Exercises: Volume of 3D solids (prisms, cylinders, cones, pyramids, spheres) - Digital SAT Math

A

Recall / Warm-Up

1.

What is the area of a circle with radius 4 cm?

A.

$8\pi$ cm²

B.

$16\pi$ cm²

C.

$32\pi$ cm²

D.

$4\pi$ cm²

2.

A rectangle has length 6 m and width 4 m. What is its area?

A.

10 m²

B.

20 m²

C.

24 m²

D.

48 m²

3.

Evaluate $3^3$ .

B

Fluency Practice

$Rectangular prism with length 5 cm, width 3 cm, and height 4 cm labeled on three edges.$

1.

A rectangular prism has length 5 cm, width 3 cm, and height 4 cm. What is its volume in cm³?

2.

A cylinder has radius 3 m and height 7 m. What is its volume? Leave your answer in terms of $\pi$ .

$Cylinder with diameter 10 cm labeled across the base and height 6 cm labeled on the side. An arrow shows r = d/2 = 5.$

3.

A cylinder has diameter 10 cm and height 6 cm. What is its volume? Leave your answer in terms of $\pi$ .

4.

A cone has radius 6 in and height 10 in. What is its volume? Leave your answer in terms of $\pi$ .

5.

A square pyramid has a base edge of 8 m and a height of 9 m. What is its volume in m³?

6.

A sphere has radius 3 cm. What is its volume? Leave your answer in terms of $\pi$ .

C

Varied Practice

D

Word Problems / Application

$Cylindrical water tank with radius 5 ft and height 12 ft labeled.$

1.

A cylindrical water tank has a radius of 5 ft and a height of 12 ft.

What is the volume of the tank in terms of $\pi$ ? Give your answer in ft³.

2.

A grain storage silo is shaped like a cylinder. It holds $500\pi$ cubic meters of grain when full. The silo's height is 20 m.

What is the radius of the silo in meters?

3.

A conical paper cup has a radius of 3 cm and a height of 8 cm.

1.

What is the volume of the cup in terms of $\pi$ ? Give your answer in cm³.

2.

A cylindrical cup has the same radius and height. How does the volume of the cylindrical cup compare to the conical cup?

A.

The cylindrical cup holds the same amount.

B.

The cylindrical cup holds twice as much.

C.

The cylindrical cup holds 3 times as much.

D.

The cylindrical cup holds $\frac{1}{3}$ as much.

4.

A spherical water balloon has a diameter of 12 cm.

What is the volume of the balloon in terms of $\pi$ ? Give your answer in cm³.

E

Error Analysis

1.

Marcus calculated the volume of a cone with radius 5 cm and height 12 cm:

$V = \pi r^2 h$
$V = \pi(5)^2(12)$
$V = 300\pi$ cm³

What error did Marcus make, and what is the correct volume?

A.

Marcus used the wrong formula — he should have used the cylinder formula. The correct volume is $600\pi$ cm³.

B.

Marcus forgot to multiply by $\frac{1}{3}$ . The correct volume is $100\pi$ cm³.

C.

Marcus used the wrong radius — he should have halved it first. The correct volume is $75\pi$ cm³.

D.

Marcus squared the height instead of the radius. The correct volume is $\frac{1}{3}\pi(25)(144)$ cm³.

2.

Sofia calculated the volume of a sphere with radius 5 cm:

$V = 4\pi r^2$
$V = 4\pi(5)^2$
$V = 100\pi$ cm³

What error did Sofia make, and what is the correct volume?

A.

Sofia should have used $r = 10$ (the diameter). The correct volume is $\frac{4}{3}\pi(10)^3$ cm³.

B.

Sofia used the surface area formula instead of the volume formula, and also dropped the $\frac{4}{3}$ factor. The correct volume is $\frac{500\pi}{3}$ cm³.

C.

Sofia should have squared the coefficient 4 as well. The correct volume is $16\pi r^2$ cm³.

D.

Sofia's formula is correct but she computed $5^2$ incorrectly. The correct volume is $100\pi$ cm³.

F

Challenge / Extension

1.

A composite solid consists of a cylinder with radius 4 cm and height 6 cm, topped with a cone of the same radius and height 3 cm. What is the total volume of the solid in terms of $\pi$ ?

2.