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.

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.

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.

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?

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.

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).