What Is a Pentagram?
A pentagram is a five-pointed star polygon drawn with five straight strokes that connect five equally spaced points, joining every second point rather than every adjacent one. In the notation for star polygons it is written {5/2} - five vertices, stepping two at a time. It is one continuous, self-intersecting path, and the crossings form a smaller regular pentagon at its centre.
The pentagram is inseparable from the pentagon: draw the five diagonals of a regular pentagon shape and those diagonals are the pentagram. Because it crosses itself, the pentagram is a star polygon, a family of non-convex figures distinct from the simple polygons whose sides never cross.
How Do You Draw a Pentagram?
Drawing a pentagram cleanly is a common first question, and there are two reliable routes.
From five points on a circle. Mark five equally spaced points around a circle - each $72°$ apart, since $360° \div 5 = 72°$. Then connect every second point: $1 \to 3 \to 5 \to 2 \to 4 \to 1$. The path closes into a star.
From a pentagon. Draw a regular pentagon, then draw all five diagonals. The diagonals overlap to form the pentagram, with a small pentagon left in the middle.
Both methods rely on the same $72°$ spacing. The "skip a point" step is what turns a plain five-sided outline into a star - connecting adjacent points ($1 \to 2 \to 3 \dots$) just redraws the pentagon.
What Are the Angles of a Pentagram?
The angles inside a pentagram are fixed and worth knowing.
Point (vertex) angle: $36°$. Each of the five sharp star points has an interior angle of $36°$. The five points together contribute $5 \times 36° = 180°$.
Adjacent points seen from the centre: $72°$. The five points sit $72°$ apart around the centre.
The inner pentagon: $108°$. The central pentagon formed by the crossings is regular, so each of its interior angles is $108°$ - the same as the outer pentagon's, since the angles in a pentagon are always $108°$ for a regular one.
Why $36°$? Each star point is the tip of an isosceles triangle sitting on a side of the inner pentagon. The pentagon's exterior angle is $72°$, and the point angle is the supplement structure of two of those, leaving $180° - 72° - 72° = 36°$ at the tip.
How Is the Pentagram Related to the Golden Ratio?
This is the pentagram's most famous property. Every line segment in a pentagram is divided by its crossing points in the golden ratio, $\varphi = \dfrac{1 + \sqrt{5}}{2} \approx 1.618$.
In a regular pentagon, the ratio of a diagonal to a side equals $\varphi$ exactly:
$$\frac{\text{diagonal}}{\text{side}} = \varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618$$
When the diagonals cross to make the pentagram, each diagonal is cut into pieces whose lengths are again in the ratio $\varphi$ to $1$. Cut any of the star's lines at a crossing, and the longer piece is $\varphi$ times the shorter - nested all the way down. This self-similar golden structure is why the ancient Pythagoreans treated the pentagram as a near-mystical figure (see the Wikipedia golden ratio article).
What Are the Properties of a Pentagram?
The regular pentagram has a fixed set of properties worth stating on their own:
It is the star polygon {5/2} - five points connected in steps of two, a single self-intersecting closed path.
Five-fold symmetry. It has 5 lines of symmetry and rotational symmetry of order 5, mapping onto itself every $72°$.
Point angle $36°$, inner pentagon angle $108°$. These angles are the same for every regular pentagram, whatever its size.
Golden-ratio segments. Every line is divided in the ratio $\varphi \approx 1.618$ at its crossings, at every scale.
It contains smaller pentagrams. The central pentagon's diagonals form a smaller pentagram, which contains a smaller one still - an infinite self-similar nesting.
What Is the Difference Between a Pentagram and a Pentagon?
These two are constantly mixed up, so it is worth pinning down. A pentagon is a simple five-sided polygon whose sides never cross; a pentagram is the five-pointed star made from that pentagon's diagonals. One is a plain shape; the other is a star.
Feature | Pentagon | Pentagram |
|---|---|---|
Sides | 5 straight sides, no crossings | 5 lines that cross themselves |
Type | Simple (convex) polygon | Star polygon {5/2} |
Interior angle | $108°$ each | $36°$ at each point |
Drawn by | Joining adjacent vertices | Joining every second vertex |
They are two views of the same five points. Join the points in order and you get a pentagon; skip every other point and you get a pentagram. The pentagon belongs among the standard types of polygon, while the pentagram sits in the star-polygon family.
Where Is the Pentagram Used?
"For 2,500 years, one star has meant recognition, protection, and proportion at once." The pentagram appears far beyond geometry class.
History and symbolism - the Pythagoreans used it as a secret badge of membership; it later became a symbol of protection and appears across cultures (see the Wolfram MathWorld pentagram entry).
Flags and heraldry - most five-pointed stars on national flags are pentagrams, drawn in a single stroke.
Art and design - the golden-ratio proportions inside a pentagram make it a recurring motif in architecture, logos, and Renaissance art.
Nature - the arrangement of seeds, petals, and starfish arms often shows five-fold symmetry, the same order the pentagram embodies.
Examples of the Pentagram
Example 1
How many points does a pentagram have, and what is the angle at each point?
A pentagram is a five-pointed star, so it has 5 points. Each point has an interior angle of $36°$.
Final answer: 5 points, each with a $36°$ angle.
Example 2
A student adds up the five point angles and expects $360°$ because "stars go all the way around."
Wrong path. Reasoning that any closed figure's angles must total $360°$, the student writes $5 \times 72° = 360°$ and labels each point $72°$.
Why it breaks. The $72°$ is the spacing between points seen from the centre, not the angle at a point. The point angle is the sharp tip, which is much smaller than $72°$.
The rescue. The five point angles of a pentagram sum to $180°$, not $360°$. Each point is $180° \div 5 = 36°$.
Final answer: each point is $36°$; the five points sum to $180°$.
Example 3
A regular pentagon has a side of 2 cm. How long is its diagonal?
The diagonal-to-side ratio is the golden ratio $\varphi$:
$$\text{diagonal} = \varphi \cdot \text{side} = 1.618 \cdot 2 \approx 3.24\ \text{cm}$$
Final answer: about $3.24\ \text{cm}$.
Example 4
What star polygon notation describes a pentagram?
It is {5/2}: five vertices, connected by stepping two vertices each time.
Final answer: {5/2}.
Example 5
Through what angle can a pentagram be rotated so it looks unchanged?
A pentagram has rotational symmetry of order 5, so it maps onto itself every $360° \div 5 = 72°$.
Final answer: $72°$ (and any multiple of it).
Example 6
A pentagram diagonal is cut at a crossing into a longer piece of $1.618$ cm and a shorter piece. If the ratio is golden, how long is the shorter piece?
The longer piece is $\varphi$ times the shorter, so shorter $= 1.618 \div \varphi = 1.618 \div 1.618 = 1$ cm.
Final answer: the shorter piece is $1$ cm.
Where Do Students Trip Up on the Pentagram?
The confusion that trips students up most is treating the pentagram like an ordinary polygon - expecting its angles to follow the simple-polygon rules that a pentagon obeys. Remembering that a pentagram crosses itself, so it is a star polygon with its own angle facts, clears most of the trouble.
Mistake 1: Confusing the point angle with the central spacing
Where it slips in: Working out the angle at a star tip.
Don't do this: Using $72°$ (the spacing around the centre) as the point angle.
The correct way: The point angle is $36°$; the $72°$ is how far apart the points sit as seen from the centre.
Mistake 2: Mixing up the pentagram and the pentagon
Where it slips in: Reading "penta-" and assuming a plain five-sided shape.
Don't do this: Calling a five-pointed star a pentagon, or giving it $108°$ interior angles.
The correct way: A pentagon is the simple five-sided shape ($108°$ angles); a pentagram is the star ($36°$ points) made from that pentagon's diagonals.
Mistake 3: Forgetting the pentagram is one continuous line
Where it slips in: Counting sides or trying to draw it in separate strokes.
Don't do this: Treating the five points as five disconnected triangles.
The correct way: A regular pentagram is a single closed path drawn without lifting the pen, which is why it is {5/2} and not five separate shapes.
The pentagram's golden proportions are not a numerology coincidence - they trace to the same irrational number $\varphi$ that governs the Fibonacci sequence, a reminder that a "decorative" star hides exact mathematics that engineers and designers still use for proportion today.
Conclusion
A pentagram is a five-pointed star, the star polygon {5/2}, formed by joining every second vertex of a regular pentagon.
Each of its five points has a $36°$ angle, the points sum to $180°$, and the inner pentagon's angles are $108°$.
Every line in a pentagram is divided at its crossings in the golden ratio $\varphi \approx 1.618$, nested at every scale.
A pentagram is a self-crossing star, not a simple pentagon, so its angle facts differ from an ordinary polygon's.
To explore the pentagram and star geometry with a teacher, explore Bhanzu's geometry tutor or high school math tutor sessions, or browse math classes online for guided shape practice.
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Practice These to Solidify Your Understanding
Work through these problems in order:
A regular pentagon has a diagonal of $4.854$ cm. Using the golden ratio, find its side length.
What is the sum of the five point angles of a pentagram?
Through what smallest angle can a pentagram be rotated to look the same, and what is its order of rotational symmetry?
Answer to Question 1: side $= \text{diagonal} \div \varphi = 4.854 \div 1.618 \approx 3$ cm. Answer to Question 2: $5 \times 36° = 180°$. Answer to Question 3: $72°$, with rotational symmetry of order 5.
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