Circular Measure, Worked Examples (easy)

These easy Circular Measure examples build the everyday moves that open almost every question in the chapter: converting an angle between degrees and radians, finding an arc length from $s=r\theta$, finding a sector area from $A=\tfrac{1}{2}r^{2}\theta$, and adding two radii to an arc to get the perimeter of a sector. Each one uses small, clean numbers so you can follow every line while your calculator does only the arithmetic.

The single most important habit is to keep the angle in radians whenever you use $s=r\theta$ or $A=\tfrac{1}{2}r^{2}\theta$, these formulas are simply wrong in degrees. Use the set the honest way: cover the solution, attempt the question in full on paper, and only then check line by line against our working.

Where your answer differs, find the exact step where the two solutions part company; that single line is usually where the real learning is.

Work through all four. Attempt each fully before you read the matching solution, and notice how the same discipline, write the formula, substitute one value at a time, then simplify, runs through conversion, arc length, sector area and perimeter alike.

Notice how little changes from one example to the next. Once the angle is in radians, arc length is $r\theta$, sector area is $\tfrac{1}{2}r^{2}\theta$, and the perimeter is just the arc with the two radii added back.

Read what the question asks for, pick the matching formula, and the arithmetic stays short and reliable.

These four examples rehearse the tools that open almost every Circular Measure question in Add Math. Keep the following points in mind as you practise more.

• The bridge for conversion is $180^{\circ}=\pi$ rad: multiply by $\dfrac{\pi}{180^{\circ}}$ for degrees-to-radians, and by $\dfrac{180^{\circ}}{\pi}$ for radians-to-degrees.
• Arc length is $s=r\theta$ and sector area is $A=\tfrac{1}{2}r^{2}\theta$, both demand $\theta$ in radians, so convert first if the angle is given in degrees.
• The perimeter of a sector is the arc plus two radii, $P=r\theta+2r$; the arc alone is never the full perimeter.
• Square the radius before multiplying in the area formula, and substitute one value at a time to keep the working clean.
• Do a quick sense check against the whole circle: an angle near $2\pi$ should give an arc near the full circumference $2\pi r$.
• Keep every substitution line, with analytic marking a clear method line still earns method marks even if the final digit slips.

When a student drops a mark on questions like these, it is nearly always a small, fixable habit, using $s=r\theta$ with the angle still in degrees, or giving only the arc when the question asks for a whole perimeter. In a one-to-one lesson our teacher watches the exact line where the slip happens and corrects it on the spot, before it settles into a routine.

Because our teachers are experienced, you work with someone who explains why the angle must be in radians, not just which button to press. Lessons are taught in English, while SPM papers are set in both Malay and English, so the notation reads the same to you either way.