Circles flashcards

$C=2\pi r=\pi d$, where r is the radius and d the diameter; it is a length in linear units. A radius of 9 in gives $18\pi$ in, about 56.5 in.

$A=\pi r^2$: square the radius, then multiply by $\pi$; the answer is in square units. A 10 m diameter means r = 5, so $A=25\pi \approx 78.5$ m². Using the diameter makes it 4 times too big.

Use the fraction $\frac{\theta }{360^{\circ }}$ of the whole: arc $=\frac{\theta }{360^{\circ }}\times 2\pi r$, sector $=\frac{\theta }{360^{\circ }}\times \pi r^2$. Radius 9, angle 40°: fraction 1/9, arc $2\pi$, sector $9\pi$.

Diameter and circumference are multiplied by k, but area is multiplied by $k^2$. Tripling the radius makes the area 9 times as large; halving it leaves one fourth of the area.

$A=\pi (R^2-r^2)$, where R is the outer radius and r the inner radius. R = 5 and r = 3 give $16\pi$. Squaring the difference of the radii (only $4\pi$) is wrong.

A tangent touches the circle at one point and is perpendicular to the radius drawn there, forming a right triangle. Radius 8 and a point 17 from the center give a tangent segment of 15.

An inscribed angle (vertex on the circle) is half its intercepted arc, while a central angle equals its arc. An arc of 110° gives an inscribed angle of 55°. An inscribed angle standing on a diameter is 90°.

One circumference per full turn: distance = turns $\times$ C, and turns = distance ÷ C. A wheel of diameter 3 ft rolling $300\pi$ ft makes $\frac{300\pi }{3\pi }=100$ turns.