Chapter 4 Formulas

See other chapter 4 deck for questions.

Surd Simplification: sqrt(ab) = sqrt(a) x sqrt(b)

a, b: Factors of the number. Mastery Tip: Always ensure 'a' is the highest perfect square factor to avoid multi-step simplification.

Rationalising Law: x / sqrt(y) x sqrt(y) / sqrt(y) = (x sqrt(y)) / y

y: The irrational denominator. Mastery Tip: We multiply by "fancy 1" (the surd over itself) so the value doesn't change, only the form.

Sector Perimeter: P = 2r + (theta / 360) x 2 pi r

r: Radius; theta: Center angle. Mastery Tip: The "2r" is often forgotten; it represents the two straight radii that close the sector.

3D Pythagoras: d^2 = l^2 + w^2 + h^2

d: Internal diagonal. Mastery Tip: This is a shortcut for a two-step process: find the floor diagonal, then use it as a base for the vertical triangle.

Trapezium Area: A = 0.5h(a + b)

h: Perpendicular height; a, b: Parallel side lengths. Mastery Tip: Never use a slanted side length for 'h'.

Cylinder Surface Area: SA = 2 pi r^2 + 2 pi r h

Component 1 (2 pi r^2): Two circular ends. Component 2 (2 pi r h): The curved rectangular side.

What is the true mathematical meaning of a surd?

An irrational number that is the root of a rational number. It cannot be written as a terminating or recurring decimal and represents an exact value.

Why is perpendicular height required for area, not slanted height?

Perpendicular height measures the shortest vertical distance between bases. Slanted sides are longer (hypotenuses) and would overstate the area.

What does a trailing zero (e.g., 4.0 cm) physically represent?

It represents the precision of the tool used. 4 cm implies the nearest cm; 4.0 cm implies the nearest mm (0.1 cm). This changes the error limit.

Why do rounding errors "accumulate"?

If you round intermediate answers, you lose precision at every step. By the final step, the error can be large enough to make the answer invalid.

Scenario: Simplifying sqrt(72) | Student: sqrt(72) = sqrt(9 x 8) = 3 sqrt(8)

Error: Line 1. Failed to find the highest square factor. Correction: sqrt(36 x 2) = 6 sqrt(2).

Scenario: Semicircle Perimeter | Student: P = (180 / 360) x 2 pi(5) = 5 pi

Error: Line 1. Missing the straight diameter to close the shape. Correction: 5 pi + 10.

Scenario: Rationalising 3 / sqrt(2) | Student: 3 / sqrt(2) x sqrt(2) / sqrt(2) = (3 sqrt(2)) / 4

Error: Line 1. sqrt(2) x sqrt(2) = 2, not 4. Correction: (3 sqrt(2)) / 2.

Scenario: Combining Surds | Student: sqrt(2) + sqrt(3) = sqrt(5)

Error: Logic Law. Unlike surds cannot be added. Correction: Result is sqrt(2) + sqrt(3).

Scenario: Squared Surds | Student: (2 sqrt(5))^2 = 4 + 5 = 9

Error: Operation. It should be multiplication (4 x 5). Correction: 20.

Scenario: Accuracy Limits | Student: 86 cm (nearest cm). Upper Limit = 91 cm

Error: Line 1. Error is half of the unit (0.5), not 5. Correction: 86.5 cm.

Scenario: 3D Pythagoras | Student: d^2 = 3^2 + 4^2 + 12^2 = 169. Answer = 13 units^2

Error: Line 2. Diagonal is a length (units), not an area (units^2). Correction: 13 units.

Scenario: Cylinder SA | Student: SA = pi r^2 + 2 pi r h

Error: Formula ID. Only one circular end was included. Correction: 2 pi r^2 + 2 pi r h.

Scenario: Distributive Law | Student: 2 sqrt(3)(sqrt(6) + 5) = 2 sqrt(18) + 5

Error: Line 1. Failed to distribute the multiplier to the second term. Correction: 2 sqrt(18) + 10 sqrt(3).

Scenario: Sector Area | Student: A = (theta / 360) x 2 pi r

Error: Formula ID. Used circumference formula for area. Correction: (theta / 360) x pi r^2.