Circle Measurement and Constructions: Chapter Review

Vocabulary and notation

Circumference and disk

Circumference $C$ is the length of a circle's boundary. The enclosed disk has area $A$; a positive radius $r$ is half the diameter $d$.

Exact and approximate measures

An expression retaining $\pi$ or radicals can be exact. A rounded decimal, or a result from rounded inputs, is approximate and uses $\approx$.

Inscribed and circumscribed polygons

An inscribed polygon has vertices on the circle. A circumscribed tangent polygon surrounds it with sides tangent to the circle.

Method of exhaustion

An informal limiting argument using valid bounds whose gap can be made smaller than every prescribed positive tolerance. A fixed finite table supplies only finite precision.

Radian

The central turn selecting an arc one radius long. Radian measure counts radii along the curve; matching length units cancel, while radians identify the angle.

Sector

The region bounded by two radii and a selected arc. A proper sector has a positive angle smaller than a full turn.

Circular segment

The region bounded by a chord and a selected arc with the same endpoints. A nondiameter chord determines minor and major segments; a diameter gives semicircles.

Annulus

The ring between concentric circles with outer radius $R$ and inner radius $r$, where $0<r<R$. Its width is $R-r$.

Regular polygon and apothem

A regular polygon has equal sides and equal interior angles. Its apothem is the perpendicular center-to-side distance, not the radius to a vertex.

Formulas and theorems

• Circle measures and pi

  $$\begin{gathered} d=2r\\ \pi=\frac{C}{d}\\ C=\pi d\\ C=2\pi r\\ A=\pi r^2\\ A=\frac12Cr \end{gathered}$$

  Use when Any circle with $r>0$: $d$ is its diameter, $C$ its circumference and $A$ its disk area. Use matching length units; area uses square units. The ratio $C/d$ is dimensionless and common to all circles.

• Recover a radius or diameter

  $$\begin{gathered} r=\frac{C}{2\pi}\\ d=\frac{C}{\pi}\\ r=\sqrt{\frac{A}{\pi}} \end{gathered}$$

  Use when Use the known circumference $C>0$ or disk area $A>0$ of the same circle. Take the positive square root; recovered lengths inherit any approximation in the input.

  e.g. For a disk of area $363\pi$ square cm, find the radius: $r=11\sqrt3$ cm.

• Circle scaling

  $$\begin{gathered} \frac{r_2}{r_1}=k\\ \frac{C_2}{C_1}=k\\ \frac{A_2}{A_1}=k^2 \end{gathered}$$

  Use when Two positive-radius circles with length factor $k>0$, comparing corresponding measurements in matching units. From an area ratio, take its positive square root to recover the length factor.

• Finite polygon bounds

  $$\begin{gathered} P<C<Q\\ Q=\frac{r}{h}P\\ I=\frac12Ph\\ U=\frac12Qr\\ I<A<U \end{gathered}$$

  Text description

  A circle centered at O lies outside the solid blue regular hexagon and inside the dashed red tangent hexagon. P labels the inner polygon and Q the outer polygon. The radius r reaches a circle point. The perpendicular h meets the inner side at midpoint M; its right angle is marked. Both polygons are finite approximations.

  Use when A convex regular inscribed polygon with at least three sides has perimeter $P$ and apothem $0<h<r$. Its concentric, parallel-sided regular tangent polygon has perimeter $Q$. Their areas are $I,U$; the radius-$r$ disk has area $A$ and circumference $C$.

• Certified enclosure and midpoint error

  $$\begin{gathered} m=\frac{L+U}{2}\\ |x-m|<\frac{U-L}{2} \end{gathered}$$

  Use when Certified strict real bounds $L<x<U$, with $L,U,x$ in the same units. To certify rounding, the entire allowed interval must give the same rounded result.

• Degree and radian conversion

  $$\begin{gathered} 180^\circ=\pi\text{ radians}\\ \theta=\frac{\pi\alpha}{180}\\ \alpha=\frac{180\theta}{\pi} \end{gathered}$$

  Use when The same central turn has numerical degree measure $\alpha$ and radian measure $\theta$. A single traversal uses $0\le\alpha\le360$ or $0\le\theta\le2\pi$.

• Selected arc length and inverse data

  $$\begin{gathered} s=\frac{\alpha}{360}\,2\pi r\\ s=r\theta\\ \theta=\frac{s}{r}\\ r=\frac{s}{\theta} \end{gathered}$$

  Use when Radius $r>0$ and the same selected arc throughout. For one traversal, $0\le\alpha\le360$ degrees and $0\le\theta\le2\pi$ radians; $0\le s\le2\pi r$. The lengths $s,r$ use matching units. Require $\theta>0$ to recover $r$.

  e.g. For radius $17$ cm and selected turn $5\pi/12$ radians, find the arc length: $s=85\pi/12$ cm.

• Matching turns and complementary arcs

  $$\begin{gathered} \frac{s_1}{r_1}=\frac{s_2}{r_2}\\ \theta_{\rm major}=2\pi-\theta_{\rm minor}\\ s_{\rm major}=2\pi r-s_{\rm minor} \end{gathered}$$

  Use when The ratio equality requires the same selected central turn in positive-radius circles, with matching length units. The remaining formulas use distinct nonopposite endpoints in one circle, with $0<\theta_{\rm minor}<\pi$ radians. Opposite endpoints give two semicircles.

• Sector area

  $$\begin{gathered} A_{\rm sector}=\frac{\alpha}{360}\pi r^2\\ A_{\rm sector}=\frac12r^2\theta\\ A_{\rm sector}=\frac12rs \end{gathered}$$

  Use when Radius $r>0$ and selected angle $\alpha$ in degrees or $\theta$ in radians. Here $0\le\alpha\le360$, $0\le\theta\le2\pi$, and $s$ is the corresponding arc length. Match length units; zero and full turns give zero area and disk area.

• Proper sector perimeter

  $$\begin{gathered} P=2r+s\\ P=r(2+\theta) \end{gathered}$$

  Text description

  An illustrative proper sector is bounded by radii OA and OB and the circular arc from A to B. Both radii are labeled r. The curved edge is labeled s and the selected central turn is labeled theta. Every boundary piece is drawn with a solid blue stroke; no chord is part of this sector boundary.

  Use when A proper sector has $r>0$ and $0<\theta<2\pi$ radians, with selected arc length $s=r\theta$. Count both radii. For a full disk, the boundary is only $2\pi r$; the two radius guides are internal.

• Minor and major segment areas

  $$\begin{gathered} A_{\rm minor}=A_{\rm sector}-T\\ A_{\rm major}=\pi r^2-A_{\rm minor} \end{gathered}$$

  Text description

  Circle O has chord AB and a selected quarter-circle minor arc from A to B. The minor segment between that arc and chord is shaded blue, with its complete boundary drawn solid blue. Dashed radii OA and OB bound an interior triangle labeled T. The radius guides belong to that triangle and are not segment boundary edges. The other side of chord AB is the major segment.

  Use when Radius $r>0$; chord $AB$ is not a diameter. Use its minor central turn $0<\theta<\pi$ radians and the corresponding minor sector. $T$ is the area of triangle $OAB$, found with a perpendicular base and height. All areas use matching square units.

• Circular segment perimeter

  $$P=s+c$$

  Use when A chord with distinct endpoints in a positive-radius circle determines the selected segment. Here $s$ is its chosen arc length and $c$ its chord length, in the same units. Major and minor segments share the chord but use different arcs. A semicircle uses its diameter as the chord.

• Complete annulus

  $$\begin{gathered} A=\pi(R^2-r^2)\\ A=\pi(R-r)(R+r)\\ P=2\pi(R+r) \end{gathered}$$

  Use when Concentric circles with $0<r<R$, in matching length units. $P$ counts both complete boundary circles, including the hole; it is not their circumference difference.

• Partial annulus

  $$\begin{gathered} A=\frac12(R^2-r^2)\theta\\ P=(R+r)\theta+2(R-r) \end{gathered}$$

  Text description

  A quarter of an annulus has common center O, outer radius R and positive inner radius r. The outer arc is solid blue and the inner arc is dashed red. Two solid straight connectors close the region. Each connector has length R minus r; the bottom one is labeled. A dashed diagonal radius reaches the outer circle and a vertical dashed radius reaches the inner circle. Theta labels the selected central turn.

  Use when Concentric circles with $0<r<R$ and selected angle $0<\theta<2\pi$ radians. Keep the region between corresponding arcs; count both arcs and two straight connectors. Match length units. At a full turn, use the complete-annulus perimeter.

• Regular polygon measurement

  $$\begin{gathered} P=ns\\ A=\frac12Pa\\ \text{central angle}=\frac{360^\circ}{n} \end{gathered}$$

  Use when A convex regular polygon with integer $n\ge3$, side $s>0$, perimeter $P$ and positive perpendicular apothem $a$. Its center triangles use consecutive vertices; all lengths use one unit. Here $a$ is the $h$ used in the finite polygon bounds.

• Inscribed regular hexagon

  $$\begin{gathered} s=r\\ a=\frac{r\sqrt3}{2}\\ P=6r\\ A=\frac{3\sqrt3}{2}r^2 \end{gathered}$$

  Use when A regular hexagon inscribed in a circle of radius $r>0$. Its consecutive central angles are $60^\circ$; $s$ is a straight chord and $a$ is the perpendicular apothem.

• Inscribed equilateral triangle

  $$\begin{gathered} s=r\sqrt3\\ a=\frac r2\\ P=3r\sqrt3\\ A=\frac{3\sqrt3}{4}r^2 \end{gathered}$$

  Text description

  The six circle points A, B, C, D, E and F occur in order. The regular hexagon is dashed red. Solid blue chords AC, CE and EA form the equilateral triangle, skipping one hexagon vertex per side. O is the common center and OA is a radius labeled r. The central angle AOC is labeled 120 degrees. All six vertices lie on the same circle.

  Use when An equilateral triangle inscribed in a radius-$r$ circle, $r>0$. Alternate vertices of a constructed regular hexagon give three $120^\circ$ central angles. The apothem $a$ is not the triangle's full altitude.

• Inscribed square

  $$\begin{gathered} s=r\sqrt2\\ a=\frac{r\sqrt2}{2}\\ P=4r\sqrt2\\ A=2r^2 \end{gathered}$$

  Text description

  The original circle has center O and diameter AC. Blue helper arcs centered at A and dashed red helper arcs centered at C all use the full opening AC. Their distinct intersections U and V determine the perpendicular bisector of AC through O. This line meets the original circle at B and D. Joining A, B, C and D gives the square. The four square vertices lie on the original circle; U and V are helper points outside it.

  Use when A square inscribed in a circle of radius $r>0$. Its diagonals are perpendicular diameters, each of length $2r$; $a$ is the perpendicular apothem. The helper-circle opening is not the square side.

Problem types, step by step

Exam traps

• Trap Declaring one finite picture exact

  Fix A finite polygon or wedge rearrangement retains a discrepancy. Exhaustion requires valid bounds with arbitrarily small gaps; repeated decimal agreement alone is insufficient.

• Trap Using a diameter as the radius

  Fix Halve it before squaring for area. Circumference is a length; area has square units and the square of a length scale factor.

• Trap Treating a radian as a length or requiring pi in it

  Fix Radian measure is an angle ratio and may be rational or decimal. Match the arc and radius units, and use only radians in $s=r\theta$ or $A=r^2\theta/2$.

• Trap Choosing the shorter route automatically

  Fix Follow the stated minor or major arc. Between distinct nonopposite endpoints, the two turns total one full turn; repeated laps need their own distance.

• Trap Subtracting a triangle from a major sector

  Fix Subtract the triangle from the minor sector, then take the disk complement for the major segment. A diameter gives a semicircle with zero central-triangle area.

• Trap Letting area arithmetic choose the perimeter

  Fix Trace the actual boundary. Count two radii for a proper sector, a chord for a segment, both circles for an annulus and both connectors for a partial ring.

• Trap Squaring the ring width

  Fix Subtract squared radii, or use $\pi(R-r)(R+r)$. Boundary circles add their lengths even though disk areas subtract.

• Trap Estimating construction marks by eye

  Fix Keep the hexagon opening fixed, exclude the previous intersection and use alternate vertices for its equilateral triangle. Construct the square perpendicular with equal-distance helper crossings.

• Trap Using a degenerate helper opening

  Fix For diameter $AC=2r$, equal helper-circle openings must exceed $r$ to give two distinct crossings. Any common opening greater than $r$ gives the same perpendicular-bisector line; at $r$ they merely touch at $O$.

• Trap Replacing an apothem or chord by the wrong length

  Fix Use a perpendicular center-to-side apothem in $A=Pa/2$ and straight chords for polygon sides. Sharing a circle does not make different polygons' sides or areas equal.