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?
17 cm
34 cm
60 cm²
60 cm
Two angles of a triangle measure 55° and 80°. What is the third angle?
35°
45°
55°
65°
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
Perimeter of a yard
Area of a floor
Volume of a box
Size of an angle
A paraeducator wants to help a third grader understand why area is measured in square units. Which activity is most effective?
Have the student memorize A = l × w and recite it
Give the student grid paper and have them cover a rectangle with unit squares, then count and arrange them in rows
Tell the student that area is just length times width
Show the formula and move on to the next problem
Sections you finish are checked off in the contents.