Polygons
A soccer ball is stitched together from 12 pentagons and 20 hexagons. A stop sign is an octagon. A honeycomb is all hexagons. Why can hexagons cover a flat floor with no gaps, while a soccer ball needs pentagons too? By the end of this page you will answer that with numbers.
What Is a Polygon?
The pieces of a polygon have names. The sides are the segments. A vertex is a corner where two sides meet (the plural is vertices). An interior angle is the opening at a vertex, measured inside the figure. A polygon always has the same number of sides, vertices, and interior angles.
Not every closed shape is a polygon. A figure fails if it is open (it has a gap), if it has a curved edge, or if its sides cross.
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…
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…
In the following polygon, which represents the floor plan of a small storage room, identify and count the following elements:
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Number of vertices.
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Number of sides.
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Number of interior angles.
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Draw a diagonal connecting vertex $A$ to vertex $C$.
Naming Polygons by Their Sides
Polygons are named by how many sides they have. The Greek word poly means "many" and gon means "angle."
Classify the following polygons according to their number of sides using the corresponding technical names (triangle, quadrilateral, pentagon, etc.):
Classify the following polygons according to their number of sides using the corresponding technical names (triangle, quadrilateral, pentagon, etc.):
Convex and Concave Polygons
In a concave polygon, some diagonals can run outside the figure. They still count as diagonals.
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…
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.
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…
A skatepark ramp has the side profile shown in the following drawing. Look at the figure and answer the following questions with reasoning:
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Is it a concave or convex polygon? Why?
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State the total number of sides, vertices and diagonals this polygon has.
Regular and Irregular Polygons
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…
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.
Perimeter
Perimeter only depends on the side lengths, never on the angles. Two polygons with very different shapes can have the same perimeter.
If you multiply every side of a polygon by the same number, the perimeter is multiplied by that same number. Double every side and the perimeter doubles.
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…
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.
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…
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.
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…
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 $24 \text{ cm}$:
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What is the total perimeter of the resulting star-shaped figure?
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Calculate the size of the interior angle of the star's points (the angles that do not touch the hexagon).
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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?
Diagonals
How can you count diagonals without drawing them all? Pick one vertex. It cannot connect to itself or to its two neighbors (those are sides), so it has $n - 3$ diagonals. There are $n$ vertices, so that gives $n(n-3)$. But every diagonal has two ends, so you counted each one twice.
| $n$ | 3 | 4 | 5 | 6 |
|---|---|---|---|---|
| diagonals | 0 | 2 | 5 | 9 |
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…
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…
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.
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,…
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:
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What element of the polygon do the strings of the "magic links" represent?
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Calculate the total number of strings there will be on the table.
The Sum of the Interior Angles
You already know that the angles of a triangle add up to $180^\circ$. Any polygon can be cut into triangles by drawing all the diagonals from one vertex.
| Sides $n$ | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Triangles | 1 | 2 | 3 | 4 | 5 | 6 |
| Angle sum | $180^\circ$ | $360^\circ$ | $540^\circ$ | $720^\circ$ | $900^\circ$ | $1080^\circ$ |
Notice that every extra side adds one more triangle, so the sum grows by $180^\circ$. This works for convex and concave polygons.
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…
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?
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 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.
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…
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).
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Is this hexagon a regular polygon? Justify your answer based on the definition.
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If the plate has 6 sides, what must the sum of all its interior angles be?
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What is the size of each of its interior angles?
Interior Angles of a Regular Polygon
In a regular polygon all the angles are equal, so you can share the total equally.
Now think about gaps. A full turn around a point is $360^\circ$. Pieces fit around a point with no gap only if their angles add up to exactly $360^\circ$. Squares work: $90 + 90 + 90 + 90 = 360$.
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…
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.
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Calculate the size of the interior angle of a regular pentagon.
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Knowing that a full turn around a point is $360^\circ$, how many degrees is the gap left free when joining three pentagons?
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…
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.
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How many sides does the original skylight have?
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If the rest of the angles remain the same, what is the size of the defective angle?
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Is it still a regular polygon? Justify your answer based on the definition of a regular polygon.
Central and Exterior Angles
Draw lines from the center of a regular polygon to each vertex. The polygon is cut into $n$ identical triangles, and the angles at the center fill a full turn.
An exterior angle is formed when you extend one side past a vertex. It sits outside the polygon, between the extended side and the next side. The interior angle and the exterior angle at a vertex make a straight line, so they add up to $180^\circ$.
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…
The aperture of a professional camera is made up of blades that, when closed, leave a hole in the shape of a regular octagon.
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Calculate the size of the central angle of this octagon.
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Calculate the size of each of its interior angles.
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If the camera had a 10-sided aperture (regular decagon), would the central angle be larger or smaller? Justify your answer.
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 ($360/n$)…
Imagine a regular polygon where the size of its central angle is a quarter of the size of its interior angle.
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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 $n$.
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What is the name of this polygon?
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What is the sum of its exterior angles? (Remember that this property is common to all convex polygons).
Triangles Inside a Polygon
Cutting a polygon into triangles also helps you find the angles of those triangles. First, a reminder of how triangles are classified.
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…
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.
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Into how many triangles is the hexagon divided?
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Are these triangles equal to each other? Classify them according to their sides.
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If the interior angle of the hexagon is $120^\circ$, what are the sizes of the angles of each of the triangles formed?
Right Triangles and the Pythagorean Theorem
Some polygon problems hide a right triangle, for example when you cut a corner off a square, or when two sides of a plot meet at $90^\circ$. In a right triangle, the two sides that form the right angle are the legs, and the side across from it, the longest one, is the hypotenuse.
You will study this theorem in depth in 8th grade. Here you only need to use it. Try changing the legs and watch the squares.
If $c^2$ is not a perfect square, use the square root key on a calculator and round. For legs $2$ and $3$: $c^2 = 4 + 9 = 13$, so $c = \sqrt{13} \approx 3.61$.
When you cut a corner off a square or rectangle in a straight line, the new side is the hypotenuse of a small right triangle. Its legs are the two pieces you removed from the original sides.
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.…
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.
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Classify the resulting polygon (the shaded area) according to its number of sides and its type (concave or convex).
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If the cuts have been made exactly through the midpoints of the sides of the square, calculate the perimeter of the new polygon.
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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…
A playground has the shape of an irregular pentagon. Its interior angles are $100^\circ$, $110^\circ$, $120^\circ$ and $90^\circ$.
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Calculate the size of the fifth angle.
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Is it possible for this playground to be a convex pentagon? Justify your answer.
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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).
Can Three Sides Make a Triangle?
Try building a triangle from three rods. If one rod is too long, the other two can't reach each other. If one is too short, the other two stick out past it and never meet.
Once you have a valid triangle, you can classify it by angles with the Pythagorean test from the previous section.
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…
You have three rigid rods to build a triangle. The first measures $15 \text{ in}$ and the second $8 \text{ in}$.
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What is the minimum length (in whole inches) that the third rod must have to be able to form the triangle?
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What is the maximum length it could have?
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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).
Quadrilaterals with Perpendicular Diagonals
Some quadrilaterals have diagonals that cross at a right angle. There is a neat shortcut for their area.
This works even when the diagonals do not cut each other in half. In a parallelogram (rectangle, rhombus, square) the diagonals always cut each other in half. So if they don't, the quadrilateral is none of those. It is just an irregular quadrilateral.
A quadrilateral with two pairs of equal consecutive sides is called a kite. Its diagonals are perpendicular, so the same area formula works for it too.
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…
In any quadrilateral, we know that the diagonals are perpendicular and have the same length, but they do not intersect at their midpoint.
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What shape is it? Draw a rough example.
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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.
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Investigate: What is the name of the quadrilateral that has two pairs of equal consecutive sides (kite shape)?
Summary
Back to the start: where three hexagons meet, the angles add up to exactly $360^\circ$, so they lie flat. On a soccer ball, one pentagon and two hexagons meet at each corner, and $108^\circ + 120^\circ + 120^\circ = 348^\circ$, which is less than $360^\circ$. That small shortage is what makes the surface curve into a ball.