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 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.

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 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.

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 $\pi \approx 3.14$).

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.

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.

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 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 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 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?

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).

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.

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?

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 $15$ metres and a height of $8$ metres. In one of the corners, the owner has installed a sprinkler that waters through an angle of $60^\circ$ with a reach of $6$ metres.

1. Calculate the total area of the plot.

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

In the design of a new roundabout in Galway, a central landscaped area shaped like a regular hexagon with a side length of $10\text{ m}$ is planned, which is inscribed inside an outer circular curb whose vertices coincide with those of the hexagon.

1. a) Before performing detailed calculations, mentally estimate roughly what percentage of the total area of the circle the hexagonal landscaped area will occupy (around $50\,\%$, $80\,\%$, or over $95\,\%$?). Justify your intuition by visually comparing both geometric figures.
2. b) Calculate the area of the circumscribed circle and the area of the regular hexagon (first determine its apothem by applying the Pythagorean theorem to one of the equilateral triangles that make it up). Use $\pi \approx 3.14$.
3. c) Find the actual percentage of the surface area that the hexagon occupies relative to the circle and compare it with your initial estimate to assess whether your prediction was reasonable.

A forest fire watchtower of the emergency service is located at the top of a hill in the Wicklow Mountains region. The tower has a direct visual range of $12\text{ km}$ all around in all directions ($360^\circ$). However, a rock formation located to the northwest completely blocks visibility across an angular sector of $60^\circ$. The prevention plan classifies the area that is visible from the tower into two categories:

• Rapid intervention zone: at a distance less than or equal to $4\text{ km}$ from the tower.

• Regional alert zone: at a distance greater than $4\text{ km}$ and up to the maximum range of $12\text{ km}$.

1. a) Model the situation by calculating the total forest area (in $\text{km}^2$) that the lookouts can effectively monitor from the tower (use $\pi \approx 3.14$).
2. b) Calculate the area corresponding to the visible "regional alert zone".
3. c) If the emergency services detect a smoke plume in the visible area, is it more likely that the source is in the rapid intervention zone or in the regional alert zone, assuming that the risk is distributed homogeneously across the terrain? Justify your answer by quantitatively comparing both areas.

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 square has been built in the shape of a regular hexagon with side length $20\text{ m}$. A circular fountain has been installed in the centre of the square, with its edge tangent to all sides of the hexagon. The local council wants to pave only the area between the fountain and the perimetre 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 centre, one of length $10\text{ cm}$ and another of length $14\text{ cm}$. The area of the logo is the isosceles trapezium formed by joining the endpoints of these two chords. Calculate the area of this trapezium 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 centre 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 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.

Cork'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.

To manufacture an agricultural discharge hopper, a workshop needs to cut a sheet of steel in the shape of an isosceles trapezoid whose bases measure $30\text{ cm}$ and $14\text{ cm}$, and whose legs measure $17\text{ cm}$ each. Two 3rd Year students propose different ways to find the area:

• Aoife's strategy: Draw the two heights from the ends of the shorter base to decompose the trapezoid into a central rectangle and two equal right triangles at the ends; calculate the height using the Pythagorean theorem and add the three partial areas.

• Seán's strategy: Extend the legs upwards until they intersect at an external point, forming a large isosceles triangle that encompasses the trapezoid and a small upper isosceles triangle; calculate the heights of both triangles using similarity, calculate their areas, and subtract them.

1. a) Fully develop Aoife's strategy and find the area of the trapezoid.
2. b) Analyse Seán's proposal: is it geometrically valid? What additional knowledge does it require that makes it an unnecessarily complex route compared to Aoife's?
3. c) Prove how Aoife's decomposition reasoning allows you to justify the classic area formula for any trapezoid in which the feet of those heights fall on the longer base, $A = \dfrac{(B + b) \times h}{2}$, in a general way by algebraically grouping the rectangle and the two triangles.

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.

Unit Challenge. How would you optimize the space of a square in your municipality? Imagine that the town hall commissions you to design the geometric remodelling of a nearby park.

1. Scale and similarity: Capture the park on Google Maps or on the Sede del Catastro. Using the measuring tool, obtain a real reference distance and transfer the map to GeoGebra by applying an appropriate scale factor $k$.

2. Geometric loci: Locate three main non-collinear entrances and find the point equidistant to them (circumcentre using perpendicular bisectors) to install a central streetlamp; check that it lies inside the park and, if not, justify another location. In addition, draw the angle bisector formed by two pathways to place a row of trees.

3. Angles, lengths, and areas: Measure the angles of the park polygon. Decompose the area into elementary plane figures (triangles, trapezoids, or circular sectors) and calculate its perimeter and its total real area in $\text{m}^2$.

Final product: A short report (one page) with the GeoGebra screenshot, the working out of the calculations, and the estimated budget if planting grass costs $7.50$ €/m$^2$.

Assessment criteria:

• Precision in the use of scales and geometric proportionality with real data.

• Correct construction and justification of the geometric loci (perpendicular bisectors and angle bisector).

• Accuracy in calculating angles, lengths, and composite areas.