How this instrument works
Draw any simple five-pointed star — a pentagram, whether perfectly regular or wildly lopsided, so long as its lines cross without any point sitting on top of another. Measure the sharp angle at each of its five tips and add all five together. The total is always exactly 180°, no matter how uneven the star looks. It's the same fixed-sum promise a triangle makes with its three interior angles, just spread across five points instead of three, and it holds for every such star you can draw.
The result is a genuine theorem, not a coincidence limited to tidy examples, and it shows up often as a classic 'prove it' problem in competition mathematics. One route to a proof leans on the exterior angle theorem, applied not to the star's outer boundary but to the five triangles formed where its own line segments cross themselves: each tip angle equals the difference of two angles in the pentagon sitting at the star's center, and once those differences are added across all five points, the pentagon's contributions cancel out and 180° is what remains.
The regular pentagram supplies a satisfying check on the whole idea. Cut a star so every tip looks identical and each point angle must equal 180° ÷ 5 = 36° exactly — a figure often quoted on its own as a fun fact about five-pointed stars. Feed four 36° angles into this calculator and it hands back a fifth of 36° as well, since 180° − (36+36+36+36)° = 36°, confirming that the general rule and the well-known regular case agree down to the decimal.
This page simply rearranges that rule to solve for whichever tip you haven't measured yet. Enter any four point angles, in degrees, radians, or turns, and it subtracts their total from 180° (π radians, or half a turn) to return the fifth. A built-in check stops the calculation if your four entries already reach or pass 180° on their own, since a real star can't have a fifth point angle of zero or less.
- Enter four measured point angles into Point angle 1 through Point angle 4, in whichever unit you prefer — degrees, radians, or turns.
- Read Point angle 5 for the missing tip, found by subtracting your four entries from 180°.
- Switch the unit dropdown if your protractor or source data reads in radians or turns rather than degrees.
- Try 36° in all four fields to confirm the regular-pentagram case: a fifth point angle of exactly 36° should come back.
- Watch for the built-in warning if your four angles already sum to 180° or beyond — that combination cannot belong to a real, simple five-pointed star.
Worked example — a regular star and two irregular ones
The regular pentagram sets every point angle to 36°: angle1 = angle2 = angle3 = angle4 = 36°, so angle5 = 180° − (36+36+36+36)° = 180° − 144° = 36°. The calculated fifth tip matches the other four exactly, which is exactly what a perfectly symmetric star should produce.
An irregular star can look quite different and the rule still holds. With angle1 = 30°, angle2 = 40°, angle3 = 50°, and angle4 = 20°, the four entries total 140°, so angle5 = 180° − 140° = 40°. A second irregular case, angle1 = angle2 = angle3 = 45° and angle4 = 25°, totals 160°, giving angle5 = 180° − 160° = 20° — three very different-looking stars, one unbroken rule.
Questions
Why do a five-pointed star's point angles always add up to 180°?
It follows from the exterior angle theorem applied to the triangles formed where the star's own segments cross one another. Each tip angle can be written as the difference between two angles of the pentagon sitting at the star's center; add all five tip angles together and the pentagon's own angles cancel out in pairs, leaving a fixed total of 180° regardless of how the star is shaped.
What is the point angle of a regular pentagram?
Exactly 36°, since a regular star splits the fixed 180° total evenly across its five identical tips: 180° ÷ 5 = 36°. This calculator confirms the figure directly — enter four 36° angles and it returns a fifth of 36° as well.
Does the 180° rule still work for an irregular, lopsided star?
Yes — as long as the star is simple, meaning its lines cross cleanly without any point overlapping another, the five point angles sum to 180° no matter how uneven they are. Two worked examples on this page use point angles of 30°/40°/50°/20° and 45°/45°/45°/25°, and both still total exactly 180° once the missing fifth angle is added in.
How is this similar to the rule that a triangle's angles sum to 180°?
Both are fixed-total theorems: a triangle's three interior angles always sum to 180°, and a simple five-pointed star's five point angles always sum to the same 180°, just distributed across two more points. Neither total depends on the specific shape drawn, only on the number of points and the geometry connecting them.
What happens if my four entered angles already reach 180° or more?
The calculator flags it rather than returning a negative or zero result, because no real, simple five-pointed star can have four point angles that already meet or exceed the full 180° total on their own. Double-check your measurements — one of the four figures is likely too large.
Can I enter the point angles in radians or turns instead of degrees?
Yes — switch the unit selector on any of the four input fields to radians or turns, and the fifth angle comes back in the matching unit. The underlying rule is unchanged: the five point angles still total π radians, or half a turn, exactly as they total 180°.
Is this the same fact as the well-known 'a star's points sum to 180°' puzzle?
Yes, this is that exact classic result, often set as a proof exercise in competition mathematics. This calculator applies the identical rule numerically: given any four point angles of a simple five-pointed star, it solves for the fifth so the full set totals 180°.