Angles, lengths and areas
To tile the floor of a hexagonal square, fence off a park or work out which slice of pizza is bigger, you need to measure angles, lengths and areas. Almost everything comes from a few ideas: the angles of a triangle add up to 180°, Pythagoras' theorem, and splitting shapes into simpler ones.
Relationships between angles
When a line crosses two parallel lines, equal angles are formed in pairs:
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 a circle
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Central angle: its vertex is the centre. It measures the same as the arc it subtends.
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Inscribed angle: its vertex is on the circumference. It measures half of the central angle that subtends the same arc.
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…
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.
Pythagoras' theorem
The diagram explains it with areas: the square built on the hypotenuse has the same area as the other two together.
Typical uses:
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Finding a side when you know the other two: shorter side $= \sqrt{c^2 - a^2}$.
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Diagonal of a rectangle $a \times b$: $\sqrt{a^2 + b^2}$.
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Height of an 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.
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…
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 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.…
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.
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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?
Areas of polygons
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…
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…
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 centre in the shape of an isosceles trapezium. The longer base measures $2,8$ metres, the shorter base measures $1,6$…
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?
Circles and circular figures
| Shape | Length | Area |
|---|---|---|
| Circumference / circle | $L = 2\pi r$ | $A = \pi r^2$ |
| Arc / sector of $n°$ | $L = \dfrac{2\pi r \cdot n}{360}$ | $A = \dfrac{\pi r^2 \cdot n}{360}$ |
| Annulus | — | $A = \pi\,(R^2 - r^2)$ |
| 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 $\pi \approx 3,14$).
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)…
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.
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.
Back to the start: To find out which slice of pizza is bigger you have to compare their areas: a $45°$ slice of a pizza with a diameter of $40\text{ cm}$ has an area of $157{,}1\text{ cm}^2$, whereas a whole quarter ($90°$) of a pizza with a diameter of $20\text{ cm}$ has only $78{,}5\text{ cm}^2$.