Exercises: Inscribed and central angles

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.

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 $\angle ABC$.

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 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 the area of the lawn that receives water, rounding your answer to the nearest square foot.

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.

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.

A windshield wiper clears a sector of a car's rear window. The wiper blade has a length of $14$ inches and rotates through an angle of $\frac{2\pi}{3}$ radians. Calculate the exact area of the glass cleared by the wiper blade.

A circular patio features a decorative quadrilateral $ABCD$ inscribed within it. The measures of two opposite angles are given by algebraic expressions: $m\angle A = (4x + 15)^\circ$ and $m\angle C = (6x - 25)^\circ$. Determine the value of $x$ and the exact measure of both angles.

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).

The city of Boston is renovating a park and plans to pave a section shaped like a circular sector. The sector has a radius of $45$ feet and a central angle of $80^\circ$. If the special paving stones cost $\$14.50$ per square foot, calculate the total cost to pave this sector.

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 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.

The Seattle Great Wheel is a giant Ferris wheel with a radius of $87.5$ feet. If a passenger cabin rotates through an angle of $\frac{7\pi}{4}$ radians before stopping, calculate the exact distance the cabin has traveled along the circumference.

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 $WXYZ$.

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.

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.

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 cyclic trapezoid $ABCD$ is inscribed in a circle, with side $AB$ parallel to side $CD$. Prove mathematically that the trapezoid must be isosceles by finding the measures of the intercepted arcs, given that the measure of the base angle $\angle A$ is $74^\circ$. In your final answer, state the measures of all other interior angles ($\angle B$, $\angle C$, and $\angle D$).

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.

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 $\angle ABD$.

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.

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.

A regular hexagon is inscribed in a circle with a radius of $12$ inches. The region inside the circle but outside the hexagon consists of six identical circular segments. Calculate the exact total area of these six segments, and then provide an approximate decimal answer rounded to the nearest hundredth of a square inch.