Volume

Volume is defined as the amount of space occupied by an object. In this lesson, we'll look at the volume of cones, cylinders, spheres, prisms and pyramids. Remember, volume is a three dimensional measurement and is, therefore, measured in cubic units.

A cylinder is a three-dimensional closed figure with congruent, parallel bases connected by the set of all segments between the two circular bases. Volume is the amount of space inside a three-dimensional object. The volume of any prism is the area of the base times the height. V = Bh. Since the base of a cylinder is a circle, the specific formula is $LaTeX: V=\pi r^2h$ . Recall the radius of a circle is the distance from the center to a point on the circle.

A cone is a three-dimensional closed figure with circular base connected to a vertex. Recall the formula for a cylinder $LaTeX: V=\pi r^2h$ , a cone has 1/3 the amount of volume as a cylinder. Therefore, the formula for the volume of a cone is $LaTeX: V=\frac{\pi r^2h}{3}$ .

A sphere is a set of points in three-dimensional space equidistant from a point called the center. The formula for a sphere is $LaTeX: V=\frac{4\pi r^3}{3}$ .

A hemisphere is the half sphere formed by a plane intersecting the center of a sphere. The formula for a hemisphere is $LaTeX: V=\frac{2\pi r^3}{3}$ .

When given the volume of a cylinder, cone, sphere, or hemisphere, plug in the volume and solve for the radius. If the volume is given in terms of $LaTeX: \pi$ , do not solve using 3.14; simply use the symbol, $LaTeX: \pi$ .

Cylinders

image of cylinder Find the volume of a cylinder(in terms of $LaTeX: \Pi$ ) whose radius is 3 cm and whose height is 7 cm.

V = equation image indicator $LaTeX: \Pi$ r²h
V = $LaTeX: \Pi$ (3²)(7)
V = $LaTeX: \Pi$ (9)(7)
V = 63 $LaTeX: \Pi$ cm³

View the video to see another example.

Right Cylinder

Oblique Cylinder

Cross-sections of cylinders

Cross sections parallel to the base will produce a circle that is congruent to the base.
Cross sections not parallel to the base but passes through the lateral side produces an oval (ellipse).
Cross sections perpendicular to the base produces a rectangle.

Cones

image of cone What is the volume of a cone (in terms of $LaTeX: \Pi$ ) whose height is 10 ft. and radius 6 ft.?

V = ¹/₃ equation image indicator $LaTeX: \Pi$ r²h
V = ¹/₃ $LaTeX: \Pi$ (6)²(10)
V = ¹/₃ $LaTeX: \Pi$ (36)(10)

V = ¹/₃ equation image indicator $LaTeX: \Pi$ (360)

V = 120 equation image indicator $LaTeX: \Pi$ ft.³

View the video to see another example.

Right Cone

Oblique Cone

Cross-sections of cylinders

Cross Sections not parallel to the base will produce an oval (ellipse).
Cross sections parallel to the base will produce similar circles.

Sphere

image of sphere Find the volume of a sphere whose radius is 12 inches.

V= ⁴/₃ equation image indicator $LaTeX: \Pi$ r³

V = ⁴/₃ equation image indicator $LaTeX: \Pi$ (12)³
V = ⁴/₃ $LaTeX: \Pi$ (1728)
V = ⁶⁹¹²/₃ $LaTeX: \Pi$ in.³

V = 2304 equation image indicator $LaTeX: \Pi$ in.³

View the video to see another example.

Cross-Sections of Spheres

Cross sections of a sphere, at any angle will produce similar circles.

Volume of Cylinders, Cones and Spheres Practice