Picture a dog on a lead, walking around a post while the lead stays perfectly taut. The path the dog traces out is a circle — every single point on that path is exactly one lead-length away from the post. That's the whole definition: a circle is made up of all points on a flat surface that are equidistant (the same distance) from one fixed centre point. The "lead length" is the radius.

Two measurements describe the size of any circle:

• Radius (r): centre → edge, the "lead length".
• Diameter (d): edge → straight through the centre → other edge. It's exactly double the radius, because it's really just two radii glued end to end through the middle.

The circumference is simply the fancy word for a circle's perimeter — the total distance you'd walk if you traced the entire outer edge once.

Here's a genuinely surprising fact: if you take any circle in the universe — a coin, a wheel, a planet's orbit — and divide its circumference by its diameter, you always get the exact same number: approximately 3.14159... This number is called π (pi), and it never repeats or terminates (it's irrational). It's not something someone invented; it's a built-in property of circles themselves.

Exam questions sometimes ask you to give an answer "in terms of π" or as an "exact value". This means: don't multiply by 3.14159 on your calculator — just leave the π symbol sitting in your final answer (e.g. 32π cm² rather than 100.5 cm²). This is actually the more accurate answer, since rounding π loses precision.

1. Identify the diameter. (If you're only given the radius, double it first.)
2. Multiply the diameter by π.
3. Round, or leave in terms of π, depending on what the question asks.

1. Identify the radius. (If you're only given the diameter, halve it first — this is the step students forget most often!)
2. Square the radius.
3. Multiply that squared value by π.

Find the area and perimeter of the semicircle shown, where the diameter across the flat edge is 16 cm. Give answers in terms of π.

Thinking it through: A semicircle is exactly half of a circle. So step 1 is always to work out the values for the full circle, and then halve where needed. But be careful — the perimeter of a semicircle is NOT just half the circumference! It's the curved arc plus the straight diameter edge across the bottom, because that flat edge is part of the boundary too.

Think of a pizza. The whole pizza is the circle. If you cut one slice out, the curved crust edge of that slice is called an arc — it's just a portion of the circumference. The entire triangular-ish slice itself — the two straight cut edges plus the curved crust — is called a sector.

Every pair of points on a circle's edge actually creates two arcs — a short way round and a long way round:

• The shorter one is the minor arc.
• The longer one is the major arc.

Similarly, two radii drawn from the centre create two sectors:

• The smaller wedge is the minor sector.
• The bigger wedge is the major sector.

Here's the one insight that makes this entire topic click: a sector is just a fraction of the full circle, and that fraction is decided by the angle θ (theta) at the centre, out of the full 360° that makes up a complete circle.

Say θ = 90°. That's a quarter of the way round a full circle (since 90/360 = ¼). So a sector with a 90° angle has exactly ¼ of the full circle's area, and its arc is exactly ¼ of the full circumference. That's the entire trick — everything else is just applying the fraction θ/360 to the formulas you already know.

1. Step 1: Divide the angle by 360 to get the fraction: θ/360
2. Step 2: Calculate the circumference of the full circle: 2πr
3. Step 3: Multiply the fraction by that circumference: (θ/360) × 2πr

1. Step 1: Divide the angle by 360 to get the fraction: θ/360
2. Step 2: Calculate the area of the full circle: πr²
3. Step 3: Multiply the fraction by that area: (θ/360) × πr²

(a) Find the area of the sector, correct to 3 significant figures.

(b) Find the length of the arc, as a multiple of π.