Circles: Circumference and Area

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

A basketball player is practicing free throws. The circle painted in the key has a diameter of $3,6$ feet. What is the length of the radius of that circle?

If a $2$ dollar coin has a radius of $12,875$ inches, what is its diameter? Express the result in inches.

Identify the name of the following colored regions inside or around the circle:

• A: Region enclosed between two radii.

• B: Region enclosed between two concentric circles (like a donut).

• C: Half of a circle.

Observe the position of the lines $r$, $s$ and $t$ with respect 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 $5$ inches, indicate whether the following points are interior, exterior, or on the circle:

• Point $P$: is $3$ inches from $C$.

• Point $Q$: is $5$ inches from $C$.

• Point $R$: is $7$ inches from $C$.

The logo of a new sportswear brand consists of a circumference 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 circumference, draw the diagram and answer:

1. a) What is the relative position of the line with respect to the circumference?
2. b) If we draw another line that passes 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 circumference?

Alex wants to put an LED light strip around the edge of his circular mirror. If the radius of the mirror is $40$ inches, how many inches of LED strip will he need to buy as a minimum? (Use $\pi \approx 3,14$).

A mountain bike wheel has a total radius of $35\text{ in}$. If the cyclist travels a distance of $1,1\text{ kilometers}$ in a straight line:

1. a) How many feet does the bicycle travel in a single turn of the wheel?
2. b) How many full turns will the wheel have made at the end of the journey?

(Fact: $1\text{ mi} = 1000\text{ ft}$; consider $\pi \approx 3,14$).

A pizzeria offers an individual pizza with a diameter of $20$ inches. Calculate the surface 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: A family pizza of $40\text{ cm}$ diameter.

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

If both offers cost the same, calculate the surface area of pizza in each option (using $\pi \approx 3,14$) and reason which of the two offers is more advantageous for the customer.

Look at the following figure which represents 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 circular ring (the bluish area) between them.

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

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

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

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

1. a) The length of the arc of grass that gets wet on the outer edge.
2. b) The total area of grass 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 travelled along an arc of a circle whose length is $23,55 \text{ meters}$. What angle, expressed in 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 circular ring that covers 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 plaza, a cable structure has been installed. One of the cables forms an inscribed angle of $35^\circ$ with respect to an arc of the plaza.

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

An engineer is designing a mechanical part consisting of a metal disc of radius $R = 12 \text{ cm}$. To lighten the part, they must make a straight cut (a chord) that is at an exact distance of $7,2 \text{ cm}$ from the center of the disc. Using the Pythagorean theorem, determine what the total length of this cut (the chord) will be in order to program the cutting machine.

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 connecting the ends 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).