Exercises: Circumference and circle

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

A basketball player is practising free throws. The circle painted in the key has a diameter of $3,6$ metres. What is the length of the radius of this 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 centimetres of LED strip he will need to buy? (Use $\pi \approx 3,14$).

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 $\pi \approx 3,14$).

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

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.

If a $2$ euro coin has a radius of $12,875$ mm, what is its diameter? Express the answer in millimetres.

Given a circle with centre 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 $C$.

For a school party, Marcos wants to calculate the surface area of a circular tablecloth that completely covers a round table whose diameter is $120\text{ cm}$. Marcos does the following calculation in his notebook:

1. a) Identify what conceptual error Marcos made when applying the area formula.
2. b) Explain the relationship between the diameter and the radius, and reason why the result obtained by Marcos is exactly four times larger than it should be.
3. c) Correct the calculation and find the real area that the tablecloth must have (in $\text{cm}^2$, using $\pi \approx 3,14$).

A municipal parks and gardens department has a flexible fence of exactly $31,4\text{ metres}$ in length to completely enclose a circular flowerbed, with no leftover or missing material.

1. a) State the corresponding mathematical relationship and calculate what the radius and the diameter of the flowerbed must measure (use $\pi \approx 3,14$).
2. b) If a sprinkler is placed in the exact centre of the flowerbed, what is the maximum distance from the sprinkler at which a flower can grow inside the enclosure?
3. c) Reason geometrically whether it would be possible to draw a straight chord inside the flowerbed connecting two points on the boundary that measures $11\text{ metres}$.

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 centre 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 centre, what would that line be called?
3. c) What is the name of the segment of the first line that lies inside the circle?

You can do it on paper and send a photo with your answer.

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.

A garden sprinkler rotates through a circular sector of $120^\circ$. If the jet of water reaches a distance of $6\text{ metres}$, 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 the design of a circular rose window, a square with a diagonal of $8\text{ decimetres}$ 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).

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

1. a) How many metres 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 $\pi \approx 3,14$).

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.

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?

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 a town square, a circular fountain with a diameter of $8\text{ metres}$ is going to be built. A stone kerb will be placed around its entire perimeter and the floor (the entire circular surface) will be waterproofed.

1. a) Before calculating with decimals, make a mental estimate of the length of the kerb by approximating $\pi \approx 3$. Reason whether the actual measurement will be greater than or less than your estimate.
2. b) Imagine a square circumscribed about the fountain (whose four sides touch the circle externally). How long is the side of that square? What is its perimeter? Justify why the perimeter of that square serves as an upper bound for the circumference of the circle.
3. c) Accurately calculate the length of the kerb and the area of the fountain's base using $\pi \approx 3,14$, and check whether both values are consistent with the deductions you made in the previous sections.

In the public bicycle sharing system of a Spanish city, the vehicle wheels have a radius of $30\text{ cm}$. A user travels in a straight line along the bike lane from one station to another, covering a total distance of $1\,130,4\text{ metres}$.

1. a) Model the situation by calculating how many metres the bicycle advances each time the wheel makes a full turn on the pavement (use $\pi \approx 3,14$).
2. b) Determine how many full turns each wheel will have completed by the end of the entire journey.
3. c) During an inspection, a mechanic notes that they replaced a $30\text{ cm}$ wheel spoke and a straight metal reinforcement (chord) that connects two points on the outer edge of the rim and measures $65\text{ cm}$. Relying on the definitions of chord and diameter, explain why the measurement of this reinforcement is physically impossible for that wheel.

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 centre of the square (the area that does not belong to any of the four circular sectors).

An engineer is designing a mechanical part consisting of a metal disc 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 centre of the disc. Using Pythagoras' Theorem, determine the total length of this cut (the chord) in order to program the cutting machine.

In an amusement park, a Ferris wheel has a radius of $15 \text{ metres}$. A sensor detects that one of the cabins has traveled along an arc of a circle whose length is $23,55 \text{ metres}$. 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.

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: analyse the type of triangle formed between the centre of the clock and the numbers 10 and 12).

In the centre of a circular roundabout with radius $R = 10\text{ metres}$, there is a pond, also circular, with radius $r = 6\text{ metres}$. The remaining strip between the pond and the edge of the roundabout (an annulus) will be covered with artificial turf. To calculate the surface area of the turf, three members of the technical team suggest different methods:

• Clara's method: She calculates the area of the entire roundabout ($A_1 = 3,14 \cdot 10^2$), the area of the pond ($A_2 = 3,14 \cdot 6^2$), and subtracts the two results: $A_1 - A_2$.

• Bruno's method: He directly subtracts the radii ($10 - 6 = 4\text{ metres}$) and calculates the area of a circle with that resulting radius: $A = 3,14 \cdot 4^2$.

• Sara's method: She factors out $\pi$ using the distributive property: $A = \pi \cdot (10^2 - 6^2) = 3,14 \cdot (100 - 36)$.

1. a) Explain which method is incorrect and show geometrically why subtracting the radii before calculating the area leads to a false result.
2. b) Compare the two valid methods and justify which one is more advisable to minimise rounding errors and save calculation steps.
3. c) If the pond were expanded to a radius of $8\text{ metres}$ while maintaining the outer radius of the roundabout at $10\text{ metres}$, apply the method you selected as optimal to calculate the new area of turf required.

A public water company needs to replace two identical drainage pipes, each with a circular cross-section and a diameter of $20\text{ cm}$, with a single new circular pipe that has exactly the same discharge capacity (that is, whose cross-sectional area equals the sum of the areas of the two old pipes combined).

1. a) A technician claims: "Since we are replacing two $20\text{ cm}$ pipes, the new one must have a diameter of $40\text{ cm}$ to discharge the same amount." Demonstrate by calculating the cross-sectional areas whether the technician's statement is correct or incorrect.
2. b) Reason algebraically from the formula $A = \pi \cdot r^2$ why doubling the diameter of any circle multiplies its area by $4$ and not by $2$.
3. c) State the equation relating the area of the new pipe to the sum of the two old ones and calculate approximately what the actual diameter of the replacement pipe must measure (round your answer to one decimal place).