Geometry Area, Volume and Dissection Arguments Worksheet Solutions (PDF)
This geometry area volume answers for Geometry covers area, volume and dissection arguments (Common Core G-GMD.A.1, G-GMD.A.3), with a free PDF to print.
Geometry · Common Core State Standards: G-GMD.A.1, G-GMD.A.3
Geometry area volume answers: exercises 1 to 10
Exercise 1 – Circumference and area
\(C = 2\pi \cdot 7 = 14\pi\).
\(A = \pi \cdot 7^{2} = 49\pi\).
14π and 49π
Exercise 2 – Area from the diameter
Radius 5.
\(A = 25\pi = 78.5\)
78.5 square units
Exercise 3 – Cylinder volume
\(V = \pi \cdot 3^{2} \cdot 10 = 90\pi\) = 282.7.
90π, about 282.7 cubic units
Exercise 4 – Cylinder from the diameter
Radius 4.
\(V = \pi \cdot 4^{2} \cdot 12 = 192\pi = 603.2\)
603.2 cubic inches
Exercise 5 – Cone volume
\(V = \frac{1}{3}\pi \cdot 6^{2} \cdot 9 = 108\pi = 339.3\)
108π, about 339.3
Exercise 6 – Cone with slant height
Height: \(\sqrt{13^{2} – 5^{2}} = 12\).
\(V = \frac{1}{3}\pi \cdot 25 \cdot 12 = 100\pi = 314.2\)
Height 12; V = 100π, about 314.2
Exercise 7 – Sphere volume
\(V = \frac{4}{3}\pi \cdot 3^{3} = 36\pi = 113.1\)
36π, about 113.1
Exercise 8 – Larger sphere
\(V = \frac{4}{3}\pi \cdot 6^{3} = 288\pi = 904.8\)
904.8 cubic units
Exercise 9 – Square pyramid
Base area \(8^{2} = 64\).
\(V = \frac{1}{3} \cdot 64 \cdot 9 = 192\)
192 cubic units
Exercise 10 – Rectangular pyramid
\(V = \frac{1}{3} \cdot (6 \cdot 10) \cdot 7 = 140\)
140 cubic units
Exercises 11 and up
Exercise 11 – Hemisphere
Half of a sphere: \(\frac{1}{2} \cdot \frac{4}{3}\pi \cdot 6^{3} = 144\pi = 452.4\)
452.4 cubic units
Exercise 12 – Find the radius
\(\pi r^{2} \cdot 8 = 200\pi\)
\(r^{2} = 25\), so \(r = 5\).
r = 5
Exercise 13 – Cavalieri
Right prism: \(40 \cdot 9 = 360\).
By Cavalieri’s principle, the slanted prism has the same volume, 360.
360 for both
Exercise 14 – Cone and cylinder
The cone has one third of the volume.
\(\frac{1}{3} \cdot 150\pi = 50\pi\)
50π
Exercise 15 – Pyramid in a cube
Cube: \(6^{3} = 216\). Pyramid: \(\frac{1}{3} \cdot 36 \cdot 6 = 72\).
\(216 \div 72 = 3\) pyramids fill the cube.
216; 72; 3 pyramids
Exercise 16 – Doubling the radius
\(\frac{4}{3}\pi(2r)^{3} = 8 \cdot \frac{4}{3}\pi r^{3}\), so the volume is 8 times as large.
Check: \(\frac{32}{3}\pi\) and \(\frac{256}{3}\pi\), a ratio of 8.
Factor of 8
Exercise 17 – Water tank
\(V = \pi \cdot 4^{2} \cdot 10 = 160\pi = 502.7\) cubic feet.
\(502.7 \cdot 7.48 = 3,760\) gallons.
502.7 cu ft; about 3,760 gallons
Exercise 18 – Ice cream cone
Cone: \(\frac{1}{3}\pi \cdot 3^{2} \cdot 10 = 30\pi\).
Hemisphere: \(\frac{2}{3}\pi \cdot 3^{3} = 18\pi\).
\(30\pi + 18\pi = 48\pi = 150.8\)
150.8 cubic centimeters
Exercise 19 – Grain silo
Cylinder: \(\pi \cdot 10^{2} \cdot 30 = 3000\pi\).
Hemisphere: \(\frac{2}{3}\pi \cdot 10^{3} = \frac{2000}{3}\pi\).
\(3000\pi + \frac{2000}{3}\pi = \frac{11000}{3}\pi = 11,519\)
About 11,519 cubic feet
Exercise 20 – Sand pile
Radius 6.
\(V = \frac{1}{3}\pi \cdot 6^{2} \cdot 5 = 60\pi = 188.5\)
188.5 cubic feet
Key facts behind this geometry area volume answers
- 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.
How to use this geometry area volume answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
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