Circles, arcs and sectors

The number that never ends

• $\pi \approx 3.14159\dots$ is the ratio of every circle's circumference 圆周 to its diameter.
• It's irrational — the digits never repeat and never end. The ancient Egyptians approximated it as $\dfrac{256}{81} \approx 3.16$.

Circumference and area 面积

• For a circle with radius $r$ (diameter $d = 2r$):

A sector 扇形 is the fraction $\tfrac{\theta}{360}$ of the whole circle, so its arc and area are that fraction of $2\pi r$ and $\pi r^2$

Arcs and sectors

• A sector is a slice of the circle between two radii, with angle $\theta$.
• It is the fraction $\dfrac{\theta}{360}$ of the whole circle:

A sector with angle $\theta$ is $\dfrac{\theta}{360}$ of the whole circle — the same fraction applies to arc length 弧长 and area.

Worked example

• $\theta = 90^{\circ}$, $r = 8$: fraction $= \dfrac{90}{360} = \dfrac{1}{4}$.
• Arc $= \dfrac{1}{4} \times 2\pi(8) = 4\pi\text{ cm}$.
• Area $= \dfrac{1}{4} \times \pi(8^2) = 16\pi\text{ cm}^2$.

A circle: circumference = pi d and area = pi r squared

Arc length is not sector perimeter

• A sector of radius 6 cm and angle $120^{\circ}$ has fraction $1/3$. Arc length $L=(1/3)2\pi(6)=4\pi$ cm; area $S=(1/3)\pi(6)^2=12\pi\text{ cm}^2$.
• Its perimeter also includes two radii: $P=L+2r=4\pi+12\approx24.6$ cm. Core sector angles are factors of $360^{\circ}$; Extended can use other angles.

A segment is a sector minus a triangle

• Extended: a segment 弓形 lies between a chord and its arc. Its area is the sector area minus the triangle, and the triangle is $\tfrac12 r^2\sin\theta$.
• With $r=10$ cm and $\theta=80^{\circ}$: sector $=\tfrac{80}{360}\pi(10)^2\approx69.81$, triangle $=\tfrac12(10)^2\sin80^{\circ}\approx49.24$, so the segment is about $20.6\text{ cm}^2$.

Segment = sector take away triangle $OAB$.

You've got it

Write from k equal to 25

• A circle has radius 5 cm.
• Its area is k pi. Find k.