2.1 Geometry: Area, Volume & Right Triangles
Key Takeaways
- The Pythagorean theorem states a² + b² = c² for any right triangle, where c is the hypotenuse — the 3-4-5, 5-12-13, 8-15-17, and 7-24-25 ratios (and their multiples) let you skip the arithmetic entirely.
- Supplementary angles sum to 180°, complementary angles sum to 90°, and the three interior angles of any triangle always sum to 180°.
- Circle area uses A = πr² and circumference uses C = 2πr; cylinder volume uses V = πr²h — these three formulas cover most 2D/3D geometry questions on the NAPT.
- Rectangular area is length × width, and rectangular volume is simply that same product times height (length × width × height).
- Vertical angles formed by two intersecting lines are always equal, and angles meeting at a single point always total 360°.
Geometry problems on the NAPT test whether you can work with angles, shapes, and solids under time pressure — the same spatial-reasoning skills used every day in Navy Nuclear Field ratings, from sizing a section of pipe to calculating how much water a tank holds. This section covers angle relationships, area and perimeter formulas for common two-dimensional (2D) shapes, volume formulas for common three-dimensional (3D) solids, and the Pythagorean theorem, with worked numeric examples throughout.
Angle Relationships
An angle is the figure formed when two rays share a common endpoint, called the vertex, and it is measured in degrees (°). Angles are classified by size:
| Angle Type | Measure |
|---|---|
| Acute | Less than 90° |
| Right | Exactly 90° |
| Obtuse | Between 90° and 180° |
| Straight | Exactly 180° |
| Reflex | Between 180° and 360° |
Beyond classifying single angles, the NAPT tests a small set of angle relationship rules that recur constantly:
- Complementary angles sum to 90°. If one angle is 30°, its complement is 60°.
- Supplementary angles sum to 180°. If one angle is 118°, its supplement is 62° (180 − 118 = 62).
- Vertical angles — the pair of opposite angles formed when two straight lines cross — are always equal to each other.
- Angles on a straight line (a linear pair) always sum to 180°.
- Angles around a point (a full circle of angles meeting at one vertex) always sum to 360°.
- Triangle angle sum: the three interior angles of any triangle — no matter its shape — always add up to 180°.
- Exterior angle theorem: an exterior angle of a triangle equals the sum of the two remote (non-adjacent) interior angles.
Worked Example
A triangle has two known interior angles of 47° and 81°. What is the third angle?
180° − 47° − 81° = 52°
Every triangle angle problem on the NAPT reduces to this same subtraction once you remember the 180° rule.
Area & Perimeter of Common 2D Shapes
Area measures the space inside a flat shape, in square units. Perimeter measures the distance around its edges, in linear units. These formulas cover the shapes most likely to appear:
| Shape | Area Formula | Perimeter / Circumference |
|---|---|---|
| Square | A = s² | P = 4s |
| Rectangle | A = l × w | P = 2(l + w) |
| Triangle | A = ½ × b × h | P = sum of the 3 sides |
| Parallelogram | A = b × h | P = 2(b + s) |
| Trapezoid | A = ½ × (b₁ + b₂) × h | P = sum of the 4 sides |
| Circle | A = πr² | C = 2πr (or πd) |
Where l = length, w = width, b = base, h = height, s = side, r = radius, d = diameter, and π (pi) ≈ 3.14.
Worked Examples
Rectangle: A shipboard storage compartment measures 12 feet by 7 feet.
- Area = 12 × 7 = 84 square feet
- Perimeter = 2(12 + 7) = 2(19) = 38 feet
Circle: A porthole has a radius of 5 inches.
- Area = π(5)² = 3.14 × 25 = 78.5 square inches
- Circumference = 2π(5) = 2 × 3.14 × 5 = 31.4 inches
Trapezoid: A sloped deck panel has parallel sides (bases) of 8 feet and 14 feet, with a height (the perpendicular distance between them) of 5 feet.
- Area = ½ × (8 + 14) × 5 = ½ × 22 × 5 = 55 square feet
Volume of Common 3D Solids
Volume measures the space a solid occupies, in cubic units:
| Solid | Volume Formula |
|---|---|
| Cube | V = s³ |
| Rectangular prism | V = l × w × h |
| Cylinder | V = πr²h |
| Sphere | V = (4/3) × π × r³ |
| Cone | V = (1/3) × π × r² × h |
Worked Examples
Rectangular prism: A storage locker measures 4 ft × 5 ft × 3 ft.
- Volume = 4 × 5 × 3 = 60 cubic feet
Cylinder: A cylindrical fuel tank has a radius of 3 feet and a height of 10 feet.
- Volume = π(3)²(10) = 3.14 × 9 × 10 = 282.6 cubic feet
Sphere: A spherical pressure vessel has a radius of 6 feet.
- Volume = (4/3) × 3.14 × (6)³ = (4/3) × 3.14 × 216 = 904.32 cubic feet
Cone: A hopper-shaped funnel has a radius of 3 feet and a height of 9 feet.
- Volume = (1/3) × 3.14 × (3)² × 9 = (1/3) × 3.14 × 9 × 9 = 84.78 cubic feet
The chart below compares the three worked volume examples side by side. Notice that the sphere's volume is more than three times the cylinder's even though both use a similar-sized radius — because sphere volume grows with the cube of the radius, while cylinder volume grows with only the square of the radius.
The Pythagorean Theorem
The Pythagorean theorem applies only to right triangles (triangles containing one 90° angle). It relates the two shorter sides, called legs (a and b), to the longest side, called the hypotenuse (c) — the side directly across from the right angle:
a² + b² = c²
Worked Examples
Finding the hypotenuse: A right triangle has legs of 6 and 8.
- 6² + 8² = c²
- 36 + 64 = c²
- 100 = c²
- c = 10
Finding a missing leg: A right triangle has a hypotenuse of 13 and one leg of 5.
- 5² + b² = 13²
- 25 + b² = 169
- b² = 144
- b = 12
Applied scenario: A gangway brace runs diagonally from the deck to a bulkhead. The brace (the hypotenuse) is 17 feet long, and its base sits 8 feet out from the wall. How high up the bulkhead does it reach?
- 8² + h² = 17²
- 64 + h² = 289
- h² = 225
- h = 15 feet
Common Pythagorean Triples
Certain whole-number combinations satisfy a² + b² = c² exactly, and the NAPT frequently uses them (or their multiples) to keep answers clean:
| Triple | Multiples that also work |
|---|---|
| 3-4-5 | 6-8-10, 9-12-15, 12-16-20 |
| 5-12-13 | 10-24-26, 15-36-39 |
| 8-15-17 | 16-30-34 |
| 7-24-25 | 14-48-50 |
Recognizing these patterns lets you skip the arithmetic on many test questions — if you see legs of 9 and 12, you already know the hypotenuse is 15 without squaring anything, because 9-12-15 is a multiple of 3-4-5.
Two angles are supplementary. One angle measures 118°. What is the measure of the other angle?
A water tank is shaped like a cylinder with a radius of 4 feet and a height of 6 feet. Using π ≈ 3.14, what is its approximate volume?
A rectangular hatch brace has a horizontal leg of 7 inches and a vertical leg of 24 inches. What is the length of the diagonal brace (the hypotenuse)?
A gangway brace forms a right triangle against a bulkhead. The brace (the hypotenuse) is 26 feet long, and its base sits 10 feet out from the wall. How high up the bulkhead does the brace reach?