Circles

One constant, one radius

Every circle is governed by a single constant. Its circumference, its area, the slice of a sector, the sweep of an arc: all are that constant scaled by the radius, taken in whole or in part. For a circle's size, one number, the radius, determines its circumference and area; arcs and sectors then use a fraction of those whole-circle quantities. Layered on top are the angle relationships unique to circles, the alternative way of measuring angles called radians, and the way a circle is written as an equation on the coordinate plane.

None of it is heavy once you see that almost every circle question is the same move: take a whole-circle quantity and keep the right fraction of it.

Circle circumference and area formulas

Two formulas, both on the reference sheet, anchor the circle. The circumference, the distance around, is $C = 2\pi r$, equivalently $\pi d$ since the diameter is $d = 2r$. The area enclosed is $A = \pi r^2$. Notice the shapes of the two: circumference grows with the radius, area with the radius squared, which is why doubling a circle's radius doubles its circumference but quadruples its area.

Figure 1. Everything about a circle flows from the radius: circumference $C = 2\pi r$ and area $A = \pi r^2$.

Worked: from radius to circumference and area. A circle has radius $5$. Its circumference is $2\pi$ times the radius: $2\pi \times 5 = 10\pi$. Its area is $\pi$ times the radius squared: $\pi \times 5^2 = 25\pi$. Answers are usually left in terms of $\pi$ unless a decimal is requested, which keeps them exact and tidy.

The most common slip is mixing up radius and diameter. If a problem hands you the diameter or the circumference, your first step is to convert it to the radius. A circumference of $12\pi$ means $2\pi r = 12\pi$, so $r = 6$, and from there everything else follows.

Arc length and sector area

Slice a circle with two radii and you get a sector (a wedge of area) bounded by an arc (a piece of the circumference). Each is the same fraction of the whole circle, and that fraction is the central angle over $360$ degrees. So an arc is that fraction of the circumference, and a sector is that fraction of the area: one idea, applied to two quantities.

Figure 2. An arc and a sector are each the central-angle fraction of the whole: $\text{arc} = \frac{\theta}{360}(2\pi r)$ and $\text{sector area} = \frac{\theta}{360}(\pi r^2)$.

Worked: find an arc and a sector. A circle of radius $8$ has a sector with central angle $90^\circ$; that is $\frac{90}{360} = \frac14$ of the circle. The arc is one-quarter of the circumference: $\frac14(2\pi \cdot 8) = \frac14(16\pi) = 4\pi$. The sector is one-quarter of the area: $\frac14(\pi \cdot 8^2) = \frac14(64\pi) = 16\pi$. Find the fraction first, then apply it to whichever whole-circle quantity is asked.

The same relationship runs backward. If a sector of a radius-$9$ circle has area $27\pi$, compare it to the whole circle's area $81\pi$: the fraction is $\frac{27}{81} = \frac13$, so the central angle is $\frac13$ of $360^\circ$, or $120^\circ$.

Central and inscribed angles

Two kinds of angle open onto a circle's arc. A central angle has its vertex at the center, and its measure equals the measure of the arc it cuts off. An inscribed angle has its vertex on the circle, and it is exactly half the central angle that subtends the same arc. This halving is the key fact, and it has a famous special case: an angle inscribed in a semicircle, one whose endpoints are the ends of a diameter, is always a right angle, because the diameter's arc is $180^\circ$ and half of that is $90^\circ$.

Figure 3. An inscribed angle is half the central angle that subtends the same arc; here, $x$ is half of $2x$.

Worked: the inscribed-angle rule. If a central angle subtending an arc measures $130^\circ$, an inscribed angle subtending that same arc measures half as much: $65^\circ$. And a triangle inscribed in a circle with one side a diameter has a right angle opposite that diameter, a quick way to spot a hidden right triangle.

Before you apply either rule, check where the angle's vertex sits. At the center: central. On the circle: inscribed. A diagram is not guaranteed to be drawn to scale, so trust the labels and the vertex, not how big an angle looks.

Radius-tangent and tangent-segment theorems

A tangent line touches a circle at exactly one point, and that single point of contact carries a fact worth memorizing cold: the radius drawn to it is perpendicular to the tangent line. The instant a tangent appears in a figure, that radius, that point, and any other marked point form a right triangle waiting to be used.

Figure 4. Radius $OA$ is perpendicular to the tangent line at $A$; $O$, $A$, and an external point $P$ form a right triangle with the right angle at $A$.

Worked: use the tangent right angle to find a length. From external point $P$, segment $PA$ is tangent to a circle with center $O$ at point $A$. If $OP = 17$ and the radius $OA = 8$, what is $PA$? The radius meets the tangent at a right angle, so triangle $OAP$ is right-angled at $A$, with $OP$ as the hypotenuse: $PA^2 + OA^2 = OP^2$, so $PA^2 = 17^2 - 8^2 = 289 - 64 = 225$ and $PA = 15$. This is an $8$-$15$-$17$ triple in disguise; recognizing the tangent right angle is what turns the figure into an ordinary Pythagorean setup.

A second tangent fact follows from symmetry: two tangent segments drawn from the same external point to the same circle are always equal in length. The two right triangles they form share the hypotenuse $OP$ and both have a leg equal to the radius, so they are congruent, and the remaining legs, the tangent segments, must match.

Worked: equal tangent segments from one external point. From external point $P$, segments $PA$ and $PB$ are both tangent to the same circle. If $PA = 3x + 2$ and $PB = 5x - 8$, what is $x$? Equal tangent segments set the two expressions equal: $3x + 2 = 5x - 8$, so $10 = 2x$ and $x = 5$. No circle equation or angle measure was needed; the equal-tangents fact alone supplies the equation.

Radians and degrees, and arc length from an angle

Degrees are not the only way to measure an angle. A radian measures an angle by the arc it spans, in units of the radius: an angle of one radian cuts off an arc exactly as long as the radius. Since the full circumference is $2\pi$ radii long, a full turn is $2\pi$ radians, which means $2\pi \text{ radians} = 360^\circ$, and the workhorse conversion is $\pi \text{ radians} = 180^\circ$.

Figure 5. One radian spans an arc equal to the radius; a full turn is $2\pi$ radians, equal to $360^\circ$.

• Degrees to radians: multiply by $\frac{\pi}{180}$. Radians to degrees: multiply by $\frac{180}{\pi}$.
• Arc length in radians is simply $r\theta$, radius times the angle, with no fraction needed.
• Common values: $180^\circ = \pi$, $90^\circ = \frac{\pi}{2}$, $60^\circ = \frac{\pi}{3}$, $45^\circ = \frac{\pi}{4}$.

Worked: convert and measure with radians. Convert $135^\circ$ to radians: multiply by $\frac{\pi}{180}$, giving $\frac{135\pi}{180} = \frac{3\pi}{4}$ radians. Convert $\frac{5\pi}{6}$ radians to degrees: multiply by $\frac{180}{\pi}$, giving $\frac{5 \cdot 180}{6} = 150^\circ$. Arc length the radian way: a radius of $6$ and an angle of $\frac{\pi}{3}$ give an arc of $6 \times \frac{\pi}{3} = 2\pi$, the same answer as the degree method, with less bookkeeping.

The standard equation of a circle

On the coordinate plane, a circle is the set of points a fixed distance from a center, and the distance formula turns that directly into an equation. A circle with center $(h, k)$ and radius $r$ is written

$(x - h)^2 + (y - k)^2 = r^2$

so the center is read straight off as $(h, k)$, with the signs flipped from what appears inside the parentheses, and the radius is the square root of the right-hand side. When a circle is given in expanded form instead, completing the square in $x$ and in $y$ restores this standard form and reveals the center and radius.

Figure 6. A circle of radius $r$ centered at $(h, k)$; every point on it satisfies $(x - h)^2 + (y - k)^2 = r^2$.

Worked: read and recover a circle's equation. Read it off: $(x - 2)^2 + (y + 3)^2 = 16$ has center $(2, -3)$, since $y + 3$ means $k = -3$, and radius $\sqrt{16} = 4$.

Recover it: given $x^2 + y^2 - 6x + 8y = 0$, group and complete the square in each variable: $(x^2 - 6x + 9) + (y^2 + 8y + 16) = 0 + 9 + 16$, so $(x - 3)^2 + (y + 4)^2 = 25$. The center is $(3, -4)$ and the radius is $\sqrt{25} = 5$. Add to both sides whatever you add to complete each square.

To complete a square, take half the coefficient of the linear term and square it: half of $-6$ is $-3$, and $(-3)^2 = 9$. Half of $8$ is $4$, and $4^2 = 16$. Those two numbers go on the left to build the squares and on the right to keep the equation balanced.

Translated and scaled circle equations

Because the standard-form equation stores the center and radius directly, transforming a circle, shifting it or resizing it, is a matter of editing exactly the piece that changed and leaving the rest of the equation alone.

Worked: shift a circle and rewrite its equation. The circle $(x - 3)^2 + (y + 2)^2 = 16$ is shifted $4$ units left. Shifting left changes only the center's $x$-coordinate; subtract $4$ from it and rebuild the $x$-term, leaving the $y$-term and the radius untouched: center $(3, -2) \to (-1, -2)$, so the new equation is $(x + 1)^2 + (y + 2)^2 = 16$. A shift changes only the numbers inside the parentheses, never the right-hand side.

Resizing works the other way: changing the radius touches only the right-hand side, since that side stores $r^2$, not $r$ itself.

Worked: read a resizing from the equation alone. The equation of a circle changes from $(x - 1)^2 + (y - 5)^2 = 9$ to $(x - 1)^2 + (y - 5)^2 = 25$. The center terms are identical, so the center did not move. The right-hand side went from $9$ to $25$: $r = \sqrt{9} = 3$ became $r = \sqrt{25} = 5$. The radius increased by $2$, not by $16$; the right side stores the radius squared, so the jump from $9$ to $25$ is not the change in radius itself.

Choosing the right circle theorem

A single circle diagram often carries more than one relationship at once: a central angle, an inscribed angle, a tangent line, and a chord can all appear in the same figure. The reliable habit is to name what is actually given, a center, a tangent point, an arc, before choosing a theorem, rather than picking whichever relationship looks visually obvious. Ask specifically: does an angle have its vertex at the center (central), on the circle itself (inscribed), or outside it (tangent or secant)? That single check selects the correct rule.

Suppose a figure shows a central angle of $100^\circ$ and, separately, an inscribed angle that subtends the same arc. Reading the vertex of each angle first confirms which rule applies: the central angle gives the arc directly, $100^\circ$, and the inscribed angle is exactly half of it, $50^\circ$, regardless of how large either angle looks when drawn.

Summary

A circle is run entirely by its radius: circumference $2\pi r$, area $\pi r^2$, and every arc and sector just the central-angle fraction, $\frac{\theta}{360}$, of one of those. Central angles equal their arcs; inscribed angles are half as large, with the semicircle giving a free right angle. A tangent is perpendicular to the radius at the point of contact, and tangent segments from one external point are equal. Radians remeasure angles in radius-lengths, with $\pi$ radians $= 180^\circ$ and arc length a clean $r\theta$. And on the plane, $(x - h)^2 + (y - k)^2 = r^2$ names the center and radius directly, recoverable by completing the square. See the fraction, read the center, and the circle gives way.