Inscribed and Central Angles
Picture a round room in a museum with one famous painting on the wall. Sofia stands in the doorway on the left, Marcus stands at a spot on the wall across the room, and both look at the painting. Whose viewing angle is bigger? And what would the angle be if the museum put a bench at the exact center of the room? By the end of this page you will know.
Central angles, arcs, and inscribed angles
Central angles and arcs
An arc smaller than $180^\circ$ is a minor arc and is named with two letters, like $\overset{\frown}{AB}$. An arc larger than $180^\circ$ is a major arc and needs three letters, like $\overset{\frown}{ACB}$, so you know which way around you go. A semicircle is exactly $180^\circ$. The whole circle is $360^\circ$, so a minor arc and its matching major arc always add up to $360^\circ$.
Arcs that sit next to each other add, just like angles: if $B$ is on arc $AC$, then $m\overset{\frown}{AC} = m\overset{\frown}{AB} + m\overset{\frown}{BC}$.
Inscribed angles
Why half? When one side of the angle passes through the center $O$, triangle $OAB$ is isosceles because $OA$ and $OB$ are both radii. The central angle is an exterior angle of that triangle, so it equals the sum of the two equal base angles, which is twice the inscribed angle. The other cases follow by adding or subtracting arcs.
A security camera is placed at the center of a circular room. It rotates to cover a central angle of $114^\circ$. If an observer is standing at the edge of the room…
A security camera is placed at the center of a circular room. It rotates to cover a central angle of $114^\circ$. If an observer is standing at the edge of the room looking at the same section of the wall, find the measure of the inscribed angle that intercepts this exact same arc.
A circular pizza is divided into three unequal slices such that the arc lengths of the slices are in the ratio $2:3:4$. If lines are drawn connecting the endpoints of…
A circular pizza is divided into three unequal slices such that the arc lengths of the slices are in the ratio $2:3:4$. If lines are drawn connecting the endpoints of these arcs to form an inscribed triangle, find the measure of all three interior angles of this triangle.
Same arc and the semicircle
Look again at the figure above. Angles $P$ and $Q$ are both $40^\circ$. That is no accident: each one is half of the same $80^\circ$ arc.
That answers the museum question. Sofia and Marcus both see the painting at half of its arc, so as long as they stand on the same side of the painting, their viewing angles are equal. The person on the bench at the center sees an angle twice as big.
The reason is quick: a diameter cuts off a $180^\circ$ arc, and half of $180^\circ$ is $90^\circ$.
In a circular garden, two pathways form inscribed angles $\angle M$ and $\angle N$ that intercept the exact same arc of a flower bed. The measure of $\angle M$ is given…
In a circular garden, two pathways form inscribed angles $\angle M$ and $\angle N$ that intercept the exact same arc of a flower bed. The measure of $\angle M$ is given by $(5x - 12)^\circ$ and the measure of $\angle N$ is $(3x + 18)^\circ$. Solve for $x$ and find the measure of the angles.
Triangle $ABC$ is inscribed in a circle such that the side $AB$ passes directly through the center of the circle, making it a diameter. If the measure of $\angle BAC$ is…
Triangle $ABC$ is inscribed in a circle such that the side $AB$ passes directly through the center of the circle, making it a diameter. If the measure of $\angle BAC$ is $28^\circ$, calculate the measure of $\angle ABC$.
Inscribed quadrilaterals
A quadrilateral whose four vertices lie on a circle is inscribed (or cyclic). Look at the figure: angle $A$ intercepts arc $BCD$, and the opposite angle $C$ intercepts arc $DAB$. Those two arcs together make the whole circle.
Why? $m\angle A + m\angle C = \tfrac12 m\overset{\frown}{BCD} + \tfrac12 m\overset{\frown}{DAB} = \tfrac12 (360^\circ) = 180^\circ$. The converse is true too: if opposite angles of a quadrilateral are supplementary, the quadrilateral can be inscribed in a circle.
A stained glass window features a quadrilateral $PQRS$ inscribed inside a perfect circle. If the measure of $\angle P$ is $108^\circ$, determine the measure of the…
A stained glass window features a quadrilateral $PQRS$ inscribed inside a perfect circle. If the measure of $\angle P$ is $108^\circ$, determine the measure of the opposite angle, $\angle R$.
A quadrilateral $WXYZ$ is inscribed in a circle. The circle is divided into four arcs by the vertices of the quadrilateral: arc $WX$, arc $XY$, arc $YZ$, and arc $ZW$.…
A quadrilateral $WXYZ$ is inscribed in a circle. The circle is divided into four arcs by the vertices of the quadrilateral: arc $WX$, arc $XY$, arc $YZ$, and arc $ZW$. The measures of these arcs are in the ratio $1:2:3:4$ respectively. Calculate the measure of each interior angle of the quadrilateral $WXYZ$.
A complex engineering component features a quadrilateral $PQRS$ inscribed in a circular housing. The measures of the interior angles are given as a system of equations:…
A complex engineering component features a quadrilateral $PQRS$ inscribed in a circular housing. The measures of the interior angles are given as a system of equations: $m\angle P = (3x + 2y)^\circ$, $m\angle Q = (x + 4y)^\circ$, $m\angle R = (2x + 3y)^\circ$, and $m\angle S = (5x - y)^\circ$. Set up a system of equations based on the properties of inscribed quadrilaterals to find the values of $x$ and $y$, and determine the measure of the largest angle in the quadrilateral.
Angles formed by chords, secants, and tangents
A secant is a line that cuts a circle at two points. A tangent touches the circle at exactly one point, and it is perpendicular to the radius drawn to that point. The vertex of an angle can sit on the circle, inside it, or outside it, and each location has its own rule.
| Where the vertex is | Angle measure |
|---|---|
| On the circle (inscribed, or tangent and chord) | $\tfrac12 \,(\text{intercepted arc})$ |
| Inside the circle (two chords cross) | $\tfrac12\,(\text{arc}_1 + \text{arc}_2)$ |
| Outside the circle (two secants, secant and tangent, or two tangents) | $\tfrac12\,(\text{far arc} - \text{near arc})$ |
Here the angle at $E$ intercepts arcs $AC = 80^\circ$ and $BD = 90^\circ$ (its vertical angle, which also looks at $BD$). So $x = \tfrac12(80 + 90) = 85^\circ$.
Why add inside and subtract outside? Draw the chord $AD$. Then $\angle AEC$ is an exterior angle of triangle $AED$, so it equals the two inscribed angles $\angle DAB + \angle ADC = \tfrac12 \overset{\frown}{BD} + \tfrac12 \overset{\frown}{AC}$. For a vertex outside, the big inscribed angle is an exterior angle of a triangle that contains the outside angle, so the outside angle is the difference of two inscribed angles.
Two secant lines intersect at a point $P$ outside a circle, forming an angle of $35^\circ$. The secants intercept two arcs on the circle. If the measure of the larger…
Two secant lines intersect at a point $P$ outside a circle, forming an angle of $35^\circ$. The secants intercept two arcs on the circle. If the measure of the larger intercepted arc is $110^\circ$, find the measure of the smaller intercepted arc.
In a circle, chords $AB$ and $CD$ intersect at a point $E$ inside the circle. The measure of arc $AC$ is $42^\circ$ and the measure of arc $BD$ is $68^\circ$. Calculate…
In a circle, chords $AB$ and $CD$ intersect at a point $E$ inside the circle. The measure of arc $AC$ is $42^\circ$ and the measure of arc $BD$ is $68^\circ$. Calculate the measure of the angle $\angle AEC$ formed by the intersection of the chords. Then, assuming arc $AD$ is twice the measure of arc $BC$, find the measure of the inscribed angle $\angle ABD$.
Arc length and sector area
A pizza slice is a piece of a circle. A slice with a central angle of $45^\circ$ is $\tfrac{45}{360} = \tfrac18$ of the whole pizza. Its crust is $\tfrac18$ of the circumference, and the slice is $\tfrac18$ of the area. That idea gives both formulas.
Distance along the Earth
On a globe, a line of longitude is a great circle with its center at the center of the Earth. The latitude of a city is the central angle from the equator up to that city. So if two cities share a longitude, the central angle between them is the difference of their latitudes. The radius of the Earth is about $3{,}959$ miles.
A local pizzeria in Chicago bakes a large pizza with a diameter of $16$ inches. The pizza is cut into $8$ identical slices. Calculate the exact arc length of the crust…
A local pizzeria in Chicago bakes a large pizza with a diameter of $16$ inches. The pizza is cut into $8$ identical slices. Calculate the exact arc length of the crust on a single slice of pizza.
An automated lawn sprinkler waters a sector of a yard. The water reaches a radius of $20$ feet, and the sprinkler rotates through a central angle of $135^\circ$. Find…
An automated lawn sprinkler waters a sector of a yard. The water reaches a radius of $20$ feet, and the sprinkler rotates through a central angle of $135^\circ$. Find the area of the lawn that receives water, rounding your answer to the nearest square foot.
Two major US cities, Seattle, WA (latitude $47.6^\circ$ N) and San Francisco, CA (latitude $37.7^\circ$ N), lie on approximately the same longitude line. Assuming the…
Two major US cities, Seattle, WA (latitude $47.6^\circ$ N) and San Francisco, CA (latitude $37.7^\circ$ N), lie on approximately the same longitude line. Assuming the Earth is a perfect sphere with a radius of $3,959$ miles, calculate the distance between the two cities along the Earth's surface. Round your answer to the nearest mile.
Radians
Degrees are an arbitrary choice: $360$ comes from ancient astronomers. A more natural unit is built from the circle itself.
The circumference is $2\pi r$, so exactly $2\pi$ of those radius-length arcs fit around a circle. That gives the key conversion.
| Degrees | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | $120^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|---|---|---|
| Radians | $\tfrac{\pi}{6}$ | $\tfrac{\pi}{4}$ | $\tfrac{\pi}{3}$ | $\tfrac{\pi}{2}$ | $\tfrac{2\pi}{3}$ | $\pi$ | $\tfrac{3\pi}{2}$ | $2\pi$ |
Why? By definition, one radian cuts off an arc of length $r$, so $\theta$ radians cut off $\theta$ times that: $s = r\theta$. For area, the fraction of the circle is $\tfrac{\theta}{2\pi}$, so $A = \tfrac{\theta}{2\pi}\cdot \pi r^2 = \tfrac12 r^2\theta$.
A grandfather clock features a pendulum that is $30$ inches long. During one full swing, the pendulum travels through an angle of $0.35$ radians. Calculate the exact…
A grandfather clock features a pendulum that is $30$ inches long. During one full swing, the pendulum travels through an angle of $0.35$ radians. Calculate the exact distance the tip of the pendulum travels in one swing.
An automotive testing facility in Michigan features a perfectly circular test track with a radius of $0.75$ miles. A prototype vehicle drives along the track for a total…
An automotive testing facility in Michigan features a perfectly circular test track with a radius of $0.75$ miles. A prototype vehicle drives along the track for a total distance of $2.4$ miles. Calculate the central angle of the curve the car has traveled, giving your answer first in radians, and then in degrees (rounded to the nearest tenth).
A circular table saw blade used in a woodworking shop has a diameter of $10$ inches and spins at a rate of $4,500$ revolutions per minute (RPM). Calculate the angular…
A circular table saw blade used in a woodworking shop has a diameter of $10$ inches and spins at a rate of $4,500$ revolutions per minute (RPM). Calculate the angular velocity of the blade in radians per second, and then determine the exact linear distance a single tooth on the edge of the blade travels in $0.15$ seconds.
Segments and composite regions
Most real problems ask for an area that is a combination of pieces: a window pane, a lawn with a curved edge, a logo. The strategy is always the same: sketch, find the radius and angles you need, then add or subtract simple areas.
A segment of a circle is the region between a chord and its arc. It is a sector minus a triangle.
Tangents come up in composite problems too. Remember that a tangent is perpendicular to the radius at the point of tangency, which gives you a right triangle.
A right triangle is inscribed in a semicircle as shown in the diagram. The diameter of the semicircle is $34$ inches, and one leg of the inscribed triangle measures $16$…
A right triangle is inscribed in a semicircle as shown in the diagram. The diameter of the semicircle is $34$ inches, and one leg of the inscribed triangle measures $16$ inches. Find the exact area of the shaded region (the area of the semicircle minus the area of the triangle).
A historic building in Philadelphia features a circular window with a radius of $24$ inches. A straight horizontal lead mullion (a chord) divides the window into two…
A historic building in Philadelphia features a circular window with a radius of $24$ inches. A straight horizontal lead mullion (a chord) divides the window into two segments, such that the central angle intercepting the chord is $120^\circ$. Find the exact area of the smaller glass segment (the region bounded by the chord and the minor arc).
A security camera is mounted on a wall $130$ feet away from the exact center of a circular restricted zone. The restricted zone has a radius of $50$ feet. The camera's…
A security camera is mounted on a wall $130$ feet away from the exact center of a circular restricted zone. The restricted zone has a radius of $50$ feet. The camera's field of view is perfectly bounded by the two tangent lines extending from the camera to the circular zone. Calculate the exact length of the arc of the restricted zone that is visible to the camera.