Angles, lengths and areas

A designer is creating a mosaic for a town square floor using tiles in the shape of a regular hexagon. Calculate the sum of all the interior angles of the hexagon and the value of each of its interior angles.

During the redevelopment works of a pedestrian square in Toledo, stone tiles shaped like regular polygons are going to be laid. The municipal architect has specified in the blueprints that each of the interior angles of these tiles must measure exactly $135^\circ$.

1. a) Set up an equation using the formula that relates the number of sides $n$ of a regular polygon to the measure of its interior angles, and solve it to find out which regular polygon it is.
2. b) The workers want to lay these tiles on the ground by joining them solely along their edges. Is it possible to tessellate the square using only this type of tile without leaving any gaps or overlapping? Justify your answer by analysing how many degrees the angles meeting at a single vertex must add up to.

A design company is creating a new logo based on a regular hexagon with side length $6$ cm. Inside the hexagon, they have drawn the diagonals connecting alternate vertices, forming a "Star of David" or hexagram. Calculate the interior angles of the triangles that form the star and determine the area of the shaded region (the central star), knowing that the apothem of the original hexagon is approximately $5.2$ cm.

Look at the following circle with centre $O$. If the central angle $\angle AOB$ measures $80^\circ$, calculate the value of the inscribed angle $\angle ACB$ subtended by the same arc.

We are designing a children's playground with a circular play area. On the plan, we have drawn a triangle inscribed in the circumference of the play area so that one of the sides of the triangle coincides exactly with the diameter of the circle, which measures $10$ meters. If another side measures $6$ meters:

1. a) Justify what type of triangle it is according to its angles.

1. b) Calculate the area of the region of the play area that lies outside the triangle (the shaded part of the figure).

A restoration workshop in Granada needs to restore a circular glass rose window with a diameter of $2\text{ m}$ (a radius of $1\text{ m}$). In the original design, a straight metal moulding connects two points $A$ and $B$ on the circumference so that the central angle subtending that arc measures exactly $120^\circ$. To secure the structure, a rod must be placed from any point $P$, located on the major arc $AB$, to the endpoints $A$ and $B$, forming the inscribed triangle $APB$.

1. a) Use the property relating an inscribed angle to the central angle that subtends the same arc to prove the exact value of the angle $\widehat{APB}$. Does this angle vary if you move point $P$ along this major arc? Justify your reasoning.
2. b) Mentally draw the radii $OA$ and $OB$ that go from the centre $O$ to the endpoints of the chord. Decompose the isosceles triangle $OAB$ into two right triangles by drawing the altitude from the centre to the chord, and apply the Pythagorean theorem (taking advantage of the fact that the acute angles measure $30^\circ$ and $60^\circ$) to calculate the distance from the centre $O$ to the chord and the exact length of the chord $AB$.
3. c) Calculate the area of the minor circular segment bounded by the chord $AB$ and the minor arc $AB$ (use $\pi \approx 3.14$). Explain in detail the subtraction of areas that yields this surface area.

To measure the height of a broken lamppost in an industrial estate in Limerick, Seán leans a $5\text{ m}$ long ladder against the top of the post, with the foot of the ladder placed $3\text{ m}$ away from the base of the post on the horizontal ground. To calculate the height $h$ of the lamppost, Seán writes in his notebook:

$$h^2 = 5^2 + 3^2 = 25 + 9 = 34 \implies h = \sqrt{34} \approx 5.83\text{ m}$$

1. a) Explain, without performing additional operations, why the numerical result obtained by Seán is physically impossible in this context.
2. b) Identify the conceptual error Seán made when applying the Pythagorean theorem.
3. c) Correct the procedure and calculate the actual height of the lamppost.

A maintenance worker wants to install a security camera in the corner of a rectangular warehouse. To do this, he leans a ladder of length $5$ metres so that its foot is exactly $1.4$ metres from the wall. However, he realizes that to get the correct viewing angle, he needs the camera to be $60$ cm higher. What minimum length should the new ladder have if he keeps the foot at the same point?

A signal mast $8$ metres high casts a shadow $6$ metres long on the ground at 11:00 am. At that same moment, the shadow of a nearby cooling tower measures $45$ metres.

1. Use similar triangles to calculate the height of the tower.

2. If the safety cable running from the top of the signal mast to the end of its shadow were taut, what length would it be?

A skylight is to be built in the ceiling of a living room in the shape of a scalene triangle. The sides of the triangle measure $13\text{ dm}$, $14\text{ dm}$ and $15\text{ dm}$. The glass used costs $€45$ per square decimetre. Calculate the height of the triangle on the side of length $14\text{ dm}$ (using a system of equations derived from Pythagoras' theorem) and the total cost of the glass required for the skylight.

The council wants to fence a rectangular park measuring $40$ metres long and $30$ metres wide. In addition, they want to build a straight path crossing the park from one corner to the opposite one (the diagonal). Use Pythagoras' theorem to calculate the length of this path.

A signmaking company is manufacturing a "Caution" sign in the shape of a rhombus. If the diagonals of the rhombus measure $16$ cm and $12$ cm respectively, calculate the area of the metal plate needed to manufacture it.

Calculate the area of a regular pentagon if we know that its side length is $10$ cm and its apothem is approximately $6.88$ cm.

An architect has designed a window for a cultural centre in the shape of an isosceles trapezium. The longer base measures $2.8$ metres, the shorter base measures $1.6$ metres and the non-parallel sides measure $1.2$ metres each. For the cost estimate, he needs to calculate the total area of the glass and the sum of the interior angles of the window to order the metal frame. What is the area of the window?

An architect is designing a window for a museum with a particular geometric shape. The design consists of an isosceles triangle whose equal sides measure $10\text{ m}$ and whose base measures $12\text{ m}$. Inside it, he wants to place a rectangular solar panel whose base rests on the base of the triangle and whose two upper vertices touch the equal sides of the triangle. If the architect wants the height of the rectangle to be exactly half of its base, calculate the dimensions of the rectangle and the area of the region of the triangle that remains uncovered by the solar panel.

In a pizzeria, a family pizza has a diameter of $40$ cm. If we cut a slice that forms a sector with a central angle of $60^\circ$, calculate the area of that piece of pizza. (Use $\pi \approx 3.14$).

A sports centre has a pool with a peculiar shape: a rectangle of $20$ m by $10$ m, with a semicircular children's area added on one of its shorter sides. Calculate the total perimeter of the pool. (Use $\pi \approx 3.14$).

In a square in your town, there is a giant tile shaped like an annulus. The radius of the outer circle is $4$ metres. The area of the walkable part (the annulus itself) is known to be $7\pi$ $m^2$. Find the radius of the inner circle and determine the length of the circular fence needed to enclose only the inner gap.