7. Solid Geometry: Surface Area, Volume, and the Pythagorean Theorem
GED® Math: Complete Review - Volume 1 (Arithmetic & Geometry) · preview lesson
Volume measures 3D space in cubic units. Surface area measures total outer area in square units.
Key formulas (these appear on the GED® formula sheet):
| Solid | Volume | Surface area |
|---|---|---|
| Rectangular prism | \(lwh\) | \(2(lw+lh+wh)\) |
| Cylinder | \(\pi r^2 h\) | \(2\pi r^2 + 2\pi rh\) |
| Cone | \(\frac{1}{3}\pi r^2 h\) | \(\pi r^2 + \pi r l\) (l = slant height) |
| Sphere | \(\frac{4}{3}\pi r^3\) | \(4\pi r^2\) |
| Pyramid | \(\frac{1}{3}Bh\) | — |
Pythagorean theorem (right triangles only):
\[
a^2 + b^2 = c^2,
\]
where \(c\) is the hypotenuse — the side opposite the right angle, always the longest side.
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.
Worked example: A ladder 10 ft long leans against a wall, with its base 6 ft from the wall. How high does it reach?
\[
6^2 + h^2 = 10^2 \Rightarrow h^2 = 100 - 36 = 64 \Rightarrow h = 8 \text{ ft.}
\]
Distance formula (Pythagorean theorem on the coordinate plane):
\[
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
\]
A rectangular box is \(4\) cm long, \(3\) cm wide, and \(5\) cm tall. What is its volume?
Volume \(= lwh = 4 \times 3 \times 5 = 60\text{ cm}^3\).
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