In plain English: 8.G.C.9 is the Common Core grade 8 math standard that asks students to know the volume formulas for cylinders (V = πr²h), cones (V = (1/3)πr²h) and spheres (V = (4/3)πr³) and use them to solve real-world and mathematical problems. Students find volumes, work backward to a missing radius or height, and add the volumes of combined solids. It is part of Grade 8 Math geometry.
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. Also written as 8.G.9 · Official standard
Volume is the amount of space inside a solid, measured in cubic units such as cm³ or m³. In grade 7 (7.G.B.6), students found the volume of a prism as base area × height. This lesson extends that idea to three solids with round parts. A cylinder has two matching circles for bases, joined by a curved side, like a can. A cone has one circular base and a curved side that narrows to a single point, like a party hat. A sphere is a perfectly round ball: every point on it is the same distance, the radius r, from its center.
Students learn three formulas: cylinder V = πr²h (the circle area πr² times the height h), cone V = (1/3)πr²h (one third of the cylinder with the same base and height) and sphere V = (4/3)πr³. They see where the formulas come from with a pouring experiment and a picture of a cone, a sphere and a cylinder of the same size, whose volumes are in the ratio 1 : 2 : 3. They then use the formulas on real objects, find a missing height or radius, and add volumes of combined shapes. Every numerical answer on this page uses π ≈ 3.14. An exact answer "in terms of π" leaves π as a symbol, for example 100π cm³.
Learning Objectives
By the end of this lesson, students will be able to:
State the volume formulas for a cylinder, a cone and a sphere, and explain what r and h stand for in each
Explain why a cone holds one third of a cylinder with the same base and height, and how a sphere compares with a cylinder that fits around it
Find the volume of cylinders, cones and spheres in real-world and mathematical problems, using π ≈ 3.14 or leaving the answer in terms of π
Work backward from a volume to find a missing height or radius
Find the volume of a solid made of two or more of these shapes by adding their volumes
Prior Knowledge Required
Students should already be comfortable with:
Volume of a rectangular prism in cubic units, V = l × w × h 5.MD.C.5
The area of a circle, A = πr², and the link between radius and diameter (d = 2r) 7.G.B.4
Volume of a right prism as the area of its base times its height 7.G.B.6
Square roots and cube roots of perfect squares and perfect cubes, such as √16 = 4 and ∛27 = 3 8.EE.A.2
Show a can of food or a round tub. Ask students how they found the volume of a box in grade 5 and of a prism in grade 7.
Warm-Up Prompt
"A can has a circular bottom with a radius of 4 cm. Using π ≈ 3.14, what is the area of the bottom? Now picture thin circles of that size stacked 12 cm high. How much space do they fill?"
The bottom has area 3.14 × 4² = 3.14 × 16 = 50.24 cm². Each centimeter of height adds another 50.24 cm³, so 12 cm of height gives 50.24 × 12 = 602.88 cm³. Point out that this is the prism rule, base area × height, with a circle for the base. Activity 1 builds a paper cylinder of exactly this size.
Direct Instruction20 minutes
Introduce each formula with a real object, and have students copy the three formulas into a labeled chart with a sketch of each solid.
Cylinder: V = πr²h. The base is a circle with area πr², and h is the height, the distance between the two bases. It is the prism rule V = B × h with B = πr².
Cone: V = (1/3)πr²h. A cone with the same base and the same height as a cylinder holds exactly one third as much. The height h is measured straight down from the tip to the center of the base. The slant height, the length of the slanted side from the tip to the edge of the base, is not used (Diagram 2).
Sphere: V = (4/3)πr³. A sphere that just fits inside a cylinder (same radius, and the cylinder's height equals the sphere's diameter 2r) fills two thirds of it (Diagram 1). A hemisphere, half of a sphere, has volume (2/3)πr³.
Radius, not diameter: every formula uses the radius. If a problem gives the diameter (the distance across through the center), halve it first.
π ≈ 3.14 on this page: work out r²h or r³ first, then multiply by 3.14. For an exact answer, stop before multiplying and write the number in front of π, for example 240π cm³.
Working backward: if you know the volume, put it in the formula and undo the steps. To find r, you may need a square root (for r²) or a cube root (for r³, the number that multiplies by itself three times to make r³).
Combined solids: a composite solid is made of two or more simple solids. Find each volume and add them.
Cylinder (T2)
A cylindrical candle is 8 cm across and 15 cm tall. How much wax does it contain? Use π ≈ 3.14.
Equation: The diameter is 8 cm, so r = 4 cm. V = πr²h = 3.14 × 4² × 15 = 3.14 × 240 = 753.6 cm³. Exact: 240π cm³.
Cone, using the height and not the slant height (Diagram 2)
A cone-shaped snack cup has a radius of 5 cm and a height of 12 cm. Its slanted side is 13 cm long. How much does it hold?
Equation: Use h = 12, not 13. V = (1/3)πr²h = (1/3) × 3.14 × 5² × 12 = (1/3) × 3.14 × 300 = 314 cm³. Exact: 100π cm³. Using 13 by mistake gives about 340.2 cm³.
Sphere (T3)
A basketball is about 24 cm across. About how much air is inside it?
Equation: r = 24 ÷ 2 = 12 cm. V = (4/3)πr³ = (4/3) × 3.14 × 12³ = (4/3) × 3.14 × 1,728 = 3.14 × 2,304 = 7,234.56 cm³, about 7,234.6 cm³. Exact: 2,304π cm³.
Finding a missing height
A cylindrical flower vase has a radius of 5 cm and holds 1,570 cm³ of water when full. How tall is the inside of the vase?
Equation: V = πr²h, so 1,570 = 3.14 × 25 × h = 78.5h. Divide: h = 1,570 ÷ 78.5 = 20 cm.
Composite solid: cone plus hemisphere
An ice cream cone is 11 cm tall with a radius of 3 cm. It is filled level with ice cream, and a hemisphere of ice cream with the same radius sits on top. How much ice cream is there?
Use Diagram 1 to compare the three formulas when the radius is r and the height is 2r: the cone is (2/3)πr³, the sphere is (4/3)πr³ and the cylinder is (6/3)πr³ = 2πr³. They are in the ratio 1 : 2 : 3, so a cone and a sphere of this size together fill the cylinder exactly. Stress that r³ means r × r × r, not 3r.
Guided Practice15 minutes
Pairs solve each problem. One partner writes the formula and substitutes; the other checks the radius and the units. They switch roles for each problem. Use π ≈ 3.14.
Guided practice problems with answers
Problem
Answer
A hockey puck is a cylinder 7.6 cm across and 2.5 cm thick. Find its volume to the nearest tenth.
r = 3.8 cm. V = 3.14 × 3.8² × 2.5 = 3.14 × 14.44 × 2.5 = 113.354, about 113.4 cm³
A pile of sand is shaped like a cone with a radius of 1.5 m and a height of 1.2 m. How much sand is in the pile, to the nearest tenth?
V = (1/3) × 3.14 × 1.5² × 1.2 = (1/3) × 3.14 × 2.7 = 2.826, about 2.8 m³
A classroom globe is 30 cm across. What is its volume?
r = 15 cm. V = (4/3) × 3.14 × 15³ = (4/3) × 3.14 × 3,375 = 14,130 cm³
A sphere has a volume of 904.32 cm³. Find its radius.
904.32 = (4/3) × 3.14 × r³, so r³ = 904.32 × 3 ÷ 4 ÷ 3.14 = 216 and r = ∛216 = 6 cm
Listen for students who square the diameter, who forget the 1/3 for the cone, and who write r³ as 3r. Ask them to say each step aloud: "radius first, then square or cube, then multiply."
Independent Practice15 minutes
Students work alone, then compare answers with a partner and settle any difference by redoing the step where they split. Use π ≈ 3.14.
Independent practice problems with answers
Problem
Answer
A round cake pan is 9 in across and 2 in deep. How much batter fills it to the top?
r = 4.5 in. V = 3.14 × 4.5² × 2 = 3.14 × 40.5 = 127.17 in³
A cone-shaped party hat has a radius of 8 cm and a height of 15 cm. How much space is inside it?
A mixing bowl is a hemisphere with a radius of 7.5 cm. How much does it hold, to the nearest tenth?
V = (2/3) × 3.14 × 7.5³ = (2/3) × 3.14 × 421.875 = 883.125, about 883.1 cm³
A cone has a volume of 37.68 in³ and a radius of 3 in. How tall is it?
37.68 = (1/3) × 3.14 × 9 × h = 9.42h, so h = 4 in
A cylinder and a cone both have a radius of 4 cm. The cylinder is 6 cm tall. How tall must the cone be to hold the same amount?
18 cm. With the same base, a cone holds one third of a cylinder of equal height, so it needs 3 times the height: 3 × 6 = 18 cm
Closure5-10 minutes
Exit ticket (π ≈ 3.14): (1) Find the volume of a cylinder with a radius of 2 cm and a height of 5 cm. (3.14 × 4 × 5 = 62.8 cm³.) (2) A cylinder holds 450 mL. How much does a cone with the same base and height hold? (150 mL, one third.) (3) In one sentence, explain why the volume of a cone uses its height and not its slant height.
Differentiation Strategies
For Struggling Students
Give a formula card with a sketch of each solid and r and h marked on it, and a fill-in frame: "radius = ____, r² or r³ = ____, × 3.14 = ____, × (1/3) or (4/3) if needed = ____"
Start with problems that give the radius, then move to problems that give the diameter
Let students leave answers in terms of π first, so they can focus on the formula before the decimal arithmetic
For Advanced Students
Ask: if the radius of a cone is doubled and the height is cut in half, what happens to its volume? (It doubles.)
Have students find the height of a cylinder that holds the same amount as a sphere of radius 6 cm, with the same radius (8 cm)
Extension (beyond this standard): the Greek mathematician Archimedes showed that a sphere fills two thirds of the cylinder that fits around it. Ask students to check this with Diagram 1 and explain it in their own words; high school geometry (HSG.GMD.A.1) asks for informal arguments like this
Assessment Guidance
What to Look For
Strong work names the formula before substituting, halves a diameter before using it, and gives units in cubic form (cm³, m³). In cone problems, check that students use the height and not the slant height. In sphere problems, check that they cube the radius and multiply by 4/3. When a problem asks for a missing dimension, look for a clear undoing of each step and a square root or cube root at the end. Watch for answers in square units and for mixing the exact form and the 3.14 form in the same answer.
02
Classroom Activities
3 Activities
1
Pour and Compare
20 minPairs
Pairs build an open paper cylinder and an open paper cone with the same base and the same height, fill the cone with dry rice and pour it into the cylinder to find how many cones fill it. They then compare a hemisphere scoop with a short cylinder.
Build
Cylinder: radius 4 cm, height 12 cm, open at the top (the same size as the warm-up can). Tape a cardstock circle on as the bottom
Cone: radius 4 cm and height 12 cm, open at the wide end. Roll cardstock into a cone, trim it until its opening fits the cylinder exactly and its tip touches the table when it stands inside the cylinder
Short cylinder: radius 4 cm and height 4 cm, to compare with a hemisphere scoop of radius 4 cm
Procedure
Fill the cone level with rice and pour it into the tall cylinder. Count how many cones fill it. Expect about 3
Compute both volumes with π ≈ 3.14: cylinder 3.14 × 4² × 12 = 602.88 cm³ and cone (1/3) × 602.88 = 200.96 cm³
Fill the hemisphere scoop level and pour it into the short cylinder. It should fill about two thirds: hemisphere (2/3) × 3.14 × 4³ ≈ 133.97 cm³, short cylinder 3.14 × 4² × 4 = 200.96 cm³
Record how close your pour counts came to 3 cones and to 2/3 of the short cylinder, and list reasons for any difference (spilled rice, a cone that is a little too tall)
Discussion Questions
The cone and the short cylinder have the same volume, 200.96 cm³, even though they look very different. Why?
Two hemispheres make a sphere. How much of the short cylinder would a whole sphere of radius 4 cm fill?
How does the pouring result explain the 1/3 in the cone formula?
Modification for Distance Learning
Students use a paper cup and a straight-sided cup of the same width, or an online volume simulation, and record their pour counts in a shared class table.
2
Object Card Sort
20 minPairs
Each pair gets 8 cards, each showing a real object and its measurements. Pairs sort the cards by shape, choose the formula, find each volume with π ≈ 3.14 (to the nearest tenth), and put the cards in order from smallest to largest volume.
The 8 Cards
Card A: a tennis ball, 6.6 cm across
Card B: a soup can, 7 cm across and 10 cm tall
Card C: a cone-shaped snow-cone cup, 8 cm across and 10 cm tall
Card D: an orange, 8 cm across
Card E: the inside of a coffee mug, 8 cm across and 9 cm tall
Card F: a kitchen funnel (cone), radius 5 cm and height 6 cm
Card G: a ping-pong ball, 4 cm across
Card H: a pillar candle, 5 cm across and 12 cm tall
Answer Key
Spheres: A 150.5 cm³, D 267.9 cm³, G 33.5 cm³
Cylinders: B 384.7 cm³, E 452.2 cm³, H 235.5 cm³
Cones: C 167.5 cm³, F 157 cm³
Order: G, A, F, C, H, D, B, E
Discussion Questions
The orange and the snow-cone cup are both 8 cm across. Which holds more, and by about how much? (The orange, by about 100 cm³.)
The coffee mug has the largest volume and the ping-pong ball the smallest. Did anyone predict a different order before computing? Which card surprised you?
Which cards gave a diameter instead of a radius, and what did you do first?
3
Design a Can
15 minGroups of 3
A drink company wants a cylindrical can that holds 355 mL (355 cm³). Each group tests three radii, works backward to the height each can needs, and argues for one design.
The Task
For radii of 3 cm, 3.5 cm and 4 cm, solve 355 = 3.14 × r² × h for h, to the nearest tenth
Sketch each can to scale on grid paper (1 square = 1 cm)
Choose the can that is easiest to hold in one hand, and explain your choice with your numbers
Answer Key
r = 3 cm: h = 355 ÷ 28.26 ≈ 12.6 cm
r = 3.5 cm: h = 355 ÷ 38.465 ≈ 9.2 cm
r = 4 cm: h = 355 ÷ 50.24 ≈ 7.1 cm
Discussion Questions
The 3 cm can is the tallest of the three. Why does a smaller radius need a greater height for the same volume?
Going from r = 3 cm to r = 4 cm makes the radius about 1.3 times as large. About how many times as large is the base area?
Challenge Variation
The company also wants a ball-shaped bottle that holds 355 mL. Groups use guess and check to find its radius to the nearest tenth of a centimeter (4.4 cm, since 4.4 cm gives about 356.6 cm³ and 4.3 cm gives about 332.9 cm³).
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Cone, a Sphere and a Cylinder of the Same Size
A cone, a sphere and a cylinder, each with radius r and height 2r (the sphere's diameter), drawn to scale. Their volumes are (2/3)πr³, (4/3)πr³ and (6/3)πr³, in the ratio 1 : 2 : 3. So the cone is one third of the cylinder, the sphere is two thirds of it, and the cone and the sphere together fill it exactly.
Diagram 2: Height Versus Slant Height in a Cone
The snack cup from the second worked example, drawn to scale (1 cm = 20 units). The height (12 cm) runs straight down from the tip to the center of the base and meets the radius (5 cm) at a right angle. The slant height (13 cm) is the slanted side. The volume formula uses the height: (1/3)(3.14)(5²)(12) = 314 cm³.
04
Homework Assignment
~30 min
8.G.C.9 Homework: Volume of Cylinders, Cones and Spheres
Directions: Use π ≈ 3.14 for every numerical answer and round to the nearest tenth when needed. Write the formula you use, show the radius you substitute, and give every answer in cubic units.
Part 1: Find the Volume (Problems 1-3)
A cylindrical rain barrel is 60 cm across and 90 cm tall. (a) Find its volume in cubic centimeters, and give the exact answer in terms of π too. (b) How many liters does it hold when full? (1,000 cm³ = 1 liter.)
A pile of road salt is shaped like a cone that is 9 m across and 3 m tall. (a) How many cubic meters of salt are in the pile, to the nearest tenth? (b) A truck carries 10 m³ per load. How many loads are needed to move the whole pile?
A volleyball is 21 cm across. (a) Find the volume of air inside it to the nearest tenth. (b) A classmate substituted 21 into the formula. Explain the mistake and how much too large the answer would be (how many times as large).
Part 2: Solve Volume Problems (Problems 4-6)
A cylindrical popcorn tin is 20 cm across and holds 7,536 cm³. How tall is the tin? Show how you work backward.
A chocolate maker melts a cylinder of chocolate with a radius of 3 cm and a height of 8 cm and pours it into sphere-shaped molds that are 3 cm across. How many chocolate balls can she make? Show both volumes.
A small farm silo is a cylinder with a radius of 2.5 m and a height of 9 m, topped by a hemisphere with the same radius. (a) Find the total volume of the silo, to the nearest tenth. (b) The farmer fills only the cylinder part. What percent of the whole silo is filled, to the nearest percent?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Formula
Correct formula written for each solid
One formula wrong or missing
Formulas missing or wrong
Radius and Height
Halves every diameter and uses the height, not a slant length
One radius or height error
Several radius or height errors
Computation
Every volume correct with π ≈ 3.14, rounded as asked
One or two computing slips
Many errors
Units and Explanations
Cubic units throughout and clear written reasons
Some units or reasons missing
No units or reasons
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Use π ≈ 3.14 wherever a numerical answer is asked for. Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A cone-shaped paper cup has a radius of 3 cm and a height of 10 cm. Which expression gives the volume of the cup?
Answer: B
A cone holds one third of the cylinder with the same base and height, so V = (1/3)πr²h = (1/3) × π × 3² × 10, which is 30π cm³. Choice A is the cylinder formula, which forgets the 1/3. Choice C uses the sphere formula and leaves out the height. Choice D uses the diameter, 6 cm, in place of the radius.
Question 2 of 20 · Multiple Choice
A round glass bead is a sphere 10 mm across. Which expression gives its volume in cubic millimeters?
Answer: D
The radius is 10 ÷ 2 = 5 mm, and the volume of a sphere is V = (4/3)πr³ = (4/3) × π × 5³. Choice A squares the radius instead of cubing it, which cannot give cubic units. Choice B cubes the diameter instead of the radius. Choice C is the volume of a hemisphere, half a sphere.
Question 3 of 20 · Multiple Choice
A cylindrical oatmeal container is 10 cm across and 18 cm tall. What is its volume? Use π ≈ 3.14.
Answer: B
The radius is 10 ÷ 2 = 5 cm, so V = 3.14 × 5² × 18 = 3.14 × 450 = 1,413 cm³. Choice A uses the diameter 10 as the radius: 3.14 × 10² × 18 = 5,652. Choice C multiplies by the diameter instead of r²: 3.14 × 10 × 18 = 565.2. Choice D uses the cone formula and takes one third of 1,413.
Question 4 of 20 · Multiple Choice
A hanging planter is shaped like a cone with a radius of 6 cm and a height of 7 cm. How much soil fills it? Use π ≈ 3.14.
Answer: A
V = (1/3) × 3.14 × 6² × 7 = (1/3) × 3.14 × 252 = 263.76 cm³. Choice B forgets the 1/3 and gives the cylinder volume, 791.28. Choice C uses the diameter 12 instead of the radius: (1/3) × 3.14 × 144 × 7 = 1,055.04. Choice D multiplies by r instead of r²: (1/3) × 3.14 × 6 × 7 = 43.96.
Question 5 of 20 · Multiple Choice
A cantaloupe is close to a sphere 18 cm across. About what is its volume? Use π ≈ 3.14.
Answer: C
The radius is 9 cm, so V = (4/3) × 3.14 × 9³ = (4/3) × 3.14 × 729 = 3,052.08 cm³. Choice A uses the diameter 18 as the radius. Choice B forgets the 4/3: 3.14 × 729 = 2,289.06. Choice D multiplies by 1/3 instead of 4/3: 763.02.
Question 6 of 20 · Multiple Choice
A cylinder holds 540 mL. A cone has the same base and the same height. How much does the cone hold?
Answer: D
A cone holds one third of the cylinder with the same base and height: 540 ÷ 3 = 180 mL. Choice A multiplies by 3 instead of dividing. Choice B takes half. Choice C takes two thirds, which is how a sphere compares with the cylinder that fits around it.
Question 7 of 20 · Multiple Choice
A ball fits exactly inside a cylinder: they have the same radius, and the cylinder's height equals the ball's diameter. The cylinder holds 330 cm³. What is the volume of the ball?
Answer: A
The ball is (4/3)πr³ and the cylinder is πr² × 2r = 2πr³, so the ball is two thirds of the cylinder: (2/3) × 330 = 220 cm³. Choice B takes one third, which is the cone of the same size. Choice C takes half. Choice D multiplies by 3/2 instead of 2/3.
Question 8 of 20 · Multiple Choice
A cone-shaped camping tent has a floor with a radius of 1.5 m. The center pole runs from the tip straight down to the center of the floor and is 2 m long. Each slanted side, from the tip to the edge of the floor, is 2.5 m long. Which calculation gives the space inside the tent?
Answer: B
The height h of a cone is the straight distance from the tip down to the center of the base, so h = 2 m (the pole), and r = 1.5 m: V = (1/3) × 3.14 × 1.5² × 2 = 4.71 m³. Choice A uses the slant height, 2.5 m, which is longer and is not used for volume. Choice C uses the diameter, 3 m, in place of the radius. Choice D swaps the radius and the height.
Question 9 of 20 · Multiple Choice
A cylindrical flour canister has a radius of 3 in and holds 310.86 in³. How tall is it? Use π ≈ 3.14.
Answer: C
310.86 = 3.14 × 3² × h = 28.26h, so h = 310.86 ÷ 28.26 = 11 in. Choice A divides by 3.14 × 3, using r instead of r². Choice B divides by 3.14 only and forgets r². Choice D divides by 3.14 × 6, using the diameter instead of r².
Question 10 of 20 · Multiple Choice
The radius of a sphere is tripled. How many times as large does its volume become?
Answer: C
The volume depends on r³. Tripling r multiplies r³ by 3 × 3 × 3 = 27, so the volume is 27 times as large. Choice A assumes the volume grows the same way as the radius. Choice B squares the 3, which is what happens to an area. Choice D multiplies 27 by the 4/3 again, but the 4/3 is part of both the old and the new volume, so it does not change the comparison.
Question 11 of 20 · Multiple Choice
A toy ball has a volume of 113.04 in³. What is its radius? Use π ≈ 3.14.
Answer: C
113.04 = (4/3) × 3.14 × r³, so r³ = 113.04 × 3 ÷ 4 ÷ 3.14 = 27 and r = ∛27 = 3 in. Choice A stops at r³ and forgets the cube root. Choice B gives the diameter, 2 × 3. Choice D divides 27 by 3 instead of taking the cube root.
Question 12 of 20 · Multiple Choice
A sharpened pencil is a cylinder 0.8 cm across and 15 cm long, with a cone-shaped tip 1.5 cm long and the same radius. What is the volume of the pencil, to the nearest tenth? Use π ≈ 3.14.
Answer: A
The radius is 0.4 cm. Cylinder: 3.14 × 0.4² × 15 = 7.536 cm³. Cone: (1/3) × 3.14 × 0.4² × 1.5 = 0.2512 cm³. Total: 7.7872, about 7.8 cm³. Choice B leaves out the tip. Choice C forgets the 1/3 for the tip: 7.536 + 0.7536 ≈ 8.3. Choice D uses the diameter 0.8 as the radius for both parts.
Question 13 of 20 · Multiple Choice
Cylinder A has a radius of 2 cm and a height of 8 cm. Cylinder B has a radius of 4 cm and a height of 2 cm. Which statement is true?
Answer: C
A: 3.14 × 2² × 8 = 3.14 × 32 = 100.48 cm³. B: 3.14 × 4² × 2 = 3.14 × 32 = 100.48 cm³. They hold the same amount (32π cm³ each). Choice A judges by height alone. Choice B sees that the radius doubled, which multiplies the base area by 4, but misses that the height is divided by 4. Choice D is wrong, since both radii and heights are given.
Question 14 of 20 · Multiple Choice
Mei uses π ≈ 3.14 and finds that a cylinder with a radius of 5 cm and a height of 8 cm has a volume of 628 cm³. What is the exact volume in terms of π?
Answer: D
r²h = 5² × 8 = 200, so the exact volume is 200π cm³, and 200 × 3.14 = 628. Choice A multiplies r × h instead of r² × h. Choice B multiplies the diameter 10 by the height. Choice C multiplies by π twice.
Question 15 of 20 · Short Answer
An above-ground pool is a cylinder 6 m across. It is filled with water to a depth of 1.2 m. How many cubic meters of water are in the pool, to the nearest tenth? About how many liters is that? (1 m³ = 1,000 liters.) Use π ≈ 3.14.
r = 3 m. V = 3.14 × 3² × 1.2 = 3.14 × 10.8 = 33.912, about 33.9 m³, which is about 33,912 liters. Use the water depth, 1.2 m, as h, not the height of the pool wall.
Question 16 of 20 · Short Answer
A paper cup at a water cooler is a cone 7 cm across and 9 cm deep. How many milliliters does it hold when full, to the nearest tenth? (1 cm³ = 1 mL.) Use π ≈ 3.14.
r = 3.5 cm. V = (1/3) × 3.14 × 3.5² × 9 = (1/3) × 3.14 × 110.25 = 115.395, about 115.4 mL.
Question 17 of 20 · Short Answer
A beach ball is 42 cm across. How much air is inside it? Give the answer in cubic centimeters and in liters (1,000 cm³ = 1 liter). Use π ≈ 3.14.
r = 21 cm. V = (4/3) × 3.14 × 21³ = (4/3) × 3.14 × 9,261 = 38,772.72 cm³, about 38.8 liters.
Question 18 of 20 · Short Answer
A cone and a cylinder have the same radius. The cone is twice as tall as the cylinder. Jordan says the cone holds two thirds as much as the cylinder. Is Jordan right? Explain using the two volume formulas.
Yes. Call the cylinder's height h, so the cone's height is 2h. Cylinder: V = πr²h. Cone: V = (1/3)πr² × 2h = (2/3)πr²h, which is two thirds of πr²h. A full answer writes both formulas with the same r and shows how the 1/3 and the 2 combine to make 2/3.
Question 19 of 20 · Short Answer
A cone-shaped funnel is 9 cm tall and holds 150.72 cm³. How wide is its opening (the diameter)? Use π ≈ 3.14.
150.72 = (1/3) × 3.14 × r² × 9 = 9.42r², so r² = 150.72 ÷ 9.42 = 16 and r = √16 = 4 cm. The opening is 8 cm across (the diameter is 2 × 4).
Question 20 of 20 · Short Answer
A storage tank is a cylinder with a hemisphere on each end. The radius is 1.2 m and the cylinder part is 5 m long. Find the volume of the tank to the nearest tenth. Use π ≈ 3.14.
Cylinder: 3.14 × 1.2² × 5 = 22.608 m³. The two hemispheres make one sphere: (4/3) × 3.14 × 1.2³ = 7.23456 m³. Total: 29.84256, about 29.8 m³.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.C.9 mean?
8.G.C.9 means students know the volume formulas for cylinders, cones and spheres and use them to solve problems. The formulas are V = πr²h for a cylinder, V = (1/3)πr²h for a cone and V = (4/3)πr³ for a sphere. Problems include real objects, missing dimensions and solids made of more than one shape.
Do students have to memorize the formulas for 8.G.C.9?
The standard says "know the formulas," so students should be able to recall and use them. Some state tests give a formula sheet and others do not, so check your state's test. Knowing where the formulas come from (the 1/3 for a cone, the 2/3 relationship for a sphere) makes them much easier to remember.
Why is the volume of a cone one third of a cylinder?
Because a cone with the same base and height as a cylinder holds exactly one third as much. Students can see this by pouring: three full cones of rice fill the cylinder. A full proof uses ideas from high school geometry, so grade 8 relies on the experiment and the pattern.
Where does the sphere volume formula come from?
It comes from comparing a sphere with the cylinder that fits around it. That cylinder has volume πr² × 2r = 2πr³, and the sphere fills two thirds of it: (2/3) × 2πr³ = (4/3)πr³. The Greek mathematician Archimedes discovered this relationship more than 2,000 years ago.
Should students use 3.14 or the π key on a calculator?
Either works if the problem says which one; this page uses π ≈ 3.14 throughout. The π key gives a slightly different decimal, so answers can differ in the tenths or hundredths place. An exact answer "in terms of π," such as 48π cm³, avoids the choice entirely.
What is the difference between height and slant height?
The height is the straight distance from the tip of a cone to the center of its base; the slant height runs along the slanted side. The volume formula uses the height. The slant height is always longer, so using it makes the volume too large.
What mistakes do students make with 8.G.C.9?
A common mistake is using the diameter in place of the radius. Others include forgetting the 1/3 in the cone formula, squaring the radius of a sphere instead of cubing it, using the slant height of a cone, and writing the answer in square units instead of cubic units.
How do you find the radius when you know the volume?
Put the volume in the formula and undo each step until r² or r³ is alone, then take the square root or the cube root (8.EE.A.2). For example, a cylinder that is 25 cm tall and holds 1,256 cm³ has r² = 1,256 ÷ (3.14 × 25) = 16, so r = 4 cm.
How does 8.G.C.9 connect to high school geometry?
It is the base for high school volume work. In high school geometry, students use the same formulas along with pyramids (HSG.GMD.A.3) and give informal arguments for why the formulas work (HSG.GMD.A.1). 8.G.C.9 is where students first learn the three formulas.
How can parents help with 8.G.C.9 at home?
Measure round objects together. Use a ruler to measure a can, a ball or a funnel, estimate which holds the most, and then compute the volumes with π ≈ 3.14 to check. Ask your child to explain which measurement is the radius and why the answer is in cubic units.
07
Related Standards
6 standards
These standards connect to 8.G.C.9: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.G.B.4Prerequisite
Know and use the formulas for the area and circumference of a circle
