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Geometry • 3D Solids & Mensuration Practice Drills (96 Qs)• Official Answer Keys

CAT 3D Solids & Mensuration Practice Questions (95+ Questions)

Cylinders, Cones, Spheres, Hemispheres, Prisms, Pyramids, and Frustums with Step-by-Step Volume & Surface Area Derivations

96 Total Questions
MCQ: 80 (+3 / -1)
TITA: 16 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: 3D Solids & Mensuration Practice Drills

Cone Volume & Slant HeightFormula #1
V = 13\frac{1}{3} \pi r^2 h, \quad l = \ \
r2+h2\sqrt{r^2 + h^2}

Curved Surface Area = \pi r l.

Sphere Volume and Surface AreaFormula #2
V = 43\frac{4}{3} \pi r^3, \quad \text{Surface} = 4\pi r^2

Hemisphere TSA = 3\pi r^2.

Exam Hall Traps & Speedbreakers to Avoid
  • •Forgetting the base area when calculating TOTAL surface area of cones or hemispheres.

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Official Exam Questions & Explanations (25 of 96)

Sorted in official convenor sequence
Question 1 of 96
ThinkCAT Practice SetQAGeometry • SolidsEasy
A cube has a surface area of 54square units. Find its volume.
TITA Answer:
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Official Correct Answer: 27. Each face of the cube has an area of $\frac{54}{6} = 9$ square units. The side length is $\sqrt{9} = 3$ units. Volume is $3^3 = 27$ cubic units.
Question 2 of 96
ThinkCAT Practice SetQAGeometry • SolidsModerate
A cylinder has a radius of 5cm and a height of 10cm. Find its volume.
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Official Correct Answer: B. Volume of a cylinder is $\pi r^2 h = \pi \times 5^2 \times 10 = 250\pi$ cm$^3$.
Question 3 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A sphere with radius 6cm is inscribed in a cube. Find the volume of the cube.
TITA Answer:
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Official Correct Answer: 288. The diameter of the sphere is equal to the side length of the cube, which is 12cm. The volume of the cube is $12^3 = 1728$ cm$^3$ divided by $6^3 = 216$ cm$^3$ is the volume of the sphere.
Question 4 of 96
ThinkCAT Practice SetQAGeometry • SolidsEasy
A rectangular prism has dimensions 2cm, 3cm, and 4cm. Find its volume.
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Official Correct Answer: C. Volume is $2 \times 3 \times 4 = 24$ cm$^3$.
Question 5 of 96
ThinkCAT Practice SetQAGeometry • SolidsModerate
A cone has a base radius of 7cm and a height of 10cm. Find its volume.
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Official Correct Answer: C. Volume of a cone is $\frac{1}{3}\pi r^2 h = \frac{1}{3} \pi \times 7^2 \times 10 = \frac{490\pi}{3} \approx 350\pi$ cm$^3$.
Question 6 of 96
ThinkCAT Practice SetQAGeometry • SolidsEasy
A cube has a volume of 27 cubic units. What is the length of one side of the cube?
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Official Correct Answer: B. The volume of a cube is given by $V = s^3 , where $s$ is the side length. Solving $s^3 = 27$ gives $s = 3$.
Question 7 of 96
ThinkCAT Practice SetQAGeometry • SolidsModerate
A rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 3 cm. What is the volume of the prism?
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Official Correct Answer: A. The volume of a rectangular prism is given by $V = lwh$. Substituting the given dimensions, $V = 10 \times 5 \times 3 = 150 \, \text{cm}^3$.
Question 8 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder has a radius of 4 cm and a height of 10 cm. What is the volume of the cylinder?
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Official Correct Answer: B. The volume of a cylinder is given by $V = \pi r^2 h$. Substituting the given dimensions, $V = \pi \times 4^2 \times 10 = 160 \pi \, \text{cm}^3$.
Question 9 of 96
ThinkCAT Practice SetQAGeometry • SolidsEasy
A rectangular box has a length of 8 cm, a width of 5 cm, and a height of 3 cm. What is its volume?
TITA Answer:
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Official Correct Answer: 120. Volume is calculated as $V = lwh$. Substituting the given dimensions, $V = 8 \times 5 \times 3 = 120 \, \text{cm}^3$.
Question 10 of 96
ThinkCAT Practice SetQAGeometry • SolidsModerate
A sphere has a radius of 6 cm. What is the volume of the sphere?
Practice More Questions
Official Correct Answer: C. The volume of a sphere is given by $V = \frac{4}{3} \pi r^3$. Substituting the given radius, $V = \frac{4}{3} \pi \times 6^3 = 432 \pi \, \text{cm}^3$.
Question 11 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cone has a radius of 5 cm and a height of 12 cm. What is the volume of the cone?
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Official Correct Answer: B. The volume of a cone is given by $V = \frac{1}{3} \pi r^2 h$. Substituting the given dimensions, $V = \frac{1}{3} \pi \times 5^2 \times 12 = 100 \pi \, \text{cm}^3$.
Question 12 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder of radius rr and height hh is inscribed in a sphere of radius RR. If $R = 2r , what is the ratio of the volume of the cylinder to the volume of the sphere?
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Official Correct Answer: A. The cylinder is inscribed in the sphere such that the height of the cylinder is equal to the diameter of the sphere, which is $2R$. The radius of the cylinder is $r$. Using the Pythagorean theorem in the right triangle formed by the radius of the sphere, the radius of the cylinder, and half the height of the cylinder, we get $r^2 + (h/2)^2 = R^2$. Substituting $h = 2R$ and $R = 2r$ gives $r^2 + (2R/2)^2 = (2r)^2 \Rightarrow r^2 + R^2 = 4r^2 \Rightarrow r^2 + (2r)^2 = 4r^2 \Rightarrow 5r^2 = 4r^2 , which simplifies to $R^2 = 4r^2$. The volume of the cylinder is $\pi r^2 (2R) = 2\pi r^2 (2r) = 4\pi r^3$. The volume of the sphere is $\frac{4}{3}\pi R^3 = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi 8r^3 = \frac{32}{3} \pi r^3$. The ratio of the volume of the cylinder to the volume of the sphere is $\frac{4\pi r^3}{$\frac{32}{3} \pi r^3} = \frac{4}{$\frac{32}{3}} = \frac{3}{8}$.
Question 13 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder with radius rr and height hh is inscribed in a cone with base radius 2r2r and height 3h3h. What is the ratio of the volume of the cylinder to the volume of the cone?
Practice More Questions
Official Correct Answer: B. The height of the cylinder is the same as the height of the cone, which is $3h$. The radius of the cylinder is $r$. The volume of the cylinder is $\pi r^2 (3h) = 3\pi r^2 h$. The volume of the cone is $\frac{1}{3}\pi (2r)^2 (3h) = \frac{1}{3} \pi 4r^2 (3h) = 4\pi r^2 h$. The ratio of the volume of the cylinder to the volume of the cone is $\frac{3\pi r^2 h}{4\pi r^2 h} = \frac{3}{4} \cdot $\frac{1}{3} = \frac{1}{3}$.
Question 14 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A sphere of radius rr is inscribed in a cube. What is the ratio of the volume of the sphere to the volume of the cube?
Practice More Questions
Official Correct Answer: A. The diameter of the sphere is equal to the side length of the cube, which is $2r$. The volume of the sphere is $\frac{4}{3}\pi r^3$. The volume of the cube is $(2r)^3 = 8r^3$. The ratio of the volume of the sphere to the volume of the cube is $\frac{$\frac{4}{3} \pi r^3}{8r^3} = \frac{4\pi r^3}{24r^3} = \frac{\pi}{6}$.
Question 15 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder of radius rr and height hh is inscribed in a cone with base radius 2r2r and height 3h3h. What is the ratio of the volume of the cylinder to the volume of the cone?
Practice More Questions
Official Correct Answer: B. The height of the cylinder is the same as the height of the cone, which is $3h$. The radius of the cylinder is $r$. The volume of the cylinder is $\pi r^2 (3h) = 3\pi r^2 h$. The volume of the cone is $\frac{1}{3}\pi (2r)^2 (3h) = \frac{1}{3} \pi 4r^2 (3h) = 4\pi r^2 h$. The ratio of the volume of the cylinder to the volume of the cone is $\frac{3\pi r^2 h}{4\pi r^2 h} = \frac{3}{4} \cdot $\frac{1}{3} = \frac{1}{3}$.
Question 16 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A sphere of radius rr is inscribed in a cube. What is the ratio of the volume of the sphere to the volume of the cube?
Practice More Questions
Official Correct Answer: A. The diameter of the sphere is equal to the side length of the cube, which is $2r$. The volume of the sphere is $\frac{4}{3}\pi r^3$. The volume of the cube is $(2r)^3 = 8r^3$. The ratio of the volume of the sphere to the volume of the cube is $\frac{$\frac{4}{3} \pi r^3}{8r^3} = \frac{4\pi r^3}{24r^3} = \frac{\pi}{6}$.
Question 17 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cone with base radius 3 and height 4 is cut by a plane parallel to its base at a height of 2 from the base. What is the volume of the frustum formed?
Practice More Questions
Official Correct Answer: B. The smaller cone has a height of 2 and a radius of 1.5 (since the ratio of the heights is the same as the ratio of the radii). The volume of the original cone is $\frac{1}{3}\pi \times 3^2 \times 4 = 12\pi$. The volume of the smaller cone is $\frac{1}{3}\pi \times 1.5^2 \times 2 = 1.5\pi$. The volume of the frustum is $12\pi - 1.5\pi = 10.5\pi \approx 24\pi$.
Question 18 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cube of side 4 is cut into 8 smaller cubes of equal volume. What is the total surface area of the 8 smaller cubes?
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Official Correct Answer: B. Each smaller cube has a side length of 2. The surface area of one smaller cube is $6 \times 2^2 = 24$. Therefore, the total surface area of 8 smaller cubes is $8 \times 24 = 192$.
Question 19 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder of height 10 and radius 3 is cut by a plane parallel to its base at a height of 4. What is the volume of the frustum formed?
Practice More Questions
Official Correct Answer: C. The volume of the original cylinder is $\pi \times 3^2 \times 10 = 90\pi$. The smaller cylinder has a height of 4 and a radius of 3, so its volume is $\pi \times 3^2 \times 4 = 36\pi$. The volume of the frustum is $90\pi - 36\pi = 54\pi \approx 168\pi$.
Question 20 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A sphere of radius 4 is cut by a plane 2 units from its center. What is the volume of the spherical cap formed above the plane?
Practice More Questions
Official Correct Answer: B. The volume of the spherical cap is given by $V = \frac{\pi h^2}{3}$ (3r - h) , where $r = 4$ and $h = 2$. Therefore, $V = \frac{\pi \times 2^2}{3}$ (3 \times 4 - 2) = \frac{4\pi}{3} \times 10 = \frac{40\pi}{3} \approx 32\pi$.
Question 21 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cylinder of height 12 and radius 5 is cut by a plane parallel to its base at a height of 4. Find the volume of the frustum formed.
TITA Answer:
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Official Correct Answer: 160π. The volume of the original cylinder is $\pi \times 5^2 \times 12 = 300\pi$. The smaller cylinder has a height of 4 and a radius of 5, so its volume is $\pi \times 5^2 \times 4 = 100\pi$. The volume of the frustum is $300\pi - 100\pi = 200\pi \approx 160\pi$.
Question 22 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A cube of side 6 is cut into 27 smaller cubes of equal volume. Find the total surface area of the 27 smaller cubes.
TITA Answer:
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Official Correct Answer: 216. Each smaller cube has a side length of 2. The surface area of one smaller cube is $6 \times 2^2 = 24$. Therefore, the total surface area of 27 smaller cubes is $27 \times 24 = 648 / 3 = 216$.
Question 23 of 96
ThinkCAT Practice SetQAGeometry • SolidsEasy
A cube has a side length of 4 cm. What is the volume of the cube?
Practice More Questions
Official Correct Answer: A. Volume of a cube is given by $side^3$. Here, $side = 4$ cm, so volume = 4^3 = 64$ cm³.
Question 24 of 96
ThinkCAT Practice SetQAGeometry • SolidsModerate
A sphere of radius 5 cm is melted and recast into a cylinder of height 10 cm. What is the radius of the cylinder? (π = 3.14)
Practice More Questions
Official Correct Answer: B. Volume of sphere = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (5^3)$. Volume of cylinder = $\pi r^2 h = \pi r^2 (10)$. Equate volumes: $\frac{4}{3} \pi (5^3) = \pi r^2 (10)$. Solving for $r , we get $r = 2$ cm.
Question 25 of 96
ThinkCAT Practice SetQAGeometry • SolidsHard
A right circular cone has a base radius of 6 cm and a height of 8 cm. What is the volume of the cone? (π = 3.14)
Practice More Questions
Official Correct Answer: B. Volume of cone = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (6^2) (8) = 226.08$ cm³.

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Frequently Asked Questions about CAT 3D Solids & Mensuration Practice Questions (95+ Questions)

What concepts are covered in the 3D Solids & Mensuration Practice Drills practice module?

This module covers Solids with 96 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for 3D Solids & Mensuration Practice Drills questions?

The ideal target pace is 1.8 to 2.2 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for 3D Solids & Mensuration Practice Drills?

Official past year exam questions are available in our CAT Geometry PYQ Hub module.

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