Geometry Area, Volume and Dissection Arguments Lesson (Free PDF)
This geometry area volume lesson for Geometry covers area, volume and dissection arguments (Common Core G-GMD.A.1, G-GMD.A.3), with a free PDF to print.
A circle has area pi r squared. A prism or cylinder has volume Bh, a pyramid or cone has one third of that, and a sphere has volume four thirds pi r cubed. Cavalieri’s principle shows that solids with equal heights and equal cross-sections at every level have equal volumes.
Geometry · Common Core State Standards: G-GMD.A.1, G-GMD.A.3
Key facts in this geometry area volume lesson
- Circle: C = 2 pi r, A = pi r squared.
- Prism and cylinder: V = Bh.
- Pyramid and cone: V = (1/3)Bh.
- Sphere: V = (4/3) pi r cubed.
- Cavalieri’s principle: same height and same cross-sectional areas give the same volume.
What are the circle formulas?
The circumference of a circle is \(C = 2\pi r\) and its area is \(A = \pi r^{2}\). You can see the area formula by cutting a circle into thin sectors and rearranging them into a shape close to a rectangle with width \(\pi r\) and height \(r\). As the sectors get thinner, the rearranged shape gets closer to a rectangle, a dissection argument for \(A = \pi r \cdot r\).
How do you find the volume of a prism or a cylinder?
The volume of any prism or cylinder is the area of the base times the height: \(V = Bh\). For a cylinder the base is a circle, so \(V = \pi r^{2}h\).
What is Cavalieri’s principle?
Cavalieri’s principle says that two solids with the same height and the same cross-sectional area at every level have the same volume. A slanted stack of coins has the same volume as a straight stack of the same coins.
How do you find the volume of a pyramid or a cone?
A pyramid or cone has one third of the volume of the prism or cylinder with the same base and height. For a cube, three congruent pyramids fit together, each with a face for its base and the opposite corner as its apex.
\[V_{\text{pyramid}} = \frac{1}{3}Bh \quad V_{\text{cone}} = \frac{1}{3}\pi r^{2}h\]
What is the volume of a sphere?
A sphere of radius \(r\) has \(V = \frac{4}{3}\pi r^{3}\). Cavalieri’s principle compares a hemisphere to a cylinder with a cone removed to prove this.
- Identify the solid and its dimensions (use radius, not diameter).
- Choose the formula: Bh for prisms and cylinders, one third Bh for pyramids and cones, four thirds pi r cubed for spheres.
- Substitute and calculate. Keep pi until the end.
- Write cubic units.
Example: A cone has radius 6 and height 9. \(V = \frac{1}{3}\pi \cdot 6^{2} \cdot 9 = 108\pi\), about 339.3 cubic units.
Geometry area volume lesson: questions and answers
What is the volume formula for a cylinder?
V = pi r squared h, the area of the circular base times the height. A cylinder with radius 3 and height 10 has volume 90 pi, about 282.7 cubic units.
Why does a cone have one third of the volume of a cylinder?
Three congruent pyramids fill a cube, and the same relationship holds for cones and cylinders with equal base and height.
What is Cavalieri’s principle?
If two solids have the same height and the same cross-sectional area at every level, they have the same volume, even when one is slanted.
How do you find the volume of a sphere?
Use V = (4/3) pi r cubed. A sphere of radius 3 has volume 36 pi, about 113.1 cubic units.
Which measure do you use in the formulas, radius or diameter?
Use the radius. If you are given a diameter, divide it by 2 first.
How to use this geometry area volume lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the geometry area volume lesson as a PDF with the button below and keep it with your notes.
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