EGR- Volume Lesson

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, and spheres. 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 $LaTeX: V=\pi r^2h$ $V=\pi r^2h$ . 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 $LaTeX: V=\pi r^2h$ $V=\pi r^2h$ , a cone has 1/3 the amount of volume as a cylinder. Therefore, the formula for the volume of a cone is $LaTeX: V=\frac{\pi r^2h}{3}$ $V=\frac{\pi r^2h}{3}$ .

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

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

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 $LaTeX: \pi$ $\pi$ , do not solve using 3.14; simply use the symbol, $LaTeX: \pi$ $\pi$ .

Cylinders

image of cylinder Find the volume of a cylinder(in terms of $LaTeX: \Pi$ $\Pi$ ) whose radius is 5 cm and whose height is 10 cm.

V = equation image indicator $LaTeX: \Pi$ $\Pi$ r²h
V = $LaTeX: \Pi$ $\Pi$ (5²)(10)
V = $LaTeX: \Pi$ $\Pi$ (25)(10)
V = 250 $LaTeX: \Pi$ $\Pi$ cm³

View the video Links to an external site. to see another example.

Cones

image of cone What is the volume of a cone (in terms of $LaTeX: \Pi$ $\Pi$ ) whose height is 15 ft. and radius 9 ft.?

V = ¹/₃ equation image indicator $LaTeX: \Pi$ $\Pi$ r²h
V = ¹/₃ $LaTeX: \Pi$ $\Pi$ (9)²(15)
V = ¹/₃ $LaTeX: \Pi$ $\Pi$ (81)(15)

V = ¹/₃ equation image indicator $LaTeX: \Pi$ $\Pi$ (1215)

V = 405 equation image indicator $LaTeX: \Pi$ $\Pi$ ft.³

View the video Links to an external site. to see another example.

Sphere

image of sphere Find the volume of a sphere whose radius is 8 inches.

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

V = ⁴/₃ equation image indicator $LaTeX: \Pi$ $\Pi$ (8)³
V = ⁴/₃ $LaTeX: \Pi$ $\Pi$ (512)
V = ²⁰⁴⁸/₃ $LaTeX: \Pi$ $\Pi$ in.³

V = 682 $LaTeX: \frac{2}{3}$ $\frac{2}{3}$ equation image indicator $LaTeX: \Pi$ $\Pi$ in.³

View the video Links to an external site. to see another example.

Try it (Remember to have the formulas and a calculator handy)

Find the diameter of a sphere with volume $LaTeX: 972\pi ft^3$ $972\pi ft^3$ .

A cylinder-shaped pond has a 12 ft diameter and the water in the pond is 4 feet deep. Water is being drained at a rate of 1 cubic ft. per minute. How long will it take to drain the pond? Use 3.14 for $LaTeX: \pi$ $\pi$ .

The height of a cone-shaped storage building is 21 m. If the radius is 10m, find the volume of the building to the nearest tents of a meter. Use 3.14 for $LaTeX: \pi$ $\pi$ .

Check It:

18 ft.
452.16 min.
$2198m^3$

Practice

If you would like to practice these types of problems and check your work before you complete your graded homework assignment, click here. Links to an external site. Make sure you check your answers and look at the course resources or ask your teacher if there are any problems you do not understand. Links to an external site.

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