Exercises: Polygons

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

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 connecting vertex $A$ to vertex $C$.

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 directional arrows and indicate which is concave and which is convex.

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

Calculate the perimeter of the following garden plot if we know 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 $180^\circ$. If we divide any quadrilateral by drawing one of its diagonals, we get two triangles. Based on this, what do you think the sum of the interior angles of any quadrilateral will be?

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 the possible diagonals starting from the vertex marked in orange. How many total diagonals do you think a pentagon has if you draw them all?

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

In a building video game, a player has designed a plot in the shape of an irregular quadrilateral to place a vegetable patch. When measuring the interior angles with 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 size of this angle knowing that the sum of the interior angles of any quadrilateral is always the same.

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

• Is it a concave or convex polygon? Why?

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

The aperture of a professional camera is made up of blades that, when closed, leave a hole in the shape of a regular octagon.

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

• Calculate the size 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 in the shape of a regular pentagon. When trying to join three pieces at the same point (common vertex), they realize that there is a small empty gap left.

• Calculate the size of 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 free when joining three pentagons?

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

• Is this hexagon a regular polygon? Justify 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 size of each of its interior angles?

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 the other players at the table, except for those sitting immediately to their left and right. If we consider the players to be the vertices of an octagon:

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

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

Calculate the perimeter of an irregular pentagon knowing that it has two sides of $8\text{ cm}$ each, another two sides that are twice as long as the previous ones, and a final side that is 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 has 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 size of the defective angle?

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

In a jewelry 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 size of the interior angle of the star's points (the angles that do not touch the hexagon).

3. If we double the side 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 $15 \text{ in}$ and the second $8 \text{ in}$.

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

2. What is the maximum length it could have?

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

Look at the following figure, which represents a logo formed by a square of side $10 \text{ cm}$ 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 have been made exactly 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?

A playground has the shape of an irregular pentagon. Its interior angles are $100^\circ$, $110^\circ$, $120^\circ$ and $90^\circ$.

1. Calculate the size 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 $16 \text{ m}$, what would be the length of the line segment joining the ends of those two sides? (Use the Pythagorean theorem).

In any quadrilateral, we know that the diagonals are perpendicular and have the same length, but they do not intersect at their midpoint.

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 neither a rhombus nor a square? If so, calculate it.

3. Investigate: What is the name of the quadrilateral that has two pairs of equal consecutive sides (kite shape)?

A graphic design team wants to create a tile in the shape of a regular hexagon with side $L$. Inside the hexagon, they draw all the diagonals that start from the same vertex.

1. Into how many triangles is the hexagon divided?

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

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

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

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

2. What is the name of this polygon?

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