---
id: 5900f4511000cf542c50ff63
title: 'Problem 228: Minkowski Sums'
challengeType: 5
forumTopicId: 301871
dashedName: problem-228-minkowski-sums
---

# --description--

<!-- TODO Use MathJax and re-write from projecteuler.net -->

Let Sn be the regular n-sided polygon – or shape – whose vertices

vk (k = 1,2,…,n) have coordinates:

xk = cos( 2k-1/n ×180° )

yk = sin( 2k-1/n ×180° )

Each Sn is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.

The Minkowski sum, S+T, of two shapes S and T is the result of

adding every point in S to every point in T, where point addition is performed coordinate-wise:

(u, v) + (x, y) = (u+x, v+y).

For example, the sum of S3 and S4 is the six-sided shape shown in pink below:

How many sides does S1864 + S1865 + … + S1909 have?

# --hints--

`euler228()` should return 86226.

```js
assert.strictEqual(euler228(), 86226);
```

# --seed--

## --seed-contents--

```js
function euler228() {

  return true;
}

euler228();
```

# --solutions--

```js
// solution required
```