Circles (radius, diameter, circumference, perimeter, area)

Welcome to the World of Circles!

Hello there! Today, we are going to explore one of the most perfect shapes in mathematics: the Circle. Whether it is the coin in your pocket, the wheels on a bus, or the tasty pizza you had for dinner, circles are everywhere! In this guide, we will learn how to measure them, understand their parts, and master the formulas you need for the HKAT. Don't worry if it seems tricky at first—once you learn the patterns, you will see that circles are actually quite "well-rounded"!

1. The Anatomy of a Circle

Before we calculate anything, we need to know the names of the "parts" of a circle. Imagine a circle as a clock face.

The Center: This is the exact middle point of the circle. Every point on the edge is the same distance from this center.

The Radius (\(r\)): This is a straight line from the center to any point on the edge.
Think of it like: The "Reach" of the circle from the middle.

The Diameter (\(d\)): This is a straight line that goes from one edge to the other, passing through the center. It is the widest part of the circle.
Think of it like: The "Doorway" that goes all the way across.

The Secret Relationship

The diameter is always twice as long as the radius. This is a very important rule to remember!
Formula: \(d = 2 \times r\)
Formula: \(r = d \div 2\)

Quick Review:
• If the radius is \(5\) cm, the diameter is \(10\) cm (\(5 \times 2\)).
• If the diameter is \(12\) cm, the radius is \(6\) cm (\(12 \div 2\)).

2. The Magic Number: Pi (\(\pi\))

There is a special number in math that is linked to every circle in the universe. It is called Pi (written as the Greek symbol \(\pi\)).

Did you know? If you take any circle and divide its Circumference (the distance around) by its Diameter, you ALWAYS get the same number: approximately \(3.14\)!

In your exams, you will usually be told which value to use for \(\pi\):
• Using decimals: \(\pi \approx 3.14\)
• Using fractions: \(\pi \approx \frac{22}{7}\)

3. Circumference: The Perimeter of a Circle

In other shapes like squares or triangles, we call the distance around the outside the "perimeter." For circles, we give it a special name: Circumference.

The Formula:
To find the circumference (\(C\)), you just need to multiply the diameter by \(\pi\).
\(C = \pi \times d\)
Or, since \(d = 2r\), you can use: \(C = 2 \times \pi \times r\)

Step-by-Step Example:

Question: Find the circumference of a circle with a diameter of \(10\) cm (Take \(\pi = 3.14\)).
Step 1: Identify what you know. (\(d = 10\))
Step 2: Write the formula. (\(C = \pi \times d\))
Step 3: Put the numbers in. (\(C = 3.14 \times 10\))
Step 4: Calculate. (\(C = 31.4\) cm)

Key Takeaway: Circumference is just the "fence" around the circle!

4. Area: The Space Inside

The Area is the amount of flat space inside the circle (like the amount of cheese on a pizza). To find the area, we use the radius.

The Formula:
\(Area = \pi \times r \times r\) (often written as \(\pi r^2\))

Common Mistake Alert!

Many students confuse \(r \times r\) (radius squared) with \(2 \times r\) (diameter).
• If \(r = 3\), then \(r \times r = 3 \times 3 = 9\).
• If \(r = 3\), then \(2 \times r = 3 \times 2 = 6\).
Be careful! For Area, you must multiply the radius by itself.

Step-by-Step Example:

Question: Find the area of a circle with a radius of \(7\) cm (Take \(\pi = \frac{22}{7}\)).
Step 1: Identify the radius. (\(r = 7\))
Step 2: Write the formula. (\(Area = \pi \times r \times r\))
Step 3: Put the numbers in. (\(Area = \frac{22}{7} \times 7 \times 7\))
Step 4: Simplify. The \(7\) on the bottom cancels one \(7\) on top. (\(Area = 22 \times 7\))
Step 5: Calculate. (\(Area = 154\) cm\(^2\))

5. Semi-circles and Quadrants

Sometimes you only have half a circle (Semi-circle) or a quarter of a circle (Quadrant). For the HKAT, these are very common!

Finding the Area:

This is easy! Just find the area of a full circle and divide by \(2\) (for semi-circles) or \(4\) (for quadrants).
• Semi-circle Area = \((\pi \times r \times r) \div 2\)
• Quadrant Area = \((\pi \times r \times r) \div 4\)

Finding the Perimeter (The "Trap" Section):

Wait! Finding the perimeter of a semi-circle is tricky. If you only calculate half of the circumference, you only have the curved part (the arc). You must remember to add the straight edge (the diameter) to close the shape!

Perimeter of a Semi-circle:
\((\pi \times d \div 2) + d\)
(Half circumference + Diameter)

Quick Review: Always look at the shape. If there are straight lines on the border, you must add them to your perimeter total!

Summary Checklist

Before your test, make sure you can answer these:
• Do I have the Radius or the Diameter? (Always check this first!)
• Am I looking for Circumference (the edge) or Area (the inside)?
• If it is a semi-circle perimeter, did I add the straight line at the end?
• Did I use the correct units? (cm for perimeter/circumference, cm\(^2\) for area)

You've got this! Circles might seem round and round, but with these formulas, you'll head straight to success!