Mistake Master
Radius in, right whole, signs zeroed
Circles pack three separate traps into one topic: a diameter that impersonates the radius, an angle fraction that hits the area when an arc was asked, and an equation whose visible signs read opposite to its center. The arithmetic is small each time. Convert first, name the whole, zero the parentheses.
§1
What this topic is about
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Circle questions run on three moves: feeding the formulas a radius, never a diameter, portioning arcs and sectors as a fraction of the right whole, and reading the circle equation's center and radius past its built-in sign trap.
§2
The formulas want the radius
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Area is $\pi r^2$ and circumference is $2\pi r$, both in terms of the radius. Problems love to hand you a diameter instead, and the choices always include the unconverted answer.
- Halve a diameter BEFORE it touches a formula.
- Backward too: $\pi r^2 = 49\pi$ gives $r = 7$, and the diameter doubles it to $14$.
- $2\pi r$ equals $\pi d$; the circumference's coefficient IS the diameter.
Worked example. A circle has diameter $16$. Find its area.
$r = \dfrac{16}{2} = 8$.
$$\pi \cdot 8^2 = 64\pi.$$ Squaring the diameter gives $256\pi$, four times too big, and $16\pi$ is the circumference, a different quantity.
§3
Arcs and sectors: a share of the right whole
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A central angle of $\theta$ degrees claims $\dfrac{\theta}{360}$ of the circle. Multiply that fraction by the CIRCUMFERENCE for an arc length, by the AREA for a sector. Crossing the two is this topic's signature error.
- Arc length $= \dfrac{\theta}{360} \cdot 2\pi r$; sector area $= \dfrac{\theta}{360} \cdot \pi r^2$.
- The denominator is $360$, the whole turn, never $180$.
- Backward: $\dfrac{\text{piece}}{\text{whole}} \cdot 360$ recovers the central angle.
Worked example. A circle has radius $6$. Find the length of a $60^\circ$ arc.
$\dfrac{60}{360} \cdot 12\pi = 2\pi$.
$6\pi$ applies the fraction to the area (a sector), and $12\pi$ is the whole circumference with the fraction forgotten.
§4
The circle equation and its sign trap
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$(x - h)^2 + (y - k)^2 = r^2$ puts the center at $(h, k)$ and stores the radius SQUARED. The form already subtracts, so $(x + 4)$ means $h = -4$, and the visible signs read opposite to the center.
- The center is whatever zeroes each parenthesis; plug a candidate in to check.
- $(x + a)$ hides $h = -a$; $(x - a)$ hides $h = a$.
- The right side is $r^2$: root it before reporting a radius.
Worked example. Find the center and radius of $(x + 6)^2 + (y - 2)^2 = 25$.
$x = -6$ and $y = 2$: the center is $(-6, 2)$.
$r = \sqrt{25} = 5$. Copying the signs gives $(6, -2)$, and reporting $25$ hands in $r^2$.
§5
Building the equation
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Writing the equation from a center and radius runs the same trap backward: subtracting a negative coordinate flips its visible sign, and the radius enters squared. A point on the circle pins $r^2$ as a squared distance from the center.
Halve diameters before the formulas see them, marry the angle fraction to the right whole (circumference for arcs, area for sectors), and read the circle equation by zeroing parentheses and rooting the right side.
§6
Three patterns that cost real points
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Three patterns recur on circle questions. They are the same ones the diagnostic routes on.
The diameter impersonates the radius.
The diameter gets squared in the area formula, doubled again in the circumference, or reported when the radius was asked, and every one of those answers is on offer.
Fix. Convert first, compute second. The formulas are radius-only; the diameter never touches them directly.
The angle fraction hits the wrong whole.
An arc question multiplies the fraction into the area, a sector question into the circumference, or the share comes out of $180$, doubling everything.
Fix. Say which quantity the question wants before multiplying: arcs come from the circumference, sectors from the area, and the whole turn is $360$.
The equation's signs and square get copied.
The center gets read with both signs flipped, and the right side gets reported as the radius when it stores its square.
Fix. Zero each parenthesis to find the center, and take a square root before any radius leaves your pencil.
Ten quick checks across the patterns: radius-diameter conversions in both directions, arcs and sectors forward and backward, and the circle equation read, built, and pinned through a point. Pick or type your answer, then check. Progress is saved.