Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Sign up

Log in

Opening subject page...

Loading your content

Practice

  • All Subjects
  • Algebra Flashcards
  • SAT Math Practice Tests
  • Math Question of the Day
  • Live Classes
  • On-Demand Courses

Varsity Tutors

  • Find a Tutor
  • Test Prep
  • Online Classes
  • K-12 Learning
  • College Search
  • VarsityTutors.com

© 2026 Varsity Tutors. All rights reserved.

← Back to Learn by Concept

Geometry · Learn by Concept

Geometry Help: Solving Problems With Volume Formulas

Review real example questions for Solving Problems With Volume Formulas in Geometry.

Question 1 / 10

0 of 10 answered

A cylindrical can has diameter 10 cm10\text{ cm}10 cm and height 12 cm12\text{ cm}12 cm. What is the volume of the can in cubic centimeters?

Select an answer to continue

All questions

Question 1

A cylindrical can has diameter 10 cm10\text{ cm}10 cm and height 12 cm12\text{ cm}12 cm. What is the volume of the can in cubic centimeters?

  1. 600π cm3600\pi\text{ cm}^3600π cm3
  2. 120π cm3120\pi\text{ cm}^3120π cm3
  3. 300π cm3300\pi\text{ cm}^3300π cm3 (correct answer)
  4. 240π cm3240\pi\text{ cm}^3240π cm3

Explanation: This problem involves finding the volume of a cylindrical can given its diameter. The solid is a right circular cylinder with diameter 10 cm and height 12 cm. Since the volume formula V = πr²h requires radius, we must first convert: radius = diameter/2 = 10/2 = 5 cm. Applying the formula: V = π(5)²(12) = π(25)(12) = 300π cubic centimeters. This represents the can's total capacity. A common mistake is using diameter directly in the formula, giving π(10)²(12) = 1200π, which is four times too large. Always convert diameter to radius before applying cylinder volume formulas.

Question 2

A concrete pillar is a cylinder with diameter 12 ft12\text{ ft}12 ft and height 5 ft5\text{ ft}5 ft. What is the volume of the solid?

  1. 180π ft3180\pi\text{ ft}^3180π ft3 (correct answer)
  2. 720π ft3720\pi\text{ ft}^3720π ft3
  3. 360π ft3360\pi\text{ ft}^3360π ft3
  4. 180π ft2180\pi\text{ ft}^2180π ft2

Explanation: This problem requires finding the volume of a concrete pillar. The solid is a cylinder with diameter 12 ft and height 5 ft. The volume formula for a cylinder is V = πr²h, but we must first convert diameter to radius: r = 12/2 = 6 ft. Applying the formula: V = π(6)²(5) = π(36)(5) = 180π ft³. The volume represents the amount of concrete needed to form the pillar. A common mistake is using diameter directly in the formula instead of radius, which would give π(12)²(5) = 720π ft³. Always convert diameter to radius by dividing by 2 before applying cylinder volume formulas.

Question 3

An ice cream cone is a right circular cone with radius 4 cm4\text{ cm}4 cm and height 12 cm12\text{ cm}12 cm. Which calculation correctly applies the volume formula for the cone?

  1. π(4)2(12)\pi(4)^2(12)π(4)2(12)
  2. 13π(4)2(12)\frac{1}{3}\pi(4)^2(12)31​π(4)2(12) (correct answer)
  3. 4π(12)4\pi(12)4π(12)
  4. 2π(4)(12)2\pi(4)(12)2π(4)(12)

Explanation: Solving problems with volume formulas involves calculating the space occupied by three-dimensional solids using appropriate mathematical expressions. The solid in this problem is a right circular cone. The correct volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. Applying the formula with r = 4 cm and h = 12 cm gives the expression (1/3)π(4)²(12), matching choice B. This expression correctly computes the volume by accounting for the conical shape's one-third factor of a cylinder's volume. A common distractor misconception is using the cylinder volume formula without the one-third, as in choice A, which overestimates the volume. To transfer this strategy, always identify the solid as a cone before selecting and applying the volume formula.

Question 4

A solid metal sphere has radius 6 cm6\text{ cm}6 cm. Which calculation correctly applies the volume formula?

  1. (4π)(62)(4\pi)(6^2)(4π)(62)
  2. 43π(63)\frac{4}{3}\pi(6^3)34​π(63) (correct answer)
  3. π(62)(6)\pi(6^2)(6)π(62)(6)
  4. 13π(62)(6)\frac{1}{3}\pi(6^2)(6)31​π(62)(6)

Explanation: This problem asks which calculation correctly applies the volume formula for a sphere. The solid is a sphere with radius 6 cm. The volume formula for a sphere is V = (4/3)πr³, where r is the radius. The correct calculation is V = (4/3)π(6³) = (4/3)π(216). This gives the volume in cubic centimeters. Option C represents the formula for a cylinder (πr²h), which is incorrect for a sphere. To solve volume problems accurately, first identify the three-dimensional shape before selecting the appropriate formula.

Question 5

A cylindrical candle has radius 5 cm5\text{ cm}5 cm and height 12 cm12\text{ cm}12 cm. Which expression represents the volume?

  1. 2π(5)(12)2\pi(5)(12)2π(5)(12)
  2. π(52)(12)\pi(5^2)(12)π(52)(12) (correct answer)
  3. 13π(52)(12)\frac{1}{3}\pi(5^2)(12)31​π(52)(12)
  4. 43π(53)\frac{4}{3}\pi(5^3)34​π(53)

Explanation: This problem asks which expression represents the volume of a cylindrical candle. The solid is a cylinder with radius 5 cm and height 12 cm. The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height. The correct expression is π(5²)(12) or π(25)(12). This formula gives the volume in cubic centimeters. Option A represents the lateral surface area formula (2πrh), not volume. To find volume, always use formulas that multiply three dimensions or include squared terms for circular shapes.

Question 6

A cone and a cylinder have the same base radius and height. If the volume of the cylinder is 432π432\pi432π cubic centimeters, what is the volume of the cone?

  1. 72π72\pi72π cubic centimeters
  2. 144π144\pi144π cubic centimeters (correct answer)
  3. 216π216\pi216π cubic centimeters
  4. 324π324\pi324π cubic centimeters

Explanation: The volume of a cylinder is Vcylinder=πr2hV_{cylinder} = \pi r^2 hVcylinder​=πr2h and the volume of a cone is Vcone=13πr2hV_{cone} = \frac{1}{3}\pi r^2 hVcone​=31​πr2h. Since they have the same base radius and height, Vcone=13Vcylinder=13×432π=144πV_{cone} = \frac{1}{3} V_{cylinder} = \frac{1}{3} \times 432\pi = 144\piVcone​=31​Vcylinder​=31​×432π=144π cubic centimeters. Choice A (72π72\pi72π) represents 16\frac{1}{6}61​ of the cylinder volume. Choice C (216π216\pi216π) represents 12\frac{1}{2}21​ of the cylinder volume. Choice D (324π324\pi324π) represents 34\frac{3}{4}43​ of the cylinder volume.

Question 7

A party cone is filled with candy. The cone has radius 6 cm6\text{ cm}6 cm and height 10 cm10\text{ cm}10 cm. Which calculation correctly applies the volume formula?

  1. V=π(6)2(10)V=\pi(6)^2(10)V=π(6)2(10)
  2. V=13π(6)2(10)V=\tfrac{1}{3}\pi(6)^2(10)V=31​π(6)2(10) (correct answer)
  3. V=43π(6)3V=\tfrac{4}{3}\pi(6)^3V=34​π(6)3
  4. V=2π(6)(10)+2π(6)2V=2\pi(6)(10)+2\pi(6)^2V=2π(6)(10)+2π(6)2

Explanation: This problem involves finding the volume of a party cone filled with candy. The solid is a cone with radius 6 cm and height 10 cm. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. The correct calculation is V = (1/3)π(6)²(10), which matches option B. This formula gives one-third the volume of a cylinder with the same base and height. Option A incorrectly uses the cylinder formula without the 1/3 factor, while option D uses the surface area formula. To solve volume problems correctly, first identify whether the solid is a cone, cylinder, or sphere before selecting the appropriate formula.

Question 8

A spherical balloon has radius 7 in7\text{ in}7 in. What is the volume of the solid?

  1. 43π(7)3 in3\tfrac{4}{3}\pi(7)^3\text{ in}^334​π(7)3 in3 (correct answer)
  2. π(7)2 in3\pi(7)^2\text{ in}^3π(7)2 in3
  3. 13π(7)2 in3\tfrac{1}{3}\pi(7)^2\text{ in}^331​π(7)2 in3
  4. 43π(7)3 in2\tfrac{4}{3}\pi(7)^3\text{ in}^234​π(7)3 in2

Explanation: This problem asks for the volume of a spherical balloon. The solid is a sphere with radius 7 inches. The volume formula for a sphere is V = (4/3)πr³, where r is the radius. Applying the formula: V = (4/3)π(7)³ = (4/3)π(343) in³. The correct answer includes the proper cubic units (in³) for volume. Option B incorrectly uses πr², which is the area of a circle, not the volume of a sphere, while option D has the wrong units (in² instead of in³). When calculating sphere volume, remember to cube the radius and multiply by (4/3)π, not just π.

Question 9

A cylindrical container with radius 5 cm and height 20 cm is filled with water to a depth of 15 cm. A solid sphere is completely submerged in the water, causing the water level to rise to exactly 18 cm. What is the radius of the sphere?

  1. 22543\sqrt[3]{\frac{225}{4}}34225​​ cm
  2. 75π3\sqrt[3]{\frac{75}{\pi}}3π75​​ cm
  3. 2254π3\sqrt[3]{\frac{225}{4\pi}}34π225​​ cm (correct answer)
  4. 3004π3\sqrt[3]{\frac{300}{4\pi}}34π300​​ cm

Explanation: The volume of water displaced equals the rise in water level times the base area of the cylinder. The water level rises from 15 cm to 18 cm, a rise of 3 cm. Volume displaced = π(5)2(3)=75π\pi(5)^2(3) = 75\piπ(5)2(3)=75π cubic cm. This equals the volume of the sphere: 43πr3=75π\frac{4}{3}\pi r^3 = 75\pi34​πr3=75π. Solving: 43r3=75\frac{4}{3}r^3 = 7534​r3=75, so r3=75×34=2254r^3 = \frac{75 \times 3}{4} = \frac{225}{4}r3=475×3​=4225​, giving r=22543r = \sqrt[3]{\frac{225}{4}}r=34225​​ cm. Choice A omits the π\piπ cancellation step. Choice B results from incorrectly setting 43πr3=75\frac{4}{3}\pi r^3 = 7534​πr3=75 instead of 75π75\pi75π. Choice D uses an incorrect volume calculation.

Question 10

An ice cream cone is a right circular cone with radius 4 cm4\text{ cm}4 cm and height 9 cm9\text{ cm}9 cm. Which expression represents the volume of the cone?

  1. 13π(42)(9)\frac{1}{3}\pi(4^2)(9)31​π(42)(9) (correct answer)
  2. π(42)(9)\pi(4^2)(9)π(42)(9)
  3. 43π(43)\frac{4}{3}\pi(4^3)34​π(43)
  4. 2π(4)(9)2\pi(4)(9)2π(4)(9)

Explanation: This problem asks for the volume expression of a cone-shaped ice cream cone. The solid is a right circular cone with radius 4 cm and height 9 cm. The volume formula for a cone is V = (1/3)πr²h, where r is the radius and h is the height. The correct expression is (1/3)π(4²)(9), which represents one-third of the volume of a cylinder with the same base and height. This formula accounts for the cone's tapering shape from base to apex. A common mistake is using the cylinder formula π(4²)(9), forgetting the factor of 1/3. Remember that a cone's volume is always one-third of a cylinder with the same dimensions.