Angles, Lengths, and Areas
Francheska wants to pave a hexagonal plaza near the Malecón, put a fence around her grandfather's lot, and decide which slice of pizza is the best deal. For all of that she needs to measure angles, lengths, and areas. Almost everything comes from a few ideas: the angles of a triangle add up to $180°$, the Pythagorean theorem, and breaking figures into simpler ones.
Types of angles and systems of measurement
An angle is measured in sexagesimal degrees. A full turn has $360°$. Each degree is divided into $60$ minutes ($1° = 60'$) and each minute into $60$ seconds ($1' = 60''$). We write $m\angle ABC$ for the measure of the angle $\angle ABC$.
| Type | Measure |
|---|---|
| Acute | between $0°$ and $90°$ |
| Right | $90°$ |
| Obtuse | between $90°$ and $180°$ |
| Straight | $180°$ |
| Reflex | between $180°$ and $360°$ |
Two angles are complementary if they add up to $90°$ and supplementary if they add up to $180°$. Two vertical angles are always equal.
Parallel lines, perpendicular lines, and transversals
When a line (the transversal) crosses two parallel lines ($\parallel$), eight angles are formed that are equal or supplementary in pairs.
To solve problems with parallel lines, follow Pólya's method: understand, plan, carry out, and review.
Perpendicular lines
Two lines are perpendicular ($\perp$) if they form a right angle when they intersect. Then all four angles formed measure $90°$: one is right and the others are vertical or supplementary to it.
The angles of a triangle
Relationship between sides and angles. In a triangle, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. Also, each side is shorter than the sum of the other two (the triangle inequality). With sides $3$, $4$, and $9$ you cannot form a triangle because $3 + 4 < 9$.
Angles and diagonals of polygons
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…
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.
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…
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.
Angles in the circle
A circle has a center $O$, a radius, a chord (joins two points on it), and a diameter (the longest chord, which passes through the center). A secant line cuts the circle at two points and a tangent touches it at only one, perpendicular to the radius at that point.
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Central angle: its vertex is the center. Its measure equals the arc $\overset{\frown}{AB}$ it intercepts.
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Inscribed angle: its vertex is on the circle. Its measure is half the arc it intercepts (or half the corresponding central angle).
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…
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.
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…
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:
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Justify what type of triangle it is according to its angles.
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Calculate the area of the play area region that lies outside the triangle.
The Pythagorean theorem
Typical uses:
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Finding a side: hypotenuse $c = \sqrt{a^2 + b^2}$; leg $b = \sqrt{c^2 - a^2}$.
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Diagonal of a rectangle $a \times b$: $\sqrt{a^2 + b^2}$.
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Height of an isosceles triangle: it is drawn to the base and the base is divided into two halves, which forms a right triangle.
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Height of an equilateral triangle with side $l$: $h = \sqrt{l^2 - (l/2)^2} = \frac{\sqrt 3}{2}\,l$.
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Apothem of a regular polygon: using the radius and half a side.
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…
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 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…
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?
Perimeters and areas of polygons
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…
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.
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 center in the shape of an isosceles trapezium. The longer base measures $2.8$ meters, the shorter base measures $1.6$…
An architect has designed a window for a cultural center in the shape of an isosceles trapezium. 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?
Circle and circular figures
| Figure | Length | Area |
|---|---|---|
| Circle (circumference / disk) | $L = 2\pi r$ | $A = \pi r^2$ |
| Arc / sector of $n°$ | $L = \dfrac{2\pi r \times n}{360}$ | $A = \dfrac{\pi r^2 \times n}{360}$ |
| Annulus (ring) | — | $A = \pi\,(R^2 - r^2)$ |
| Circular segment | — | sector − triangle |
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…
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 center 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…
A sports center 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 $4$ meters. The area of the walkable part (the annulus itself)…
In a square 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.
Calculate the area of a circular segment with an angle of $90^\circ$ belonging to a circle of radius $12$ cm.
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…
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.
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Calculate the total area of the plot.
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Calculate the area of the plot that remains unwatered.
Combined problems
Hard problems mix these ideas. Almost always they are solved by finding a hidden right triangle or by decomposing the figure (adding or subtracting areas).
Shadows and similarity. If two triangles have the same angles, their sides are proportional. At the same time of day, an object and its shadow form similar triangles: $\dfrac{\text{height}_1}{\text{shadow}_1} = \dfrac{\text{height}_2}{\text{shadow}_2}$. A $2$ m pole with a $1.5$ m shadow and a building with a $30$ m shadow: $h = \dfrac{2 \cdot 30}{1.5} = 40$ m.
Chords. The distance from the center to a chord is a leg: $d = \sqrt{R^2 - (\text{chord}/2)^2}$.
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.…
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.
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Use similar triangles to calculate the height of the tower.
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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?
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 $10\text{ m}$…
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…
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 municipality 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…
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…
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…
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 $\text{RD\$}\ 45$ 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…
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…
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.
Back to the beginning: a $45°$ slice of a pizza with a $40$ cm diameter has $157.1$ cm², while a quarter ($90°$) of a pizza with a $20$ cm diameter has only $78.5$ cm². The first is bigger. To compare slices, always compare areas.