How to Find the Volume of 3D Shapes | Mathematics

January 12, 2016 thetasctest

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.

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

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