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 bond" (a colored string) with all the other players at the table, except with the ones sitting right next to them on their left and right. If we consider the players to be the vertices of an octagon:

1. a) What element of the polygon do the strings of the "magic bonds" represent?

1. b) Calculate the total number of strings there will be on the table.

The figure shows, for player P₁, their two neighbors (red points, with whom there is no bond) and their magic bonds (orange lines).

The Parks and Gardens department of a city council in Ireland is designing a lawn area shaped like a convex polygon. To plan the sprinkler installation, a technician notices that, from one of the corners (a single vertex), exactly $4$ straight lines can be drawn to other vertices without being the sides of the enclosure (that is, exactly $4$ diagonals).

1. a) Model the situation and deduce how many sides the plot has in total. Explain in detail how the number of diagonals starting from a vertex is related to the total number of sides $n$.
2. b) The total perimeter of the plot is $49\text{ m}$ and the workers have measured that all its sides have the same length. Can you be mathematically certain that the lawn area is a regular polygon? Reason your answer and give an example of a quadrilateral where all sides are equal but the figure is not regular.
3. c) If the plot were indeed a regular polygon, how long would each side be and what would the sum of all its interior angles be?

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

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.

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.

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?

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 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).

In the design of pavements and pedestrian walkways in several Irish cities, such as Dublin, Cork and Galway, tiles shaped like regular polygons are used. For a paving to cover the ground completely without leaving gaps or overlapping when meeting at the same vertex, the sum of the interior angles meeting at that vertex must be exactly equal to $360^\circ$.

An urban planning firm wants to pave a square and analyses three proposals for regular tiles:

1. a) Proposal A (regular pentagons): Calculate the measure of the interior angle of a regular pentagon and demonstrate numerically why it is impossible to pave the square using exclusively this type of tile.
2. b) Proposal B (regular hexagons): Calculate the measure of the interior angle of a regular hexagon and determine whether it allows tiling the square without gaps. If so, how many tiles will meet exactly at each common vertex?
3. c) Proposal C (regular octagons): A manufacturer claims that the square can be paved by combining regular octagons with another smaller regular shape to fill the remaining gaps. Calculate the interior angle of a regular octagon, demonstrate that two octagons together do not complete a full turn nor do three of them fit at a vertex, and deduce what type of well-known regular polygon fits exactly into the gap to add up to $360^\circ$.

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.

Aoife wants to calculate the measure of each interior angle of a regular pentagon for a school craft project and writes down in her notebook:

"Since it is a pentagon, it has $5$ sides. The sum of its interior angles is $180^\circ \times 5 = 900^\circ$. As it is regular, I divide $900^\circ$ by $5$ and get that each interior angle measures $180^\circ$".

1. a) Find the two geometric reasoning errors that Aoife made. Explain why it is impossible for an interior angle of a closed polygon to measure $180^\circ$.
2. b) Correct the procedure and calculate the correct value of the sum of the interior angles and the measure of each interior angle of the regular pentagon.

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.

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.

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

1. a) What is the minimum length (in whole centimetres) the third rod must have in order to form the triangle?

1. b) What is the maximum length (in whole centimetres) it could have?

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

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).