Cylinders, cubes, and cones. Oh my! Today we will discuss the basics of three-dimensional (3D) shapes and learn how to find their respective volumes in preparation for the TASC Test Assessing Secondary Completion™ Mathematics subtest.

But first, let’s review the difference between two-dimensional and three-dimensional shapes.

**A two-dimensional shape has length and width, but lacks height (depth).** Examples of two-dimensional shapes include:

- Circles
- Triangles
- Squares
- Rectangles
- Pentagons
- Octagons

**A three-dimensional shape has length, width, and height.** Examples of three-dimensional shapes include:

- Spheres
- Prisms
- Cubes
- Cylinders
- Pyramids
- Cones

Take a look at this visual representation of two-dimensions versus three-dimensions from SkillsYouNeed.com:

**Volume Formulas for Three-dimensional Shapes **

Each three-dimensional shape has a different volume formula. Review the formulas per geometric shape below:

**Cube.**

Volume (V) = *a*^{3}, where *a* is the side of the cube.

**Rectangular prism.**

V = *l* ⋅ *w* ⋅ *h*, where *l* is the length, *w* is the width, and *h* is the height.

**Pyramid.**

V = , where *l* is the length, *w* is the width, and *h* is the height.

**Sphere.**

V = ⋅ π ⋅ r^{3}, where *r* is the radius.

**Cone.**

V = ⋅ π ⋅ r^{2}, where *r* is the radius and *h* is the height.

**Cylinder.**

V = π ⋅ r^{2} ⋅ *h*, where *r* is the radius and *h* is the height.

**Practice Problem **

Let’s practice finding the volume of a three-dimensional shape using the following example adapted from StudyZone.org:

**Problem:** A small can of soup has a radius of 2.5 cm and a height of 10 cm. A family sized can has a radius of 5 cm and a height of 14 cm. Which contains more soup: one family sized can, or two small cans?

**Solution:**

**Step one:**Find the correct formula. Since we know that a can of soup is in the shape of a cylinder, we must use the formula:

V = π ⋅ r^{2} ⋅ *h*, where *r* is the radius and *h* is the height

**Step two:**Pull all valued information from the word problem:*Small can:*radius = 2.5 cm, height = 10 cm*Family sized can:*radius = 5 cm, height = 14 cm

**Step three:**Find the volume of two small cans of soup and a family sized can of soup:- Two mall cans: V = π ⋅ (2.5
^{2}) ⋅ 10

- Two mall cans: V = π ⋅ (2.5

V = (3.14) ⋅ (6.25) ⋅ 10

V = 196.25 cm^{3}

V of 2 cans = 2(196.25) = **392.5 cm ^{3}**

- Family sized can: V = π ⋅ (5
^{2}) ⋅ 14

V = (3.14) ⋅ (25) ⋅ 14

V = **1099 cm ^{3}**

**Step four:**Subtract 1099 – 392.5 to get the difference in volume.

**Answer:** The family sized can has more soup than two small cans of soup, as the family sized can is 706.5 cm^{3 }larger than two small cans.

Continue your volume practice at StudyZone.org and KhanAcademy.org.