3.5 Geometry and Measurement
Key Takeaways
- Perimeter is the distance around a shape (linear units); area is the space inside (square units); volume is the space within a solid (cubic units)
- Key formulas: rectangle area l × w, triangle area ½ × b × h, circle area πr², rectangular prism volume l × w × h, with π ≈ 3.14
- Angles are measured in degrees: a right angle is 90°, a straight angle is 180°, and a triangle's angles sum to 180°
- US customary length: 12 in = 1 ft, 3 ft = 1 yd, 5,280 ft = 1 mile; metric units scale by powers of 10 (100 cm = 1 m)
- Measurement instruction relies on rulers, pattern blocks, measuring cups, and visual formula derivations rather than memorization alone
Shapes and Angles
Geometry on the ParaPro stays concrete and measurable: identify shapes, work with angles, and compute perimeter, area, and volume. Knowing properties of common figures is the starting point.
| Shape | Properties |
|---|---|
| Triangle | 3 sides, 3 angles summing to 180° |
| Rectangle | 4 sides, 4 right angles, opposite sides equal |
| Square | 4 equal sides, 4 right angles (a special rectangle) |
| Parallelogram | 4 sides, opposite sides parallel and equal |
| Circle | All points equidistant from a center |
Angles
An angle measures the turn between two rays, in degrees. The categories appear directly on the test:
| Angle type | Measure |
|---|---|
| Acute | Less than 90° |
| Right | Exactly 90° (the "square corner") |
| Obtuse | Between 90° and 180° |
| Straight | Exactly 180° |
Two useful facts: the three angles of any triangle add to 180°, and the four angles of any quadrilateral add to 360°.
Worked Example: A triangle has angles of 40° and 75°. The third angle is 180° − 40° − 75° = 65°, because all three must total 180°.
Perimeter, Area, and Volume
Keeping these three straight — and knowing which units each uses — is the single biggest scoring opportunity in this section.
Perimeter is the distance around a 2-D shape, measured in plain (linear) units.
- Rectangle: P = 2l + 2w Square: P = 4s Triangle: P = a + b + c
Area is the space inside a 2-D shape, measured in square units.
- Rectangle: A = l × w Square: A = s² Triangle: A = ½ × b × h Circle: A = πr²
Volume is the space inside a 3-D solid, measured in cubic units.
- Rectangular prism: V = l × w × h Cube: V = s³ Cylinder: V = πr²h
Worked Example: A rectangle is 8 cm long and 5 cm wide.
- Perimeter = 2(8) + 2(5) = 16 + 10 = 26 cm (linear units).
- Area = 8 × 5 = 40 cm² (square units). A frequent ParaPro trap is reporting area in plain units; area is always squared.
Circles
| Term | Meaning | Formula |
|---|---|---|
| Radius (r) | Center to edge | — |
| Diameter (d) | Across through center | d = 2r |
| Circumference | Distance around | C = 2πr or πd |
| Area | Space inside | A = πr² |
Use π ≈ 3.14 (often provided). For a circle of radius 5: area = 3.14 × 5² = 3.14 × 25 = 78.5 square units.
Measurement and Unit Conversions
The ParaPro expects fluency in both the US customary and metric systems, plus the ability to convert within each.
Length:
| US customary | Metric |
|---|---|
| 12 in = 1 ft | 10 mm = 1 cm |
| 3 ft = 1 yd | 100 cm = 1 m |
| 5,280 ft = 1 mile | 1,000 m = 1 km |
Weight / mass and capacity:
| Customary | Metric |
|---|---|
| 16 oz = 1 lb | 1,000 g = 1 kg |
| 2,000 lb = 1 ton | 1,000 mg = 1 g |
| 8 fl oz = 1 cup | 1,000 mL = 1 L |
| 4 quarts = 1 gallon | — |
Metric conversions are easy because every step is a power of 10 — just shift the decimal. Customary conversions need the equivalences above.
Worked Example: A bookshelf is 4 feet wide. How many inches is that? Since 1 ft = 12 in, multiply: 4 × 12 = 48 inches. To go the other way (inches → feet) you would divide.
Helping Students Measure
Measurement is hands-on, so application answers favor physical tools and concrete reasoning:
- Provide rulers, tape measures, and measuring cups so students physically measure before reading a formula.
- Use pattern blocks and grid paper so area becomes "count the squares" before A = l × w.
- Derive formulas visually — show that a rectangle's area is rows × columns of unit squares.
- Keep a conversion chart posted for reference rather than expecting rote recall.
- Anchor work in real contexts: measuring the classroom, doubling a recipe, or finding how much paper covers a bulletin board.
- Emphasize labeling units every time, since the perimeter-vs-area unit confusion is the most common student error.
Strong answers build understanding from manipulatives up, not from formulas down.
Recap
Perimeter (linear units) goes around, area (square units) fills a flat shape, and volume (cubic units) fills a solid; memorize the rectangle, triangle, and circle formulas with π ≈ 3.14; know customary and metric equivalences; and teach measurement with rulers, grid paper, and unit labeling rather than bare formulas.
What is the area of a rectangle with length 12 cm and width 5 cm?
Two angles of a triangle measure 55° and 80°. What is the third angle?
A road is 3 miles long. Since 1 mile = 5,280 feet, the road is ___ feet long.
Type your answer below
Match each measurement to the correct type of unit.
Match each item on the left with the correct item on the right
A paraeducator wants to help a third grader understand why area is measured in square units. Which activity is most effective?