Circumference and circle

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 centre $O$, identify the elements labelled with letters. State which corresponds to the radius, which to the diameter, and which to the chord.

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

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

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 $C$.

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.

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.

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 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 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 $\pi \approx 3.14$).

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 $\pi \approx 3.14$).

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 $6\text{ metres}$, 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 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).