Exercises: Use volume formulas to solve - Common Core Math - HS Geometry

A

Recall / Warm-Up

1.

Which formula gives the volume of a cone with base radius $r$ and height $h$ ?

A.

$V = \pi r^2 h$

B.

$V = \frac{1}{3}\pi r^2 h$

C.

$V = \frac{4}{3}\pi r^3$

D.

$V = \frac{1}{3}Bh$

2.

A square pyramid has a square base with side length $s$ and height $h$ . Which expression correctly represents the base area $B$ to use in the volume formula $V = \frac{1}{3}Bh$ ?

A.

$B = s$

B.

$B = 4s$

C.

$B = s^2$

D.

$B = \frac{1}{2}s^2$

3.

A container holds 2 m³ of liquid. How many liters is this? (Use: 1 m³ = 1,000 L)

A.

2 liters

B.

200 liters

C.

2,000 liters

D.

2,000,000 liters

B

Fluency Practice

1.

A cylinder has radius 4 cm and height 11 cm. Find its volume. Express your answer in terms of $\pi$ (e.g., write "176π cm³").

2.

A cylindrical water tank has a diameter of 6 m and a height of 5 m. Find its volume in cubic meters, then convert to liters. Round to the nearest liter. (Use: 1 m³ = 1,000 L)

3.

A square pyramid has a base side length of 9 cm and a height of 8 cm. What is its volume in cm³?

4.

A sphere has a radius of 6 cm. Find its volume. Express your answer in terms of $\pi$ .

$A grain silo made of a cylinder (height 12 m, radius 5 m) topped by a cone (height 3 m, same radius). Dimension labels mark each component.$

5.

A grain silo consists of a cylinder topped with a cone. The cylinder has radius 5 m and height 12 m. The cone has the same radius and a height of 3 m. Find the total volume of the silo in terms of $\pi$ .

6.

A cone with radius 6 cm and height 10 cm is carved out of the top of a cylinder with the same radius and height. What volume of material remains? Express your answer in terms of $\pi$ .

C

Varied Practice

1.

A basketball has a diameter of 24 cm. Which expression gives its volume?

A.

$\frac{4}{3}\pi(24)^3$

B.

$\frac{4}{3}\pi(12)^3$

C.

$\pi(12)^2(24)$

D.

$\frac{1}{3}\pi(12)^2(24)$

2.

A cylindrical can has a volume of 500 cm³ and a radius of 4 cm. Find the height of the can, rounded to the nearest tenth of a centimeter. (Use $\pi \approx 3.14159$ )

$A composite solid: cylinder (height 10 cm, radius 4 cm) with a cone (height 4 cm, same radius) sitting on top. Dimensions labeled.$

3.

A decorative object consists of a solid cylinder with radius 4 cm and height 10 cm, with a cone of the same radius and height 4 cm sitting on top (not removed — the cone is added on top). What is the total volume of the object?

A.

$\frac{4}{3}\pi(4)^3 + \pi(4)^2(10) = \frac{256}{3}\pi + 160\pi \approx 771$ cm³

B.

$\frac{1}{3}\pi(4)^2(4) + \pi(4)^2(10) = \frac{64}{3}\pi + 160\pi = \frac{544}{3}\pi \approx 570$ cm³

C.

$\pi(4)^2(10) - \frac{1}{3}\pi(4)^2(4) = 160\pi - \frac{64}{3}\pi = \frac{416}{3}\pi \approx 436$ cm³

D.

$\pi(4)^2(14) = 224\pi \approx 703$ cm³

4.

A sphere has volume $V = 1{,}000\pi$ cm³. Fill in each step to find the radius. $V = \frac{4}{3}\pi r^3$ , so $1{,}000\pi = \frac{4}{3}\pi r^3$ . Dividing both sides by $\frac{4}{3}\pi$ : $r^3 =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ . Therefore $r =$ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ cm.

r³ value:

radius in cm:

D

Word Problems

$A storage silo: cylinder (height 15 m, radius 4 m) topped by a cone (height 5 m, same radius). Dimension labels on each component.$

1.

A storage silo at a grain facility consists of a cylinder with radius 4 m and height 15 m, topped with a cone of the same radius and height 5 m.

Find the total volume of the silo in cubic meters. Express your answer in terms of $\pi$ .

2.

A public swimming pool is cylindrical with an interior diameter of 20 m and a depth of 2.5 m.

1.

Find the volume of water the pool holds in cubic meters. Express your answer in terms of $\pi$ .

2.

Convert the volume to liters. Round to the nearest liter. (Use 1 m³ = 1,000 L)

3.

An engineer is designing a spherical tank to hold exactly 10,000 liters of liquid. (1 liter = 1,000 cm³ = 0.001 m³, so 10,000 L = 10 m³)

Find the minimum interior radius of the spherical tank in meters, rounded to two decimal places. (Use $\pi \approx 3.14159$ )

4.

Yara is making a conical party hat with a base diameter of 20 cm and a slant height of 26 cm. She wants to know how much air the hat can hold.

Find the volume of the conical hat to the nearest cubic centimeter. (Hint: use the Pythagorean theorem to find the vertical height first. Use $\pi \approx 3.14159$ )

E

Error Analysis

1.

A student computed the volume of a cylinder with diameter 10 cm and height 8 cm as follows:

$V = \pi(10)^2(8) = 800\pi \approx 2{,}513 \text{ cm}^3$

The student's answer is incorrect. What mistake did the student make, and what is the correct volume?

A.

The student used the diameter (10 cm) instead of the radius (5 cm) in the formula. The correct volume is $\pi(5)^2(8) = 200\pi \approx 628$ cm³.

B.

The student forgot the $\frac{1}{3}$ factor. Cylinders use $V = \frac{1}{3}\pi r^2 h$ , so the correct answer is $\frac{800\pi}{3} \approx 838$ cm³.

C.

The student squared the height instead of the radius. The correct formula gives $\pi(10)(8^2) = 640\pi \approx 2{,}011$ cm³.

D.

There is no mistake — the answer is correct.

2.

A student found the volume of a composite solid: a cylinder (radius 6 cm, height 10 cm) with a cone of the same radius and height carved out from the top.

Student's work:
$V_{\text{cylinder}} = \pi(6)^2(10) = 360\pi \text{ cm}^3$
$V_{\text{cone}} = \frac{1}{3}\pi(6)^2(10) = 120\pi \text{ cm}^3$
$V_{\text{total}} = 360\pi + 120\pi = 480\pi \approx 1{,}508 \text{ cm}^3$

The student's answer is incorrect. What mistake did the student make, and what is the correct volume?

A.

The student used the wrong formula for the cone; it should be $\pi r^2 h$ , not $\frac{1}{3}\pi r^2 h$ .

B.

The student added the cone volume instead of subtracting it. Since the cone is removed, the correct volume is $360\pi - 120\pi = 240\pi \approx 754$ cm³.

C.

The student computed the cone's volume incorrectly; it should be $\frac{1}{3}\pi(6)^3 = 72\pi$ cm³.

D.

The student did not account for the fact that the cone and cylinder have the same dimensions. They should use a different radius for the cone.

F

Challenge / Extension

1.

A friend claims: 'If you double the radius of a sphere, the volume also doubles.' Is this correct? Explain using the volume formula $V = \frac{4}{3}\pi r^3$ , and state the correct factor by which the volume changes.

2.