Circles, Arcs & Sectors Flashcards

All 12 cards in this deck

What is the formula for the circumference of a circle?

$C = 2\pi r = \pi d$

What is the formula for the area of a circle?

$A = \pi r^2$

True or false? The area of a circle of diameter $10$ cm is $\pi \times 10^2$.

False. The formula uses the radius, so it is $\pi \times 5^2 = 25\pi$ cm$^2$.

What does it mean to give a circle answer "in terms of $\pi$"?

Leave $\pi$ as a symbol in the exact answer instead of using a decimal value.
e.g. area of radius $5$ circle $= 25\pi$ cm$^2$, not $78.5$ cm$^2$.

What are the steps to find the radius of a circle from its area?
e.g. area $= 36\pi$ cm$^2$

1. Write $\pi r^2 =$ area: $\pi r^2 = 36\pi$
2. Divide by $\pi$: $r^2 = 36$
3. Square root: $r = 6$ cm

What are the steps to find the shaded area of a ring between two circles with the same centre?
e.g. radii $3$ cm and $4$ cm

1. Area of large circle: $\pi \times 4^2 = 16\pi$
2. Area of small circle: $\pi \times 3^2 = 9\pi$
3. Subtract: $16\pi - 9\pi = 7\pi$ cm$^2$

What is the formula for the arc length of a sector of angle $\theta$ and radius $r$?

Arc length $= \dfrac{\theta}{360} \times 2\pi r$

What is the formula for the area of a sector of angle $\theta$ and radius $r$?

Sector area $= \dfrac{\theta}{360} \times \pi r^2$

What lengths make up the perimeter of a sector?
e.g. a sector of radius $5$ cm with arc length $8$ cm

The arc length plus the two radii.
e.g. perimeter $= 8 + 5 + 5 = 18$ cm

True or false? To find an arc length you multiply $\dfrac{\theta}{360}$ by the area of the whole circle.

False. You multiply $\dfrac{\theta}{360}$ by the circumference, $2\pi r$.

What are the steps to find the angle of a sector from its arc length?
e.g. radius $18$ cm, arc length $4\pi$ cm

1. Form the equation $\dfrac{x}{360} \times 2\pi r =$ arc: $\dfrac{x}{360} \times 36\pi = 4\pi$
2. Divide arc by circumference: $\dfrac{4\pi}{36\pi} = \dfrac{1}{9}$
3. Multiply by $360$: $x = 40^\circ$

What are the steps to find the area of a segment of a circle?
e.g. radius $8$ cm, angle $90^\circ$

1. Sector area $= \dfrac{90}{360} \times \pi \times 8^2 = 16\pi$ $(=50.2...)$
2. Triangle area $= \tfrac12 \times 8 \times 8 \times \sin 90^\circ = 32$
3. Subtract: $50.2... - 32 = 18.3$ cm$^2$ (3 s.f.)