Circle Area & Circumference

 

Why Are Circles Different from Other 2D Shapes?

A circle consists of all points equidistant from a single center point.

The circumference of a circle is its perimeter.

$\pi$ (pi) ≈ 3.14159 links a circle’s diameter to its circumference.

Diameter (d) is twice the radius (r):

$$d = 2r$$

Answers may be required in terms of $\pi$ (exact value) or rounded to decimal places/significant figures.

 

Formulae for Circles

Circumference of a Circle:

where:

• r = radius

• d = diameter

Area of a Circle:

$$A = \pi r^2$$

Arc Length of a Sector:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the angle of the sector.

Area of a Sector:

$$\text{Sector Area} = \frac{\theta}{360} \times \pi r^2$$

 

Key Tip: Area is always in square units ($cm^2$, $m^2$), while circumference is a linear measure ($cm$, $m$).

Example: Area and Perimeter of this Sector

 

 

Finding the Area of a Sector

Given: A semicircle with diameter = 24 cm.

 

Step 1: Find the Radius

$$r = \frac{24}{2} = 12cm$$

Step 2: Find the Area of the Sector

Final Answer: $48\pi cm^2$

 

Finding the Perimeter of a Sector

The perimeter includes:

1. The arc length (half the circumference of the full circle)

2. Two radius length (the straight edge)

Step 1: Find the Circumference of the Full Circle

Step 2: Find the Arc Length (Curved Part of the Perimeter)

$$\text{Arc Length} = \frac{120}{360}(24\pi) = 8\pi$$

 

Step 3: Find the Full Perimeter

$$\text{Perimeter} = \text{Arc Length} + 2\text{Radius}$$

$$= 8\pi + 24$$

Final Answer: $8\pi + 24 cm$