📏 Geometry Assessment

Mathematics Training Certification — Five 9s Communications

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Welcome to Geometry Assessment Training

Course Duration: 30–40 minutes

Geometry is the math of shapes, angles, and space — and it shows up constantly in low-voltage work: squaring a layout, bending conduit offsets, sizing conduit fill, estimating cable runs around a room, and checking box fill. This course reviews the core geometry you need, with worked examples and on-the-job connections, then checks your understanding.

What You'll Learn:

  • Classify angles and use complementary, supplementary, vertical, and parallel-line angle relationships
  • Find triangle angles and area, and use the Pythagorean theorem
  • Calculate the circumference and area of a circle
  • Find interior angles of polygons and the perimeter and area of rectangles
  • Calculate the volume and surface area of common solids

Course Structure

  1. Angles and Lines
  2. Triangles
  3. Circles
  4. Polygons
  5. Surface Area and Volume
  6. Final Assessment — 10 questions, 80% (8 of 10) to pass

Each section ends with a knowledge check: you get two attempts, with feedback after each one, and you finish the check to unlock the next section. You'll enter your name and email just before you submit the final assessment. Your results are then emailed automatically to the training administrator, with a copy to you, whether or not you pass.

Section 1: Angles and Lines

An angle is formed where two lines or rays meet, measured in degrees (°). Most layout, bending, and mounting work comes down to reading and matching angles.

Types of angles

AngleMeasure
AcuteLess than 90°
RightExactly 90° (shown with a small square)
ObtuseBetween 90° and 180°
StraightExactly 180° (a straight line)

Angle pairs

  • Complementary angles add up to 90°. The complement of 35° is 55°.
  • Supplementary angles add up to 180°. The supplement of 35° is 145°.
  • Vertical angles are the opposite angles formed where two lines cross. They are always equal.

Parallel lines cut by a transversal

When a line (the transversal) crosses two parallel lines, it creates eight angles in two matching sets:

  • Corresponding angles (same position at each crossing) are equal.
  • Alternate interior angles (inside the parallel lines, on opposite sides of the transversal) are equal.
  • Same-side interior angles (inside the parallel lines, on the same side) are supplementary — they add to 180°.
Example: a transversal makes a 65° angle with the first parallel line. Corresponding angle at the second line: 65° Same-side interior angle: 180° − 65° = 115°
On the job: A conduit offset uses two bends of the same angle. The runs before and after the offset are parallel, so the slanted section is a transversal and the two bends are alternate interior angles — that's why they must match. For a 30° offset, the distance between bends is twice the offset depth, because 1 ÷ sin 30° = 2.
Knowledge Check 1
Two parallel lines are cut by a transversal. What is the relationship between corresponding angles?
Attempts: 0/2

Section 2: Triangles

A triangle has three sides and three angles, and the angles always add up to 180°. If you know two angles, subtract them from 180° to find the third.

Area of a triangle

A = ½ × base × height

The height is measured perpendicular (at 90°) to the base, not along a slanted side.

Example: base 8 ft, height 5 ft A = ½ × 8 × 5 = 20 ft²

Right triangles and the Pythagorean theorem

A right triangle has one 90° angle. The side opposite it is the hypotenuse (the longest side), and the other two sides are the legs:

a² + b² = c²
Finding the hypotenuse: legs 5 and 12 c² = 5² + 12² = 25 + 144 = 169 c = √169 = 13 Finding a leg: hypotenuse 10, one leg 6 b² = 10² − 6² = 100 − 36 = 64 b = √64 = 8

Common whole-number right triangles are worth memorizing: 3-4-5, 6-8-10, and 5-12-13.

On the job: To check that a layout corner is square, measure 3 ft along one wall and 4 ft along the other — the diagonal between the marks should be exactly 5 ft. For more accuracy over a larger area, use a multiple such as 6-8-10 or 9-12-15.
Knowledge Check 2
To check a layout corner, you mark 6 ft along one wall and 8 ft along the other. What should the diagonal between the marks measure if the corner is square?
Attempts: 0/2

Section 3: Circles

A circle is every point at the same distance from a center point.

  • Radius (r): center to edge
  • Diameter (d): edge to edge through the center — d = 2r
  • π (pi): about 3.14159 — the ratio of any circle's circumference to its diameter

Circumference (distance around)

C = 2πr = πd
Example: diameter 6 in C = π × 6 = 6π ≈ 18.85 in

Area (space inside)

A = πr²

Square the radius first, then multiply by π. If you're given the diameter, halve it to get the radius.

Example: radius 3 in A = π × 3² = 9π ≈ 28.27 in² Example: diameter 10 in → radius 5 in A = π × 5² = 25π ≈ 78.54 in²
On the job: Conduit fill is a circle-area calculation. A cable with a 0.25-in outside diameter has a cross-sectional area of π × (0.125)² ≈ 0.049 in². Fill tables add up these areas and compare the total with the conduit's internal area.
Knowledge Check 3
What is the area of a circle with radius 5?
Attempts: 0/2

Section 4: Polygons

A polygon is a closed shape with straight sides. A regular polygon has all sides and all angles equal.

SidesNameInterior angle sumEach angle (regular)
3Triangle180°60°
4Quadrilateral360°90°
5Pentagon540°108°
6Hexagon——
8Octagon——

Interior angle sum

Sum = (n − 2) × 180°

where n is the number of sides. Any polygon can be divided into (n − 2) triangles from one corner, and each triangle contributes 180°.

Each interior angle of a regular polygon

Each angle = (n − 2) × 180° ÷ n
Example: regular pentagon (n = 5) Sum = (5 − 2) × 180° = 540° Each = 540° ÷ 5 = 108°

The exterior angles of any polygon always add up to 360°.

Perimeter and area of rectangles and squares

  • Perimeter of a rectangle: P = 2 × (length + width); of a square: P = 4 × side
  • Area of a rectangle: A = length × width; of a square: A = side²
Example: a room 14 ft × 10 ft Perimeter = 2 × (14 + 10) = 48 ft (useful for a cable run around the walls) Area = 14 × 10 = 140 ft² (useful for ceiling tile or coverage estimates)
Knowledge Check 4
What is the sum of the interior angles of a hexagon?
Attempts: 0/2

Section 5: Surface Area and Volume

Surface area is the total area of all the outside faces of a solid (square units). Volume is the space inside it (cubic units).

SolidVolumeSurface area
Cube (side s)V = s³SA = 6s²
Rectangular box (l × w × h)V = l × w × hSA = 2(lw + lh + wh)
Cylinder (radius r, height h)V = πr²hSA = 2πr² + 2πrh
Sphere (radius r)V = (4/3)πr³SA = 4πr²
Example: box 4 in × 4 in × 2.5 in V = 4 × 4 × 2.5 = 40 in³ SA = 2(16 + 10 + 10) = 72 in² Example: cylinder, radius 2 in, height 5 in V = π × 2² × 5 = 20π ≈ 62.8 in³

Watch the units: area uses square units (ft², in²) and volume uses cubic units (ft³, in³).

On the job: Electrical box fill under NEC 314.16 is a volume calculation. Each conductor takes a set volume — 2.00 in³ for 14 AWG and 2.25 in³ for 12 AWG — and the total can't exceed the box's marked volume in cubic inches.
Knowledge Check 5
What is the volume of a cube with side length 3?
Attempts: 0/2

Final Assessment

Answer all 10 questions. You need 80% (8 of 10) to pass. These questions are different from the section knowledge checks. Enter your name and email below the questions, then submit.

1. What is the area of a triangle with base 10 and height 6?
2. Complementary angles add up to how many degrees?
3. What is the circumference of a circle with diameter 8?
4. What is each interior angle of a regular octagon?
5. A right triangle has a hypotenuse of 13 and one leg of 5. How long is the other leg?
6. What is the area of a rectangle with length 7 and width 4?
7. Vertical angles are:
8. What is the perimeter of a square with side 5?
9. The angles in a triangle add up to:
10. What is the volume of a sphere with radius 3? (V = 4/3 πr³)

Assessment Complete

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