Exercises: 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.

The municipality wants to fence a rectangular park measuring $40$ meters long and $30$ meters wide. In addition, they want to build a straight path crossing the park from one corner to the opposite one (the diagonal). Use the Pythagorean 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.

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 $\pi \approx 3.14$).

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

A sports complex 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 $\pi \approx 3.14$).

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.

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 $5$ meters so that its foot is exactly $1.4$ meters 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 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.

An architect has designed a window for a cultural center in the shape of an isosceles trapezoid. The longer base measures $2.8$ meters, the shorter base measures $1.6$ meters and the non-parallel sides measure $1.2$ meters 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?

We are designing a children's playground with a circular play area. On the plan, we have drawn a triangle inscribed in the circle of the play area such 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. Justify what type of triangle it is according to its angles.

2. Calculate the area of the play area region that lies outside the triangle.

In a plaza in your town, there is a giant tile shaped like an annulus. The radius of the outer circle is $4$ meters. 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.

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

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?

Calculate the area of a circular segment with an angle of $90^\circ$ belonging to a circle of radius $12$ cm.

A plot of land is in the shape of a rhomboid with a base of $15$ meters and a height of $8$ meters. In one of the corners, the owner has installed a sprinkler that waters through an angle of $60^\circ$ with a reach of $6$ meters.

1. Calculate the total area of the plot.

2. Calculate the area of the plot that remains unwatered.

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 technology park, a plaza has been built in the shape of a regular hexagon with side length $20\text{ m}$. A circular fountain has been installed in the center of the plaza, with its edge tangent to all sides of the hexagon. The municipal government wants to pave only the area between the fountain and the perimeter of the hexagon using special tiles. Calculate the area to be paved using $\pi \approx 3.1416$ and rounding the result to two decimal places.

A graphic design company is creating a new logo based on the properties of a circle. They have drawn a circle of radius $R = 8\text{ cm}$. Inside it, they have drawn two parallel chords located on opposite sides of the center, one of length $10\text{ cm}$ and another of length $14\text{ cm}$. The area of the logo is the isosceles trapezoid formed by joining the endpoints of these two chords. Calculate the area of this trapezoid and the exact distance separating the two chords.

A surveillance drone has a field of view of $90^\circ$. If the drone is located at a height of $90\text{ m}$ directly above the center point of a rectangular plot, the distance from the drone to any vertex of the plot is $150\text{ m}$, and from its position the vertices of one of the longer sides of the plot are seen under that $90^\circ$ angle, calculate the length of the plot. In addition, determine the total area of the rectangular plot of land.

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 decimeter. 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.

A city's "Cathedral Rose Window" consists of an outer circle with a diameter of $6\text{ m}$. Inside it, four equal circles are drawn whose diameters coincide with four perpendicular radii of the outer circle, forming a four-petaled flower. Calculate the area of the four petals formed by the overlap of these four circles.

A traffic sign has the shape of a regular octagon. We know that the maximum distance between two opposite vertices (the diameter of the circumscribed circle) is $80\text{ cm}$. A worker must paint the sign red, except for a central square whose vertices coincide with the midpoints of four of the sides of the octagon (forming an interior cross). Calculate the area that must be painted red and the interior angle of that octagon.