3D Shapes - Advanced — Practice Quiz

A Geometry cheat sheet for 3D Shapes - Advanced — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Ellipsoid Volume

$$V = \frac{4}{3}\pi abc$$

Where: \(a, b, c\) = semi-axes

Torus Volume

$$V = 2\pi^2 Rr^2$$

Where: \(R\) = major radius, \(r\) = minor radius

Torus Surface Area

$$SA = 4\pi^2 Rr$$

Frustum Volume (Pyramid or Cone)

$$V = \frac{h}{3}(B_1 + B_2 + \sqrt{B_1 B_2})$$

Where: \(B_1, B_2\) = base areas

Regular Tetrahedron Volume

$$V = \frac{1}{12}a^3\sqrt{2}$$

Where: \(a\) = edge length

Cube

$$V = a^3, \quad SA = 6a^2, \quad d = a\sqrt{3}$$

Where: \(d\) = space diagonal

Practice quiz

  1. If a torus has a major radius $R = 5 \text{ cm}$ and a minor radius $r = 2 \text{ cm}$, what is its volume?

    • $20\pi^2 \text{ cm}^3$
    • $40\pi^2 \text{ cm}^3$
    • $80\pi^2 \text{ cm}^3$
    • $100\pi^2 \text{ cm}^3$

    Answer: $40\pi^2 \text{ cm}^3$

  2. Calculate the surface area of a torus with a major radius $R = 10 \text{ m}$ and a minor radius $r = 3 \text{ m}$.

    • $60\pi^2 \text{ m}^2$
    • $90\pi^2 \text{ m}^2$
    • $120\pi^2 \text{ m}^2$
    • $180\pi^2 \text{ m}^2$

    Answer: $120\pi^2 \text{ m}^2$

  3. An ellipsoid has semi-axes of lengths $a = 3 \text{ units}$, $b = 4 \text{ units}$, and $c = 5 \text{ units}$. What is its volume?

    • $20\pi \text{ cubic units}$
    • $40\pi \text{ cubic units}$
    • $60\pi \text{ cubic units}$
    • $80\pi \text{ cubic units}$

    Answer: $80\pi \text{ cubic units}$

  4. What is the volume of a regular tetrahedron with an edge length $a = 6 \text{ cm}$?

    • $6\sqrt{2} \text{ cm}^3$
    • $12\sqrt{2} \text{ cm}^3$
    • $18\sqrt{2} \text{ cm}^3$
    • $36\sqrt{2} \text{ cm}^3$

    Answer: $18\sqrt{2} \text{ cm}^3$

  5. A cube has an edge length of $a = 4 \text{ m}$. What are its volume and surface area?

    • $V = 16 \text{ m}^3, SA = 32 \text{ m}^2$
    • $V = 64 \text{ m}^3, SA = 96 \text{ m}^2$
    • $V = 64 \text{ m}^3, SA = 64 \text{ m}^2$
    • $V = 96 \text{ m}^3, SA = 64 \text{ m}^2$

    Answer: $V = 64 \text{ m}^3, SA = 96 \text{ m}^2$

  6. If a cube has an edge length of $a = 5 \text{ cm}$, what is the length of its space diagonal?

    • $5 \text{ cm}$
    • $5\sqrt{2} \text{ cm}$
    • $5\sqrt{3} \text{ cm}$
    • $10 \text{ cm}$

    Answer: $5\sqrt{3} \text{ cm}$

  7. A frustum of a pyramid has a height $h = 9 \text{ m}$ and base areas $B_1 = 4 \text{ m}^2$ and $B_2 = 16 \text{ m}^2$. Calculate its volume.

    • $60 \text{ m}^3$
    • $72 \text{ m}^3$
    • $84 \text{ m}^3$
    • $96 \text{ m}^3$

    Answer: $84 \text{ m}^3$

  8. A cube has a volume of $216 \text{ cm}^3$. What is its edge length?

    • $4 \text{ cm}$
    • $5 \text{ cm}$
    • $6 \text{ cm}$
    • $7 \text{ cm}$

    Answer: $6 \text{ cm}$

  9. Which of the following statements is true regarding a torus?

    • Its volume depends on the square of the major radius, while its surface area depends on the square of the minor radius.
    • Both its volume and surface area depend on the product of the major and minor radii.
    • Its volume is proportional to $Rr^2$, and its surface area is proportional to $Rr$.
    • Its volume is proportional to $R^2r$, and its surface area is proportional to $R^2r^2$.

    Answer: Its volume is proportional to $Rr^2$, and its surface area is proportional to $Rr$.

  10. Consider an ellipsoid with semi-axes $a=1, b=1, c=\frac{3}{4}\sqrt{2}$ and a regular tetrahedron with edge length $a=3$. Which shape has a larger volume?

    • The ellipsoid has a larger volume.
    • The tetrahedron has a larger volume.
    • Their volumes are equal.
    • Cannot be determined without more information.

    Answer: The ellipsoid has a larger volume.

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