Training Geometry Surface Area and Volume
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Surface Area and Volume

20 min Geometry

Surface Area and Volume

Surface area and volume extend two-dimensional measurement into three dimensions. Surface area tells you how much material is needed to cover the outside of a solid (like wrapping a gift), while volume tells you how much space the solid occupies (like filling a container with water).

This lesson covers the formulas for prisms, cylinders, pyramids, cones, and spheres. The patterns are logical — a cone is one-third of a cylinder with the same base and height, just as a pyramid is one-third of a prism.

Learning these formulas equips you for practical problems in construction, packaging, manufacturing, and engineering.

3D Solid Formulas
SolidSurface AreaVolume
Rectangular Prism (box)$SA = 2(lw + lh + wh)$$V = lwh$
Cube$SA = 6s^2$$V = s^3$
Cylinder$SA = 2\pi r^2 + 2\pi rh$$V = \pi r^2 h$
Cone$SA = \pi r^2 + \pi r l$$V = \frac{1}{3}\pi r^2 h$
Sphere$SA = 4\pi r^2$$V = \frac{4}{3}\pi r^3$
Example 1

Find the volume and surface area of a box with $l=10$, $w=6$, $h=4$.

$V = 10 \times 6 \times 4 = 240$

$SA = 2(60 + 40 + 24) = 2(124) = 248$

Example 2

A cylinder has $r=5$ and $h=12$. Find the volume.

$$V = \pi(5)^2(12) = 300\pi \approx 942.48$$

Example 3

Find the volume of a sphere with radius 6.

$$V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi \approx 904.78$$

Example 4

A cone has $r = 4$ and $h = 9$. Find the volume.

$$V = \frac{1}{3}\pi(4)^2(9) = \frac{1}{3}\pi(144) = 48\pi \approx 150.80$$

Example 5

A cube has volume 343. Find the side length and surface area.

$s = \sqrt[3]{343} = 7$.

$SA = 6(7^2) = 294$.

Practice Problems

1. Volume of a box: $l=8$, $w=5$, $h=3$.
2. Surface area of a cube with side 4.
3. Volume of a cylinder: $r=3$, $h=10$.
4. Volume of a sphere: $r=9$.
5. Volume of a cone: $r=6$, $h=8$.
6. Surface area of a cylinder: $r=4$, $h=7$.
7. A tank is a cylinder with $r=3$ ft, $h=8$ ft. How many cubic feet does it hold?
8. Find the side of a cube with volume 512.
9. Surface area of a sphere with diameter 10.
10. A pool is 25 m long, 10 m wide, 2 m deep. How many cubic meters of water does it hold?
11. A cone has volume $100\pi$. If $r=5$, find $h$.
12. SA of a box: $l=12$, $w=4$, $h=3$.
13. How many liters in a cylinder with $r=10$ cm, $h=20$ cm? (1 L = 1000 cm³)
14. A hemisphere has $r=8$. Find its volume.
15. A ball has circumference $20\pi$. Find $r$ and $V$.
Show Answer Key

1. $120$

2. $96$

3. $90\pi \approx 282.74$

4. $972\pi \approx 3{,}053.63$

5. $96\pi \approx 301.59$

6. $2\pi(16) + 2\pi(28) = 88\pi \approx 276.46$

7. $72\pi \approx 226.19$ ft³

8. $s = 8$

9. $r = 5$; $SA = 100\pi \approx 314.16$

10. $500$ m³

11. $100\pi = \frac{1}{3}\pi(25)h \implies h = 12$

12. $2(48 + 36 + 12) = 192$

13. $V = \pi(100)(20) = 2000\pi \approx 6{,}283$ cm³ $\approx 6.28$ L

14. $\frac{1}{2} \cdot \frac{4}{3}\pi(512) = \frac{1024\pi}{3} \approx 1{,}072.33$

15. $2\pi r = 20\pi \implies r = 10$; $V = \frac{4}{3}\pi(1000) \approx 4{,}188.79$

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