# How to Find the Volume of 3D Shapes | Mathematics

January 12, 2016

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) = a3, where a is the side of the cube.

• Rectangular prism.

V = lwh, 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 ⋅ r3, where r is the radius.

• Cone.

V =  ⋅ π ⋅ r2, where r is the radius and h is the height.

• Cylinder.

V = π ⋅ r2h, 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 ⋅ r2h, 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 ⋅ (2.52) ⋅ 10

V = (3.14) ⋅ (6.25) ⋅ 10

V = 196.25 cm3

V of 2 cans = 2(196.25) = 392.5 cm3

• Family sized can:   V = π ⋅ (52) ⋅ 14

V = (3.14) ⋅ (25) ⋅ 14

V = 1099 cm3

• 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 cm3 larger than two small cans.