pi$ cubic feet. Choice B is the total volume, choice C is the current volume, and choice D incorrectly uses radius instead of radius squared in the calculation.

Question 11

1. $2\,160\text{ in}^3$ (correct answer)
2. $1\,440\text{ in}^3$
3. $600\text{ in}^3$
4. $360\text{ in}^3$

Explanation: When you encounter a problem asking for the volume of a rectangular prism (or box), you're working with one of the most fundamental 3D geometry formulas. Volume measures how much space is inside a three-dimensional object, and for rectangular prisms, you simply multiply length × width × height. For this shipping box, you have all three dimensions: 18 inches long, 12 inches wide, and 10 inches high. Multiply these together: $18 × 12 × 10 = 2{,}160 \text{ cubic inches}$. Notice that volume is always expressed in cubic units ($\text{in}^3$, $\text{ft}^3$, etc.) because you're multiplying three linear measurements. Let's examine why the other answers are incorrect. Answer B ($1{,}440\text{ in}^3$) results from multiplying only two dimensions correctly and making an error with the third—perhaps calculating $18 × 12 × 8$ instead of $18 × 12 × 10$. Answer C ($600\text{ in}^3$) comes from adding the dimensions instead of multiplying them ($18 + 12 + 10 = 40$, then possibly multiplying by something incorrectly). Answer D ($360\text{ in}^3$) appears to come from multiplying only some of the dimensions, like $18 × 10 = 180$, then doubling it. Remember: for any rectangular prism volume problem, always multiply all three dimensions—length × width × height. Double-check that you're multiplying (not adding) and that your final answer includes cubic units. This formula appears frequently on the GED, so master it completely.

Question 12

1. $1\,510\text{ cm}^2$
2. $477\text{ cm}^2$ (correct answer)
3. $380\text{ cm}^2$
4. $301\text{ cm}^2$

Explanation: When you encounter a surface area problem for a cylinder, you need to visualize what surfaces make up the complete shape: two circular bases (top and bottom) plus the curved side that wraps around. The surface area formula for a cylinder is $SA = 2\pi r^2 + 2\pi rh$, where the first term represents the two circular bases and the second term represents the curved lateral surface. With $r = 4$ cm and $h = 15$ cm:

• Area of both bases: $2\pi r^2 = 2(3.14)(4^2) = 2(3.14)(16) = 100.48$ cm²
• Lateral surface area: $2\pi rh = 2(3.14)(4)(15) = 376.8$ cm²
• Total surface area: $100.48 + 376.8 = 477.28$ cm²

Question 13

1. $235\text{ cm}^3$ (correct answer)
2. $471\text{ cm}^3$
3. $706\text{ cm}^3$
4. $848\text{ cm}^3$

Explanation: When you encounter volume problems involving three-dimensional shapes, you need to identify the shape and apply the correct formula. This cone problem requires the volume formula for a cone: $V = \frac{1}{3}\pi r^2 h$. Let's substitute the given values: radius = 5 cm, height = 9 cm, and $\pi = 3.14$. $V = \frac{1}{3} \times 3.14 \times 5^2 \times 9$ $V = \frac{1}{3} \times 3.14 \times 25 \times 9$ $V = \frac{1}{3} \times 706.5$ $V = 235.5 \text{ cm}^3$ Rounding to the nearest whole number gives us $235 \text{ cm}^3$, which is answer A. Now let's examine why the other answers are wrong. Answer B ($471 \text{ cm}^3$) represents a common error where students forget to divide by 3 in the cone formula, essentially calculating $\frac{2}{3}\pi r^2 h$ instead. Answer C ($706 \text{ cm}^3$) occurs when students completely omit the $\frac{1}{3}$ factor and use the cylinder formula $\pi r^2 h$ instead. Answer D ($848 \text{ cm}^3$) suggests using an incorrect formula altogether, possibly confusing cone volume with surface area calculations. The key strategy for cone volume problems is remembering that a cone's volume is exactly one-third of a cylinder with the same base and height. Always double-check that you've included the $\frac{1}{3}$ factor in your calculation, as this is the most frequent mistake students make on these problems.

Question 14

1. $729\text{ cm}^3$ (correct answer)
2. $216\text{ cm}^3$
3. $125\text{ cm}^3$
4. $64\text{ cm}^3$

Explanation: This problem tests your understanding of surface area and volume formulas for cubes, and how to work backwards from given information to find what you need. A cube has 6 identical square faces. If each face has side length $s$, then each face has area $s^2$, making the total surface area $6s^2$. Since the surface area is $486\text{ cm}^2$, you can set up the equation: $6s^2 = 486$. Dividing both sides by 6 gives $s^2 = 81$, so $s = 9\text{ cm}$. Now that you know the side length, you can find the volume using $V = s^3 = 9^3 = 729\text{ cm}^3$. This confirms that A is correct. Looking at the wrong answers: B ($216\text{ cm}^3$) equals $6^3$, which suggests someone incorrectly used $s = 6$ as the side length—this comes from mistakenly thinking $s^2 = 36$ instead of $s^2 = 81$. C ($125\text{ cm}^3$) equals $5^3$, indicating the error of using $s = 5$. D ($64\text{ cm}^3$) equals $4^3$, showing the mistake of using $s = 4$. These wrong answers likely result from computational errors when solving $6s^2 = 486$ or from incorrectly taking the square root of 81. Remember: when working with cube problems, always identify what information you're given and what formulas connect that to what you need to find. Surface area problems often require you to find the side length first, then use it to calculate volume.

Question 15

1. $113\text{ cm}^3$ (correct answer)
2. $226\text{ cm}^3$
3. $452\text{ cm}^3$
4. $904\text{ cm}^3$

Explanation: When you encounter a sphere volume problem, you're working with three-dimensional geometry. The key is identifying what information you have and applying the correct volume formula. For any sphere, the volume formula is $V = \frac{4}{3}\pi r^3$, where $r$ is the radius. Since the problem gives you a diameter of 6 cm, you first need to find the radius: $r = \frac{6}{2} = 3$ cm. Now substitute into the formula: $V = \frac{4}{3} \times 3.14 \times 3^3 = \frac{4}{3} \times 3.14 \times 27$. Calculate step by step: $3.14 \times 27 = 84.78$, then $\frac{4}{3} \times 84.78 = \frac{339.12}{3} = 113.04$ cm³. Rounding gives us 113 cm³. Looking at the wrong answers: Answer B (226 cm³) appears to result from forgetting the $\frac{4}{3}$ coefficient and just calculating $\pi r^3$, then doubling it. Answer C (452 cm³) likely comes from using the diameter instead of radius in the formula, calculating $\frac{4}{3}\pi \times 6^3$ but making arithmetic errors. Answer D (904 cm³) represents using the full diameter cubed without the proper fractional coefficient. Study tip: Always write down the sphere volume formula first, then carefully identify whether you're given radius or diameter. Most errors on sphere problems come from confusing these two measurements or forgetting the $\frac{4}{3}$ coefficient that makes spheres different from cylinders.