3.8 Geometry Basics
Key Takeaways
- Area formulas to memorize: rectangle = l × w; triangle = ½ × b × h; circle = πr²; use π ≈ 3.14
- Volume formulas: rectangular prism = l × w × h; cylinder = πr²h; report volume in cubic units
- Perimeter is the distance around a shape; a circle's perimeter is its circumference, C = 2πr or πd
- The Pythagorean theorem a² + b² = c² finds a missing side of a right triangle, where c is the hypotenuse
- Triangle angles sum to 180°; complementary angles sum to 90°; supplementary angles sum to 180°
Why Geometry Is on the TEAS
Geometry questions live in the Measurement and Data area and ask you to find the perimeter, area, or volume of common shapes, to use basic angle facts, and to apply the Pythagorean theorem. In healthcare these skills size a wound or pressure injury (area), estimate fluid in a container (volume), and underpin imaging and body-surface-area dosing. The exam supplies the value π ≈ 3.14; the rest is selecting the correct formula and substituting carefully.
Common Shapes and Their Properties
| Shape | Key Properties |
|---|---|
| Triangle | 3 sides; interior angles sum to 180° |
| Rectangle | 4 sides; opposite sides equal; 4 right angles |
| Square | 4 equal sides; 4 right angles (a special rectangle) |
| Circle | All points equidistant from the center; radius = half the diameter |
| Parallelogram | Opposite sides parallel and equal |
| Trapezoid | Exactly one pair of parallel sides |
Perimeter and Circumference
Perimeter is the total distance around a flat shape, measured in length units (cm, in). For a circle this distance is called the circumference.
| Shape | Formula | Example |
|---|---|---|
| Rectangle | P = 2l + 2w | l=5, w=3 → P = 2(5)+2(3) = 16 |
| Square | P = 4s | s=4 → P = 16 |
| Triangle | P = a + b + c | 3, 4, 5 → P = 12 |
| Circle (circumference) | C = 2πr or πd | r=5 → C = 2π(5) ≈ 31.4 |
Area
Area is the space inside a flat shape, measured in square units (cm², in²). Memorize this short list — it covers nearly every TEAS area item.
| Shape | Formula | Example |
|---|---|---|
| Rectangle | A = l × w | l=5, w=3 → A = 15 |
| Square | A = s² | s=4 → A = 16 |
| Triangle | A = ½ × b × h | b=6, h=4 → A = 12 |
| Circle | A = πr² | r=5 → A = 25π ≈ 78.5 |
| Parallelogram | A = b × h | b=8, h=5 → A = 40 |
| Trapezoid | A = ½(b₁ + b₂) × h | bases 4, 6; h=3 → A = 15 |
Worked Example (circle area): Find the area of a circle with radius 6 cm using π ≈ 3.14.
A = πr² = 3.14 × 6² = 3.14 × 36 = 113.04 cm². A common error is multiplying π by the radius before squaring — always square the radius first.
Volume
Volume is the space inside a 3-D solid, measured in cubic units (cm³, in³). Recall that 1 mL = 1 cm³, linking volume to liquid dosing.
| Solid | Formula | Example |
|---|---|---|
| Rectangular prism (box) | V = l × w × h | 8 × 5 × 3 → V = 120 cm³ |
| Cube | V = s³ | s=4 → V = 64 cm³ |
| Cylinder | V = πr²h | r=2, h=10 → V ≈ 125.6 cm³ |
Worked Example (box volume): A medicine box measures 8 cm × 5 cm × 3 cm.
V = l × w × h = 8 × 5 × 3 = 120 cm³. Because the result is a volume, the unit is cubic centimeters.
Angle Relationships
Angles are measured in degrees (°). The TEAS tests a few fixed sums you should memorize.
- Right angle: exactly 90° (a square corner).
- Straight angle: exactly 180° (a straight line).
- Complementary angles: two angles that add to 90°.
- Supplementary angles: two angles that add to 180°.
- Triangle angle sum: the three interior angles of ANY triangle add to 180°.
Worked Example (missing angle): A right triangle has one angle of 35°. The right angle is 90°, so the third angle is
180° - 90° - 35° = 55°.
The Pythagorean Theorem
For any right triangle, the squares of the two legs add to the square of the hypotenuse (the side opposite the right angle, always the longest): a² + b² = c², where c is the hypotenuse. Use it to find a missing side when the triangle has a right angle.
Worked Example: A right triangle has legs of 3 and 4. Find the hypotenuse.
c² = a² + b² = 3² + 4² = 9 + 16 = 25, soc = √25 = 5. The 3-4-5 triangle is a classic set worth memorizing; another is 5-12-13.
Putting Units in the Right Place
A reliable way to avoid careless errors is to track units: length problems give plain units (cm), area problems give square units (cm²), and volume problems give cubic units (cm³). If your answer's units don't match what the question asks for, you used the wrong formula.
Recap
Match the shape to its formula: perimeter (and circumference) for distance around, area in square units for flat space, volume in cubic units for solid space, using π ≈ 3.14 and squaring the radius before multiplying. Lean on the fixed angle facts — 90° complementary, 180° supplementary, 180° triangle sum — and reach for a² + b² = c² whenever a right triangle hides a missing side.
Calculate the area of a circle with radius 6 cm. (Use π ≈ 3.14)
A wound dressing must cover a rectangular area 9 cm long and 4 cm wide. What is the area to be covered?
A right triangle has legs measuring 6 cm and 8 cm. What is the length of the hypotenuse?
Two angles are complementary. If one angle measures 32°, the other measures ___ degrees.
Type your answer below
Order the steps to find the volume of a cylinder with radius 3 cm and height 10 cm (π ≈ 3.14).
Arrange the items in the correct order