Angles, Lengths, and Areas
You and a friend are splitting pizzas. Your slice is a 45° wedge of a 16-inch pizza. Your friend's is a quarter (90°) of a 10-inch pizza. Whose slice is bigger? To settle it, and to tile a hexagonal plaza or fence a park, you need to measure angles, lengths, and areas. Almost everything comes from a few ideas: angles in a triangle add to 180°, the Pythagorean Theorem, similar triangles, and cutting figures into simpler pieces.
Angle Relationships
A line that crosses two other lines is called a transversal. When the two lines are parallel, the angles it makes come in equal pairs.
The same idea explains why the angles of a triangle add to $180°$. Draw a line through one vertex parallel to the opposite side. The alternate interior angles are equal, so the three angles of the triangle fit together on one straight line.
A designer is creating a mosaic for the floor of a plaza 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 the floor of a plaza 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.
Angles in a Circle
Two kinds of angles show up in circles:
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A central angle has its vertex at the center. Its measure equals the measure of the arc it cuts off.
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An inscribed angle has its vertex on the circle. Its measure is half of the central angle that cuts off the same arc.
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…
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.
The Pythagorean Theorem
Drag the sliders and watch the squares:
Typical uses:
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Missing hypotenuse: $c = \sqrt{a^2 + b^2}$.
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Missing 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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Distance between two points when you can sketch a right triangle.
Special right triangles
Two right triangles come up so often that their side ratios are worth memorizing. Both come from the Pythagorean Theorem.
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45°-45°-90°: the legs are equal ($s$), and the hypotenuse is $\sqrt{s^2 + s^2} = s\sqrt{2}$.
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30°-60°-90°: cut an equilateral triangle of side $2s$ in half. You get a right triangle with short leg $s$, hypotenuse $2s$, and long leg $\sqrt{(2s)^2 - s^2} = s\sqrt{3}$.
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So the height of an equilateral triangle with side $\ell$ is $h = \dfrac{\sqrt{3}}{2}\,\ell \approx 0.866\,\ell$.
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…
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 maintenance worker wants to install a security camera in the corner of a rectangular warehouse. To do this, he leans a ladder $5$ long so that its foot is exactly…
A maintenance worker wants to install a security camera in the corner of a rectangular warehouse. To do this, he leans a ladder $5$ long so that its foot is exactly $1,4$ 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 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…
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.
Similar Triangles and Shadows
Two figures are similar if they have the same shape: corresponding angles are equal and corresponding sides are proportional. For triangles, two equal angles are enough (AA similarity).
The ratio of corresponding sides is the scale factor $k$.
Shadows are the classic use. The sun's rays are parallel, so every vertical object makes the same angle with its shadow. Add the right angle at the ground and you get two triangles with two equal angles.
Some problems say "apply Thales's theorem" to a shadow problem. It just means this: use the proportion from similar triangles.
Sometimes the triangles share the shadow's tip, as in the figure.
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.…
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.
A signal mast $8$ feet high casts a shadow of $6$ feet on the ground at 11:00 a.m. At that same moment, the shadow of a nearby cooling tower measures $45$ feet. 1. Use…
A signal mast $8$ feet high casts a shadow of $6$ feet on the ground at 11:00 a.m. At that same moment, the shadow of a nearby cooling tower measures $45$ feet.
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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?
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…
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 $150\text{ m}$ on flat ground. From the base of the first stake, which measures $2\text{ m}$, 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 $3,5\text{ m}$ and the distance from the second stake to the top of the mountain (projected on the ground) is $1,2\text{ km}$, calculate the real 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.
Areas of Triangles and Quadrilaterals
A signage company is manufacturing a "Caution" sign in the shape of a rhombus. If the diagonals of the rhombus measure $16$ inches and $12$ inches respectively,…
A signage company is manufacturing a "Caution" sign in the shape of a rhombus. If the diagonals of the rhombus measure $16$ inches and $12$ inches respectively, calculate the area of the metal plate needed to manufacture it.
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 $1,6$…
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?
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…
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 foot. 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.
Regular Polygons
A regular polygon has equal sides and equal angles. Its apothem $a$ is the segment from the center to the midpoint of a side, and it is perpendicular to that side.
Hexagons are special. The six triangles are equilateral, so the radius equals the side. The apothem is the height of one of those triangles: $a = \sqrt{s^2 - (s/2)^2} = \dfrac{\sqrt3}{2}\,s$.
If a circle is inscribed in a regular polygon (it touches all the sides), its radius is the apothem. If the circle goes through the vertices, its radius is the distance from the center to a vertex.
Calculate the area of a regular pentagon if we know that its side measures $10$ in and its apothem measures approximately $6,88$ in.
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 design company is creating a new logo based on a regular hexagon with a side of $6$ inches. Inside the hexagon, they have drawn the diagonals connecting alternate…
A design company is creating a new logo based on a regular hexagon with a side of $6$ inches. 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 $5,2$ inches.
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…
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.
Circumference and Area of a Circle
| Figure | Length | Area |
|---|---|---|
| Circle | circumference $C = 2\pi r = \pi d$ | $A = \pi r^2$ |
| Ring (annulus) | — | $A = \pi\,(R^2 - r^2)$ |
A ring is a big circle (radius $R$) with a smaller circle (radius $r$) removed from the middle, so you subtract the two areas.
Unless the problem says otherwise, you can leave answers in terms of $\pi$ (such as $64\pi$), or use $\pi \approx 3.14$ when asked to approximate.
A sports center has a swimming pool with a peculiar shape: a rectangle of $20$ ft by $10$ ft, and on one of its short sides a semicircular area for children has been…
A sports center has a swimming pool with a peculiar 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 $\pi \approx 3,14$).
In a plaza 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…
In a plaza 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.
Arcs, Sectors, and Segments
A central angle of $n°$ cuts off the fraction $\dfrac{n}{360}$ of the whole circle. That fraction applies to the circle's circumference (the arc) and to its area (the sector).
In a pizzeria, a family pizza has a diameter of $40$ inches. If we cut a slice that forms a circular sector with a central angle of $60^\circ$, calculate the area of…
In a pizzeria, a family pizza has a diameter of $40$ inches. 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 $\pi \approx 3,14$).
Calculate the area of a circular segment of central angle $90^\circ$ belonging to a circle of radius $12$ inches. Hint: Remember that the area of the circular segment is…
Calculate the area of a circular segment of central angle $90^\circ$ belonging to a circle of radius $12$ inches.
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 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…
A plot of land has the shape of a 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.
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Calculate the total area of the plot.
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Calculate the area of the plot that remains unwatered.
Back to the pizzas: the $45°$ slice of the 16-inch pizza (radius 8) has area $\frac18 \cdot 64\pi = 8\pi \approx 25.1$ in.² The quarter of the 10-inch pizza (radius 5) has area $\frac14 \cdot 25\pi \approx 19.6$ in.² Your slice is bigger.
Chords and Shapes Inside Circles
A chord is a segment with both endpoints on a circle.
Two more facts from earlier pages are useful here:
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A triangle inscribed in a circle with a diameter as one side is a right triangle (the angle opposite the diameter is inscribed in a semicircle).
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A regular polygon inscribed in a circle splits into $n$ identical triangles from the center. Each has two sides equal to the radius $R$ and a central angle of $\dfrac{360°}{n}$.
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…
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:
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Justify what type of triangle it is according to its angles.
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Calculate the area of the region of the play area that lies outside the triangle.
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…
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 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 inner cross). Calculate the area that must be painted red and the interior angle of this octagon.
Composite Figures
For irregular regions, split the figure into shapes you know, then add the areas of pieces that make up the region, or subtract the areas of pieces that are cut out.
Overlapping curved pieces often form a lens. Take a square of side $r$ and draw two quarter circles of radius $r$, centered at opposite corners.
Why? Adding the two quarter-circle areas counts the lens twice and covers the whole square once. So the sum minus the square leaves exactly one copy of the lens: $2\cdot\tfrac14\pi r^2 - r^2 = \left(\tfrac{\pi}{2} - 1\right)r^2$.
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…
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.
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…
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).