Angles, Lengths, and Area
A pizza shop sells a 14-inch pizza cut into 8 slices and a 10-inch pizza cut into 4 slices. Which slice gives you more pizza? To answer that, or to tile a hexagonal plaza or fence a park, you need angles, lengths, and areas. Almost everything comes from a few big ideas: the angles of a triangle add up to $180^\circ$, the Pythagorean Theorem, similar triangles, and cutting a complicated figure into simple pieces.
Angle Relationships
When a line (a transversal) crosses two parallel lines $r$ and $s$, the eight angles that form come in just two sizes. Angles of the same color below are equal, and a blue angle plus an orange angle always make a straight line, $180^\circ$.
Next, angles inside polygons. You already know that the three angles of a triangle add up to $180^\circ$, and that the base angles of an isosceles triangle are equal. Every other polygon can be cut into triangles.
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…
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.
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 arc it cuts off, so it is half the central angle on the same arc.
Watch the green vertex slide along the circle: the inscribed angle never changes.
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 areas.
Typical uses:
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Find a missing side: hypotenuse $c = \sqrt{a^2 + b^2}$, leg $a = \sqrt{c^2 - b^2}$.
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Diagonal of a rectangle $a \times b$: $d = \sqrt{a^2 + b^2}$. For a square of side $s$, $d = s\sqrt{2}$.
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Height of an equilateral triangle of side $s$: $h = \sqrt{s^2 - (s/2)^2} = \dfrac{\sqrt{3}}{2}\,s$.
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Apothem of a regular polygon: a right triangle made of the apothem, half a side, and the distance from the center to a vertex.
Some common whole-number right triangles ("Pythagorean triples"): $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$, $8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$.
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 $5$ feet 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 $5$ 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 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 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.
Similar Triangles and Shadows
How tall is a tree, a flagpole, or a mountain? You can't hold a tape measure up to it, but you can measure a shadow. The sun's rays are parallel, so every vertical object makes the same angle with its shadow. That means the triangles formed are similar.
Each triangle has a right angle at the base of the object and the same sun angle at the tip of the shadow, so AA applies. One of the practice problems calls this idea Thales's theorem; in US geometry we simply say the triangles are similar.
Sometimes the small triangle sits inside the big one: a stake and a tower whose shadows end at the same point. The two triangles then share the angle at the shadow tip, and again AA works. Be careful: the tower's shadow is the whole distance from the tip to the tower's base.
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 in the morning. At that same moment, the shadow of a nearby cooling tower measures $45$…
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.
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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 $150\text{ ft}$ 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.
Areas of Polygons
The height of a parallelogram or a trapezoid is always perpendicular to the bases, not the slanted side.
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…
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.
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.
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?
Circles, Arcs, Sectors, and Segments
| Figure | Length | Area |
|---|---|---|
| Circle | circumference $C = 2\pi r$ | $A = \pi r^2$ |
| Arc / sector with central angle $\theta$ | arc length $= \dfrac{\theta}{360} \cdot 2\pi r$ | $A = \dfrac{\theta}{360} \cdot \pi r^2$ |
| Annulus (ring) | — | $A = \pi\,(R^2 - r^2)$ |
| Segment (between a chord and its arc) | — | sector $-$ triangle |
A sector is just a fraction of the whole circle: $\frac{\theta}{360}$. Arc length and sector area use that same fraction.
Back to the pizzas. The 14-inch pizza has $r = 7$ in., and each of its 8 slices has area $\frac{49\pi}{8} \approx 19.24$ in.². The 10-inch pizza has $r = 5$ in., and each of its 4 slices has area $\frac{25\pi}{4} \approx 19.63$ in.². The slice of the smaller pizza is slightly bigger.
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…
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 $\pi \approx 3,14$).
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…
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.
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…
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.
Composite Figures
Real shapes (pools, windows, plazas) are rarely a single textbook figure. The strategy is always the same:
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Add areas when the figure is made of pieces that sit side by side.
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Subtract when a piece has been cut out of a bigger figure.
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For perimeter, add only the lengths of the outer boundary. A side where two pieces are glued together is not part of it.
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…
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 $\pi \approx 3,14$).
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 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…
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.
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Calculate the total area of the plot.
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Calculate the area of the plot that remains unwatered.
Challenge Problems: Four Tools
Harder problems combine everything above. Here are four moves that unlock many of them.
A rectangle inside a triangle
If a rectangle sits on the base of a triangle with its top corners touching the sides, the little triangle above the rectangle is similar to the whole triangle (they share the top angle, and the rectangle's top is parallel to the base). Use that to link the rectangle's width and height.
Chords of a circle
A line from the center perpendicular to a chord bisects it. That creates a right triangle: radius (hypotenuse), half the chord, and the distance from the center to the chord.
Regular polygons and circles
For a regular polygon:
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The inscribed circle (touching every side) has radius equal to the apothem $a$.
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The circumscribed circle (through every vertex) has radius equal to the distance from the center to a vertex.
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In a regular hexagon of side $s$, that distance equals $s$, and $a = \frac{\sqrt{3}}{2}s$.
A hexagram inside a hexagon
Join every other vertex of a regular hexagon and you get two big equilateral triangles that overlap in a six-pointed star (a hexagram). To measure the star, work with angles first, then subtract the leftover pieces from the hexagon.
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…
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 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.
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.
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.
Overlapping Circles: Petals and Lenses
A rose window, a flower logo, a four-leaf clover: many designs are made of circles that overlap. The region two circles share is called a lens. To find its area you cannot just add or subtract whole circles. Instead, find the pieces that you can compute (sectors, squares, triangles) and see how they fit together.
Here is a classic picture. A big circle has two perpendicular diameters. Four smaller circles use the four radii on those diameters as their own diameters. Neighboring small circles overlap in a petal, and the four petals form a flower around the center.
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).