Polygons

Imagine you are designing icons for a new mobile app. For the design to be clean, the programmer asks you to identify which of these shapes are polygons and which are not. Briefly justify your answer by indicating which feature fails in the shapes that are not polygons.

In the following polygon, which represents the floor plan of a small storage room, identify and count the following elements:

• Number of vertices.

• Number of sides.

• Number of interior angles.

• Draw a diagonal joining vertex $A$ with vertex $C$ (first copy the polygon into your notebook).

You can do it on paper and send a photo with your answer.

A diagonal is a line segment that joins two non-consecutive vertices of a polygon. Look at the pentagon (5 sides) in the drawing and draw all possible diagonals coming from the vertex marked in orange. How many diagonals in total do you think a pentagon has if you draw them all?

You can do it on paper and send a photo with your answer.

A group of engineers is designing a new communications antenna in the shape of a convex heptagon. To secure the structure, they need to install guy wires connecting all vertices to each other, but which are not the sides of the antenna itself. How many guy wires (diagonals) will they have to install in total? Draw the polygon and trace all the diagonals coming from a single vertex to help you reason through it.

You can do it on paper and send a photo with your answer.

Eight friends sit around a circular table to play a role-playing game. Each player must have a "magic link" (a colored string) with all other players at the table, except for those seated directly to their left and right. If we consider the players as the vertices of an octagon:

• What element of the polygon do the "magic link" strings represent?

• Calculate the total number of strings that will be on the table.

Classify the following polygons according to their number of sides using the corresponding technical names (triangle, quadrilateral, pentagon, etc.):

A polygon is convex if all its interior angles are less than $180^\circ$ and concave if it has at least one greater than $180^\circ$. Look at these two signpost arrows and state which one is concave and which one is convex.

A skate ramp has the side profile shown in the following drawing. Look at the figure and answer the following questions with reasons:

• Is it a concave or convex polygon? Why?

• Indicate the total number of sides, vertices, and diagonals this polygon has.

Look at the following figure, which represents a logo formed by a square with side length $10 \text{ cm}$ from which two triangles have been "cut out" at its top corners.

1. Classify the resulting polygon (the shaded area) according to its number of sides and its type (concave or convex).

2. If the cuts were made right through the midpoints of the sides of the square, calculate the perimeter of the new polygon.

3. What is the sum of the interior angles of this new polygon?

For a polygon to be regular, all its sides must have the same length and all its angles must be equal. Look at the following shapes from a video game and determine which of them satisfies the conditions to be a regular polygon.

For a robotics project, a metal plate must have the shape of a hexagon. The manufacturer says that the plate is "equiangular" (all its interior angles are equal), but it is not "equilateral" (its sides do not have the same length).

• Is this hexagon a regular polygon? Reason your answer based on the definition.

• If the plate has 6 sides, what must the sum of all its interior angles be?

• What is the value of each of its interior angles?

Calculate the perimeter of the following garden plot knowing that its sides measure the amounts indicated in the drawing. Remember that the perimeter is the sum of the lengths of all its sides.

We know that the sum of the angles of any triangle is always $180^\circ$. In the quadrilateral $ABCD$ in the figure, the diagonal $AC$ (dashed line) has been drawn, dividing it into two triangles: $ABC$ (in orange) and $ACD$ (in blue). Based on this, what do you think the interior angles of any quadrilateral add up to? Justify your answer.

In a construction video game, a player has designed a plot in the shape of an irregular quadrilateral to plant a vegetable patch. When measuring the interior angles using an in-game tool, they get the following values: $115^\circ$, $82^\circ$, and $96^\circ$. The fourth angle is hidden due to an interface error. Calculate the value of this angle knowing that the sum of the interior angles of any quadrilateral is always the same.

The aperture diaphragm of a professional camera is formed by blades that, when closed, leave an opening in the shape of a regular octagon.

• Calculate the measure of the central angle of this octagon.

• Calculate the measure of each of its interior angles.

• If the camera had a 10-sided aperture (regular decagon), would the central angle be larger or smaller? Justify your answer.

A decorator is designing a modern mosaic using pieces shaped like a regular pentagon. When trying to join three pieces at a single point (common vertex), they realise that a small gap remains empty.

• Calculate the interior angle of a regular pentagon.

• Knowing that a full turn around a point is $360^\circ$, how many degrees is the gap left over when joining three pentagons?

Calculate the perimeter of an irregular pentagon knowing that it has two sides of $8\text{ cm}$ each, two other sides measuring double the previous ones, and a final side measuring half the sum of the other four. Would it be possible for this pentagon to be regular? Explain why.

An architect is designing a skylight in the shape of a regular polygon. It is known that the sum of all its interior angles is exactly $1080^\circ$. However, due to a manufacturing error, one of the interior angles turned out to be $5^\circ$ larger than it should be.

1. How many sides does the original skylight have?

2. If the rest of the angles remain the same, what is the measure of the defective angle?

3. Is it still a regular polygon? Reason your answer based on the definition of a regular polygon.

A children's playground is shaped like an irregular pentagon. Its interior angles are $100^\circ$, $110^\circ$, $120^\circ$, and $90^\circ$.

1. Calculate the measure of the fifth angle.

2. Is it possible for this playground to be a convex pentagon? Justify your answer.

3. If the $90^\circ$ angle is between two sides of $12 \text{ m}$ and $16 \text{ m}$, how long would the line segment joining the ends of those two sides be? (Use Pythagoras' Theorem).

In a jewellery workshop, a pendant is being designed by joining two types of pieces: a regular hexagon and six equilateral triangles. If the perimeter of the central hexagon is $24 \text{ cm}$:

1. What is the total perimeter of the resulting star-shaped figure?

2. Calculate the value of the interior angle at the points of the star (the angles that do not touch the hexagon).

3. If we double the side length of the hexagon, what happens to the total perimeter of the star? Does it double, triple, or increase in another proportion?

You have three rigid rods to build a triangle. The first measures $15 \text{ cm}$ and the second $8 \text{ cm}$.

1. What is the minimum measure (in whole centimeters) that the third rod must have to be able to form the triangle?

2. What is the maximum measure it could have?

3. If the third rod measured exactly $17 \text{ cm}$, what type of triangle would you get according to its angles? (Hint: Check if the relationship $a^2 = b^2 + c^2$ holds).

In a quadrilateral, we know that the diagonals are perpendicular and equal in length, but they do not bisect each other.

1. What shape is it? Draw a rough example.

2. If the diagonals measure $14 \text{ cm}$, can we calculate the area of the quadrilateral even if it is not a rhombus or a square? If so, calculate it.

3. Research: What is the name of the quadrilateral that has two pairs of equal adjacent sides (kite-shaped)?

You can do it on paper and send a photo with your answer.

A graphic design team wants to create a tile shaped like a regular hexagon of side length $L$. Inside the hexagon, they draw all the diagonals originating from the same vertex.

1. Into how many triangles is the hexagon divided?

2. Are these triangles congruent to each other? Classify them according to their sides.

3. If the interior angle of the hexagon is $120^\circ$, what are the measures of the angles of each of the formed triangles?

Imagine a regular polygon where the value of its central angle is one quarter of the value of its interior angle.

1. Set up an equation relating the central angle ($360/n$) and the interior angle ($180 \times (n-2) / n$) to find the number of sides $n$.

2. What is this polygon called?

3. What is the sum of its exterior angles? (Remember that this property is common to all convex polygons).