Circle and Circumference

Identify the names of the following coloured regions inside or around the circle:

• A: Region bounded by two radii.

• B: Region bounded by two concentric circles (like a doughnut).

• C: Half of a circle.

In the following drawing of a circle with center $O$, identify the elements labeled with letters. State which corresponds to the radius, which to the diameter, and which to the chord.

If a $2$ pesos coin has a radius of $12.875$ mm, what is its diameter? Express the answer in millimeters.

Look at the position of the lines $r$, $s$ and $t$ relative to the circle with center $C$. Classify each line as tangent, secant or exterior.

Given a circle with center at point $C$ and a radius of $5$ cm, state whether the following points are inside, outside or on the circle:

• Point $P$: is $3$ cm from $C$.

• Point $Q$: is $5$ cm from $C$.

• Point $R$: is $7$ cm from $C$.

The logo of a new sportswear brand consists of a circle of radius $r = 5\text{ cm}$ crossed by a straight line. If the line passes at a distance of $3\text{ cm}$ from the center of the circle, draw a sketch and answer:

1. a) What is the position of the line relative to the circle?
2. b) If we draw another line passing at exactly $5\text{ cm}$ from the center, what would that line be called?
3. c) What is the name of the segment of the first line that lies inside the circle?

Álex wants to put a strip of LED lights around the edge of his circular mirror. If the radius of the mirror is $40$ cm, what is the minimum length in centimeters of LED strip he will need to buy? (Use $\pi \approx 3.14$).

The wheel of a mountain bike has a total radius of $35\text{ cm}$. If the cyclist travels a distance of $1.1\text{ kilometers}$ in a straight line:

1. a) How many meters does the bicycle travel in a single revolution of the wheel?
2. b) How many complete revolutions will the wheel have made by the end of the ride?

(Given: $1\text{ km} = 1000\text{ m}$; use $\pi \approx 3.14$).

A basketball player is practicing free throws. The circle painted in the key has a diameter of $3.6$ meters. What is the length of the radius of this circle?

A pizzeria offers an individual pizza with a diameter of $20$ cm. Calculate the area of the pizza to find out how much space it will take up in the box. (Use $\pi \approx 3.14$).

A pizzeria has two offers for its artisan pizzas:

• Offer A: One family pizza with a diameter of $40\text{ cm}$.

• Offer B: Two medium pizzas with a diameter of $20\text{ cm}$ each.

If both offers cost the same, calculate the surface area of pizza in each option (using $\pi \approx 3.14$) and explain which of the two offers is better value for the customer.

Observe the following figure representing a circular spinner with a central bearing and an outer ring. The radius of the inner circle is $2\text{ cm}$ and the radius of the outer circle is $5\text{ cm}$. Calculate the area of the annulus (the bluish area) between the two.

A garden sprinkler rotates through a circular sector of $120^\circ$. If the jet of water reaches a distance of $6\text{ meters}$, calculate:

1. a) The arc length of lawn that gets wet on the outer edge.
2. b) The total area of lawn watered by the sprinkler.

(Use $\pi \approx 3.14$ and round to two decimal places).

In an amusement park, a Ferris wheel has a radius of $15 \text{ meters}$. A sensor detects that one of the cabins has traveled along an arc of a circle whose length is $23.55 \text{ meters}$. Through what angle, expressed in sexagesimal degrees, has the Ferris wheel turned since the cabin was at the lowest point? Consider the value of $\pi \approx 3.14$.

A circular silver bracelet has an outer diameter of $8 \text{ cm}$ and an inner diameter of $7 \text{ cm}$. If we want to gold-plate only the top part of the bracelet (a sector of the annulus covering an angle of $120^\circ$), what is the surface area that will be gold-plated? Round the result to two decimal places.

In a circular square, a structure with cables has been installed. One of the cables forms an inscribed angle of $35^\circ$ subtending an arc of the square.

1. a) Calculate the measure of the central angle subtending the same arc.
2. b) If the central angle were $110^\circ$, what would be the measure of any inscribed angle subtending the same arc?

Look at a wall clock at $10:00$ in the morning. The hands (minute and hour) form a central angle. If we draw a chord joining the tips of both hands, assuming both measure $10 \text{ cm}$, what is the length of this chord? (Hint: analyze the type of triangle formed between the center of the clock and the numbers 10 and 12).

An engineer is designing a mechanical part consisting of a metal disk of radius $R = 12 \text{ cm}$. To make the part lighter, a straight cut (a chord) must be made at an exact distance of $7.2 \text{ cm}$ from the center of the disk. Using the Pythagorean theorem, determine the total length of this cut (the chord) in order to program the cutting machine.

In the design of a circular rose window, a square with a diagonal of $8\text{ decimeters}$ has been inscribed, as shown in the figure. Calculate the area of the glass surface (circle) that lies outside the square (the shaded areas in the drawing).

A "No entry" road sign is circular with a diameter of $60\text{ cm}$. Inside it, there is a white rectangular strip measuring $50\text{ cm}$ long and $10\text{ cm}$ wide. Calculate the area of the red part of the sign (the region of the circle not occupied by the white rectangle).

In the design of a new logo for a tech company, a square with side length $L = 10 \text{ cm}$ has been drawn. An arc of a circle with a radius equal to half the side of the square ($5 \text{ cm}$) has been drawn at each of its vertices, such that the four arcs touch at the midpoints of the sides. Calculate the area of the shaded region remaining in the center of the square (the area that does not belong to any of the four circular sectors).