Exercises: Angles, lengths and areas

A designer is creating a mosaic for the floor of a square 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 city council wants to fence a rectangular park that is $40$ feet long and $30$ feet wide. In addition, they want to build a straight path that crosses the park from one corner to the opposite one (the diagonal). Use the Pythagorean theorem to calculate the length of this path.

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

In a pizzeria, a family pizza has a diameter of $40$ in. If we cut a slice that forms a circular sector with a central angle of $60^\circ$, calculate the area of that slice 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 $\angle ACB$ that subtends the same arc.

A sports center has a swimming pool with an unusual shape: a rectangle of $20$ ft by $10$ ft, and on one of its short sides a semicircular area for children has been added. Calculate the total perimeter of the pool. (Use $\pi \approx 3,14$).

At a certain time of day, a vertical post $2,5$ feet high casts a shadow on the ground of $1,5$ feet. At that same moment, the shadow of a tree measures $6$ feet. Applying Thales's theorem, calculate the height of the tree.

Calculate the area of a regular pentagon if we know that its side measures $10$ in and its apothem measures approximately $6,88$ in.

A maintenance worker wants to install a security camera in the corner of a rectangular warehouse. To do this, he leans a ladder $5$ feet long so that its foot is exactly $1,4$ feet from the wall. However, he realizes that to get the correct viewing angle, he needs the camera to be $60$ inches higher. What minimum length should the new ladder have if he keeps the foot in the same place?

A design company is creating a new logo based on a regular hexagon with a side of $6$ in. Inside the hexagon, they have drawn the diagonals connecting alternate vertices, forming a "Star of David" or hexagram. Calculate the value of the interior angles of the triangles that form the star and determine the area of the shaded part (the central star) knowing that the apothem of the original hexagon is approximately $5,2$ in.

An architect has designed a window for a cultural center in the shape of an isosceles trapezoid. The larger base measures $2,8$ feet, the smaller base measures $1,6$ feet and the non-parallel sides measure $1,2$ feet each. For the budget, 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 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 $10$ feet. If another side measures $6$ feet:

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

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

In a square in your city, there is a giant paving stone in the shape of a circular ring (annulus). The radius of the outer circle is $4$ feet. It is known that the area of the walkable part (the ring itself) is $7\pi$ $m^2$. Calculate the radius of the inner circle and determine the length of the circular fence that would need to be put up to surround only the inner hole.

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

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 of amplitude $90^\circ$ belonging to a circle of radius $12$ in.

Hint: Remember that the area of the circular segment is obtained by subtracting the area of the triangle formed by the chord from the area of the corresponding circular sector.

A plot of land has the shape of a rhomboid (parallelogram) with a base of $15$ feet and a height of $8$ feet. In one of the corners, the owner has installed a sprinkler that waters at an angle of $60^\circ$ with a range of $6$ feet.

1. Calculate the total area of the plot.

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

In a surveying training exercise, a group of students needs to calculate the height of a mountain. To do this, they place two vertical stakes separated by a distance of $150\text{ ft}$ on flat ground. From the base of the first stake, which measures $2\text{ ft}$, they observe that its shadow cast by the sun ends at exactly the same point as the shadow of the mountain peak. If the shadow of the stake measures $3,5\text{ ft}$ and the distance from the second stake to the top of the mountain (projected on the ground) is $1,2\text{ mi}$, calculate the actual height of the mountain and the straight-line distance a drone would have to travel from the base of the first stake to the top of the mountain.

An architect is designing a large 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, he wishes to place a rectangular solar panel whose base rests on the base of the triangle and its 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 a side of $20\text{ m}$. In the center of the plaza, a circular fountain has been installed whose edge is tangent to all the sides of the hexagon. The city council wants to pave with special tiles only the surface area between the fountain and the perimeter of the hexagon. 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 the circle. They have drawn a circle with a radius of $R = 8\text{ cm}$. Inside it, they have drawn two parallel chords located on either side of the center, one $10\text{ cm}$ and the other $14\text{ cm}$ in length. The area of the logo is the isosceles trapezoid formed by joining the ends of these two chords. Calculate the area of this trapezoid and the exact distance separating the two chords.

A surveillance drone has a viewing angle of $90^\circ$. If the drone is at a height of $120\text{ m}$ above the central point of a rectangular plot, and from its position the vertices of one of the shorter sides of the plot are seen under this angle of $90^\circ$, calculate the width of the plot. Furthermore, if the distance from the drone to any vertex of the plot is $150\text{ m}$, 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 $13\text{ in}$, $14\text{ in}$ and $15\text{ in}$. The glass used costs $45\text{ \$}$ per square inch. Calculate the altitude of the triangle to the side of $14\text{ in}$ (using a system of equations derived from the Pythagorean theorem) and the total cost of the glass needed for the skylight.

The "Cathedral Rose Window" of a city is formed by an outer circle of diameter $6\text{ m}$. Inside it, four equal semicircles are inscribed whose diameters coincide with the perpendicular radii of the outer circle, forming a sort of four-petaled flower. Calculate the area of the central region bounded by the intersection of these four semicircles (the shaded area in the figure).

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 inner cross). Calculate the area that must be painted red and the interior angle of this octagon.