Problem p02312 (C++)

Buggy submission (status: Runtime Error):
#include <bits/stdc++.h>

using namespace std;

const double EPS = 1e-8, PI = acos(-1);

inline bool eq(double a, double b) { return fabs(b - a) < EPS; }

using Point = complex<double>;

Point operator*(const Point &p, const double &d) {
  return Point(real(p) * d, imag(p) * d);
}

istream &operator>>(istream &is, Point &p) {
  double a, b;
  is >> a >> b;
  p = Point(a, b);
  return is;
}

ostream &operator<<(ostream &os, Point &p) {
  os << fixed << setprecision(10) << p.real() << " " << p.imag();
}

Point rotate(double theta, const Point &p) {
  return Point(cos(theta) * p.real() - sin(theta) * p.imag(),
               sin(theta) * p.real() + cos(theta) * p.imag());
}

double radian_to_degree(double r) { return (r * 180.0 / PI); }

double degree_to_radian(double d) { return (d * PI / 180.0); }

double get_angle(const Point &a, const Point &b, const Point &c) {
  const Point v(b - a), w(c - b);
  double alpha = atan2(v.imag(), v.real()), beta = atan2(w.imag(), w.real());
  if (alpha > beta)
    swap(alpha, beta);
  double theta = (beta - alpha);
  return min(theta, 2 * acos(-1) - theta);
}

namespace std {
bool operator<(const Point &a, const Point &b) {
  return a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag();
}
} // namespace std

struct Line {
  Point a, b;

  Line() {}

  Line(Point a, Point b) : a(a), b(b) {}

  Line(double A, double B, double C) // Ax + By = C
  {
    if (eq(A, 0))
      a = Point(0, C / B), b = Point(1, C / B);
    else if (eq(B, 0))
      b = Point(C / A, 0), b = Point(C / A, 1);
    else
      a = Point(0, C / B), b = Point(C / A, 0);
  }

  friend ostream &operator<<(ostream &os, Line &p) {
    return os << p.a << " to " << p.b;
  }

  friend istream &operator>>(istream &is, Line &a) { return is >> a.a >> a.b; }
};

struct Segment : Line {
  Segment() {}

  Segment(Point a, Point b) : Line(a, b) {}
};

struct Circle {
  Point p;
  double r;

  Circle() {}

  Circle(Point p, double r) : p(p), r(r) {}
};

using Points = vector<Point>;
using Polygon = vector<Point>;
using Segments = vector<Segment>;
using Lines = vector<Line>;
using Circles = vector<Circle>;

double cross(const Point &a, const Point &b) {
  return real(a) * imag(b) - imag(a) * real(b);
}

double dot(const Point &a, const Point &b) {
  return real(a) * real(b) + imag(a) * imag(b);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C
int ccw(const Point &a, Point b, Point c) {
  b = b - a, c = c - a;
  if (cross(b, c) > EPS)
    return +1; // "COUNTER_CLOCKWISE"
  if (cross(b, c) < -EPS)
    return -1; // "CLOCKWISE"
  if (dot(b, c) < 0)
    return +2; // "ONLINE_BACK"
  if (norm(b) < norm(c))
    return -2; // "ONLINE_FRONT"
  return 0;    // "ON_SEGMENT"
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool parallel(const Line &a, const Line &b) {
  return abs(cross(a.b - a.a, b.b - b.a)) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool orthogonal(const Line &a, const Line &b) {
  return abs(dot(a.a - a.b, b.a - b.b)) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A
Point projection(const Line &l, const Point &p) {
  double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
  return l.a + (l.a - l.b) * t;
}

Point projection(const Segment &l, const Point &p) {
  double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
  return l.a + (l.a - l.b) * t;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_B
Point reflection(const Line &l, const Point &p) {
  return p + (projection(l, p) - p) * 2.0;
}

bool Intersect(const Line &l, const Point &p) {
  return abs(ccw(l.a, l.b, p)) != 1;
}

bool intersect(const Line &l, const Line &m) {
  return abs(cross(l.b - l.a, m.b - m.a)) > EPS ||
         abs(cross(l.b - l.a, m.b - l.a)) < EPS;
}

bool intersect(const Segment &s, const Point &p) {
  return ccw(s.a, s.b, p) == 0;
}

bool intersect(const Line &l, const Segment &s) {
  return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}

double distance(const Line &l, const Point &p);

bool intersect(const Circle &c, const Line &l) {
  return distance(l, c.p) <= c.r + EPS;
}

bool intersect(const Circle &c, const Point &p) {
  return abs(abs(p - c.p) - c.r) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_B
bool intersect(const Segment &s, const Segment &t) {
  return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
         ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

int intersect(const Circle &c, const Segment &l) {
  if (norm(projection(l, c.p) - c.p) - c.r * c.r > EPS)
    return 0;
  const double d1 = abs(c.p - l.a), d2 = abs(c.p - l.b);
  if (d1 < c.r + EPS && d2 < c.r + EPS)
    return 0;
  if (d1 < c.r - EPS && d2 > c.r + EPS || d1 > c.r + EPS && d2 < c.r - EPS)
    return 1;
  const Point h = projection(l, c.p);
  if (dot(l.a - h, l.b - h) < 0)
    return 2;
  return 0;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_A&lang=jp
int intersect(Circle c1, Circle c2) {
  if (c1.r < c2.r)
    swap(c1, c2);
  double d = abs(c1.p - c2.p);
  if (c1.r + c2.r < d)
    return 4;
  if (eq(c1.r + c2.r, d))
    return 3;
  if (c1.r - c2.r < d)
    return 2;
  if (eq(c1.r - c2.r, d))
    return 1;
  return 0;
}

double distance(const Point &a, const Point &b) { return abs(a - b); }

double distance(const Line &l, const Point &p) {
  return abs(p - projection(l, p));
}

double distance(const Line &l, const Line &m) {
  return intersect(l, m) ? 0 : distance(l, m.a);
}

double distance(const Segment &s, const Point &p) {
  Point r = projection(s, p);
  if (intersect(s, r))
    return abs(r - p);
  return min(abs(s.a - p), abs(s.b - p));
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_D
double distance(const Segment &a, const Segment &b) {
  if (intersect(a, b))
    return 0;
  return min(
      {distance(a, b.a), distance(a, b.b), distance(b, a.a), distance(b, a.b)});
}

double distance(const Line &l, const Segment &s) {
  if (intersect(l, s))
    return 0;
  return min(distance(l, s.a), distance(l, s.b));
}

Point crosspoint(const Line &l, const Line &m) {
  double A = cross(l.b - l.a, m.b - m.a);
  double B = cross(l.b - l.a, l.b - m.a);
  if (abs(A) < EPS && abs(B) < EPS)
    return m.a;
  return m.a + (m.b - m.a) * B / A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_C
Point crosspoint(const Segment &l, const Segment &m) {
  double A = cross(l.b - l.a, m.b - m.a);
  double B = cross(l.b - l.a, l.b - m.a);
  if (abs(A) < EPS && abs(B) < EPS)
    return m.a;
  return m.a + (m.b - m.a) * B / A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D
pair<Point, Point> crosspoint(const Circle &c, const Line l) {
  Point pr = projection(l, c.p);
  Point e = (l.b - l.a) / abs(l.b - l.a);
  if (eq(distance(l, c.p), c.r))
    return {pr, pr};
  double base = sqrt(c.r * c.r - norm(pr - c.p));
  return {pr + e * base, pr - e * base};
}

pair<Point, Point> crosspoint(const Circle &c, const Segment &l) {
  Line aa = Line(l.a, l.b);
  if (intersect(c, l) == 2)
    return crosspoint(c, aa);
  auto ret = crosspoint(c, aa);
  if (dot(l.a - ret.first, l.b - ret.first) < 0)
    ret.second = ret.first;
  else
    ret.first = ret.second;
  return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_E
pair<Point, Point> crosspoint(const Circle &c1, const Circle &c2) {
  double d = abs(c1.p - c2.p);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = atan2(c2.p.imag() - c1.p.imag(), c2.p.real() - c1.p.real());
  Point p1 = c1.p + Point(cos(t + a) * c1.r, sin(t + a) * c1.r);
  Point p2 = c1.p + Point(cos(t - a) * c1.r, sin(t - a) * c1.r);
  return {p1, p2};
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_F
pair<Point, Point> tangent(const Circle &c1, const Point &p2) {
  return crosspoint(c1, Circle(p2, sqrt(norm(c1.p - p2) - c1.r * c1.r)));
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_G
Lines tangent(Circle c1, Circle c2) {
  Lines ret;
  if (c1.r < c2.r)
    swap(c1, c2);
  double g = norm(c1.p - c2.p);
  if (eq(g, 0))
    return ret;
  Point u = (c2.p - c1.p) / sqrt(g);
  Point v = rotate(PI * 0.5, u);
  for (int s : {-1, 1}) {
    double h = (c1.r + s * c2.r) / sqrt(g);
    if (eq(1 - h * h, 0)) {
      ret.emplace_back(c1.p + u * c1.r, c1.p + (u + v) * c1.r);
    } else if (1 - h * h > 0) {
      Point uu = u * h, vv = v * sqrt(1 - h * h);
      ret.emplace_back(c1.p + (uu + vv) * c1.r, c2.p - (uu + vv) * c2.r * s);
      ret.emplace_back(c1.p + (uu - vv) * c1.r, c2.p - (uu - vv) * c2.r * s);
    }
  }
  return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_B
bool is_convex(const Polygon &p) {
  int n = (int)p.size();
  for (int i = 0; i < n; i++) {
    if (ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]) == -1)
      return false;
  }
  return true;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_A
Polygon convex_hull(Polygon &p) {
  int n = (int)p.size(), k = 0;
  if (n <= 2)
    return p;
  sort(p.begin(), p.end());
  vector<Point> ch(2 * n);
  for (int i = 0; i < n; ch[k++] = p[i++]) {
    while (k >= 2 && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < 0)
      --k;
  }
  for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = p[i--]) {
    while (k >= t && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < 0)
      --k;
  }
  ch.resize(k - 1);
  return ch;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_C
enum { OUT, ON, IN };

int contains(const Polygon &Q, const Point &p) {
  bool in = false;
  for (int i = 0; i < Q.size(); i++) {
    Point a = Q[i] - p, b = Q[(i + 1) % Q.size()] - p;
    if (a.imag() > b.imag())
      swap(a, b);
    if (a.imag() <= 0 && 0 < b.imag() && cross(a, b) < 0)
      in = !in;
    if (cross(a, b) == 0 && dot(a, b) <= 0)
      return ON;
  }
  return in ? IN : OUT;
}

bool merge_if_able(Segment &s1, Segment s2) {
  if (abs(cross(s1.b - s1.a, s2.b - s2.a)) > EPS)
    return false;
  if (ccw(s1.a, s2.a, s1.b) == 1 || ccw(s1.a, s2.a, s1.b) == -1)
    return false;
  if (ccw(s1.a, s1.b, s2.a) == -2 || ccw(s2.a, s2.b, s1.a) == -2)
    return false;
  s1 = Segment(min(s1.a, s2.a), max(s1.b, s2.b));
  return true;
}

void merge_segments(vector<Segment> &segs) {
  for (int i = 0; i < segs.size(); i++) {
    if (segs[i].b < segs[i].a)
      swap(segs[i].a, segs[i].b);
  }
  for (int i = 0; i < segs.size(); i++) {
    for (int j = i + 1; j < segs.size(); j++) {
      if (merge_if_able(segs[i], segs[j])) {
        segs[j--] = segs.back(), segs.pop_back();
      }
    }
  }
}

vector<vector<int>> segment_arrangement(vector<Segment> &segs,
                                        vector<Point> &ps) {
  vector<vector<int>> g;
  int N = (int)segs.size();
  for (int i = 0; i < N; i++) {
    ps.emplace_back(segs[i].a);
    ps.emplace_back(segs[i].b);
    for (int j = i + 1; j < N; j++) {
      const Point p1 = segs[i].b - segs[i].a;
      const Point p2 = segs[j].b - segs[j].a;
      if (cross(p1, p2) == 0)
        continue;
      if (intersect(segs[i], segs[j])) {
        ps.emplace_back(crosspoint(segs[i], segs[j]));
      }
    }
  }
  sort(begin(ps), end(ps));
  ps.erase(unique(begin(ps), end(ps)), end(ps));

  int M = (int)ps.size();
  g.resize(M);
  for (int i = 0; i < N; i++) {
    vector<int> vec;
    for (int j = 0; j < M; j++) {
      if (intersect(segs[i], ps[j])) {
        vec.emplace_back(j);
      }
    }
    for (int j = 1; j < vec.size(); j++) {
      g[vec[j - 1]].push_back(vec[j]);
      g[vec[j]].push_back(vec[j - 1]);
    }
  }
  return (g);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_C
Polygon convex_cut(const Polygon &U, Line l) {
  Polygon ret;
  for (int i = 0; i < U.size(); i++) {
    Point now = U[i], nxt = U[(i + 1) % U.size()];
    if (ccw(l.a, l.b, now) != -1)
      ret.push_back(now);
    if (ccw(l.a, l.b, now) * ccw(l.a, l.b, nxt) < 0) {
      ret.push_back(crosspoint(Line(now, nxt), l));
    }
  }
  return (ret);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_A
double area2(const Polygon &p) {
  double A = 0;
  for (int i = 0; i < p.size(); ++i) {
    A += cross(p[i], p[(i + 1) % p.size()]);
  }
  return A;
}

double area2(const Polygon &p, const Circle &c) {
  if (p.size() < 3)
    return 0.0;
  function<double(Circle, Point, Point)> cross_area =
      [&](const Circle &c, const Point &a, const Point &b) {
        Point va = c.p - a, vb = c.p - b;
        auto f = cross(va, vb), ret = 0.0;
        if (eq(f, 0.0))
          return ret;
        if (max(abs(va), abs(vb)) < c.r + EPS)
          return f;
        if (distance(Segment(a, b), c.p) > c.r - EPS)
          return c.r * c.r * arg(vb * conj(va));
        auto u = crosspoint(c, Segment(a, b));
        vector<Point> tot{a, u.first, u.second, b};
        for (int i = 0; i + 1 < tot.size(); i++) {
          ret += cross_area(c, tot[i], tot[i + 1]);
        }
        return ret;
      };
  double A = 0;
  for (int i = 0; i < p.size(); i++) {
    A += cross_area(c, p[i], p[(i + 1) % p.size()]);
  }
  return A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_B
double convex_diameter(const Polygon &p) {
  int N = (int)p.size();
  int is = 0, js = 0;
  for (int i = 1; i < N; i++) {
    if (p[i].imag() > p[is].imag())
      is = i;
    if (p[i].imag() < p[js].imag())
      js = i;
  }
  double maxdis = norm(p[is] - p[js]);

  int maxi, maxj, i, j;
  i = maxi = is;
  j = maxj = js;
  do {
    if (cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) >= 0) {
      j = (j + 1) % N;
    } else {
      i = (i + 1) % N;
    }
    if (norm(p[i] - p[j]) > maxdis) {
      maxdis = norm(p[i] - p[j]);
      maxi = i;
      maxj = j;
    }
  } while (i != is || j != js);
  return sqrt(maxdis);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_5_A
double closest_pair(Points ps) {
  if (ps.size() <= 1)
    throw(0);

  auto compare_y = [&](const Point &a, const Point &b) {
    return imag(a) < imag(b);
  };
  vector<Point> beet(ps.size());

  function<double(int, int)> rec = [&](int left, int right) {
    if (right - left <= 1)
      return 1e18;
    int mid = (left + right) >> 1;
    auto x = real(ps[mid]);
    auto ret = min(rec(left, mid), rec(mid, right));
    inplace_merge(begin(ps) + left, begin(ps) + mid, begin(ps) + right,
                  compare_y);
    int ptr = 0;
    for (int i = left; i < right; i++) {
      if (abs(real(ps[i]) - x) >= ret)
        continue;
      for (int j = 0; j < ptr; j++) {
        auto luz = ps[i] - beet[ptr - j - 1];
        if (imag(luz) >= ret)
          break;
        ret = min(ret, abs(luz));
      }
      beet[ptr++] = ps[i];
    }
    return ret;
  };
  return rec(0, (int)ps.size());
}

int main() {
  int N;
  double R;
  cin >> N >> R;

  Circle c(Point(0, 0), R);
  Polygon p(N);
  for (auto &s : p)
    cin >> s;
  cout << fixed << setprecision(10) << area2(p, c) * 0.5 << endl;
}


Fixed submission:
#include <bits/stdc++.h>

using namespace std;

const double EPS = 1e-8, PI = acos(-1);

inline bool eq(double a, double b) { return fabs(b - a) < EPS; }

using Point = complex<double>;

Point operator*(const Point &p, const double &d) {
  return Point(real(p) * d, imag(p) * d);
}

istream &operator>>(istream &is, Point &p) {
  double a, b;
  is >> a >> b;
  p = Point(a, b);
  return is;
}

ostream &operator<<(ostream &os, Point &p) {
  os << fixed << setprecision(10) << p.real() << " " << p.imag();
}

Point rotate(double theta, const Point &p) {
  return Point(cos(theta) * p.real() - sin(theta) * p.imag(),
               sin(theta) * p.real() + cos(theta) * p.imag());
}

double radian_to_degree(double r) { return (r * 180.0 / PI); }

double degree_to_radian(double d) { return (d * PI / 180.0); }

double get_angle(const Point &a, const Point &b, const Point &c) {
  const Point v(b - a), w(c - b);
  double alpha = atan2(v.imag(), v.real()), beta = atan2(w.imag(), w.real());
  if (alpha > beta)
    swap(alpha, beta);
  double theta = (beta - alpha);
  return min(theta, 2 * acos(-1) - theta);
}

namespace std {
bool operator<(const Point &a, const Point &b) {
  return a.real() != b.real() ? a.real() < b.real() : a.imag() < b.imag();
}
} // namespace std

struct Line {
  Point a, b;

  Line() {}

  Line(Point a, Point b) : a(a), b(b) {}

  Line(double A, double B, double C) // Ax + By = C
  {
    if (eq(A, 0))
      a = Point(0, C / B), b = Point(1, C / B);
    else if (eq(B, 0))
      b = Point(C / A, 0), b = Point(C / A, 1);
    else
      a = Point(0, C / B), b = Point(C / A, 0);
  }

  friend ostream &operator<<(ostream &os, Line &p) {
    return os << p.a << " to " << p.b;
  }

  friend istream &operator>>(istream &is, Line &a) { return is >> a.a >> a.b; }
};

struct Segment : Line {
  Segment() {}

  Segment(Point a, Point b) : Line(a, b) {}
};

struct Circle {
  Point p;
  double r;

  Circle() {}

  Circle(Point p, double r) : p(p), r(r) {}
};

using Points = vector<Point>;
using Polygon = vector<Point>;
using Segments = vector<Segment>;
using Lines = vector<Line>;
using Circles = vector<Circle>;

double cross(const Point &a, const Point &b) {
  return real(a) * imag(b) - imag(a) * real(b);
}

double dot(const Point &a, const Point &b) {
  return real(a) * real(b) + imag(a) * imag(b);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_C
int ccw(const Point &a, Point b, Point c) {
  b = b - a, c = c - a;
  if (cross(b, c) > EPS)
    return +1; // "COUNTER_CLOCKWISE"
  if (cross(b, c) < -EPS)
    return -1; // "CLOCKWISE"
  if (dot(b, c) < 0)
    return +2; // "ONLINE_BACK"
  if (norm(b) < norm(c))
    return -2; // "ONLINE_FRONT"
  return 0;    // "ON_SEGMENT"
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool parallel(const Line &a, const Line &b) {
  return abs(cross(a.b - a.a, b.b - b.a)) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_A
bool orthogonal(const Line &a, const Line &b) {
  return abs(dot(a.a - a.b, b.a - b.b)) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_A
Point projection(const Line &l, const Point &p) {
  double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
  return l.a + (l.a - l.b) * t;
}

Point projection(const Segment &l, const Point &p) {
  double t = dot(p - l.a, l.a - l.b) / norm(l.a - l.b);
  return l.a + (l.a - l.b) * t;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_1_B
Point reflection(const Line &l, const Point &p) {
  return p + (projection(l, p) - p) * 2.0;
}

bool Intersect(const Line &l, const Point &p) {
  return abs(ccw(l.a, l.b, p)) != 1;
}

bool intersect(const Line &l, const Line &m) {
  return abs(cross(l.b - l.a, m.b - m.a)) > EPS ||
         abs(cross(l.b - l.a, m.b - l.a)) < EPS;
}

bool intersect(const Segment &s, const Point &p) {
  return ccw(s.a, s.b, p) == 0;
}

bool intersect(const Line &l, const Segment &s) {
  return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS;
}

double distance(const Line &l, const Point &p);

bool intersect(const Circle &c, const Line &l) {
  return distance(l, c.p) <= c.r + EPS;
}

bool intersect(const Circle &c, const Point &p) {
  return abs(abs(p - c.p) - c.r) < EPS;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_B
bool intersect(const Segment &s, const Segment &t) {
  return ccw(s.a, s.b, t.a) * ccw(s.a, s.b, t.b) <= 0 &&
         ccw(t.a, t.b, s.a) * ccw(t.a, t.b, s.b) <= 0;
}

int intersect(const Circle &c, const Segment &l) {
  if (norm(projection(l, c.p) - c.p) - c.r * c.r > EPS)
    return 0;
  const double d1 = abs(c.p - l.a), d2 = abs(c.p - l.b);
  if (d1 < c.r + EPS && d2 < c.r + EPS)
    return 0;
  if (d1 < c.r - EPS && d2 > c.r + EPS || d1 > c.r + EPS && d2 < c.r - EPS)
    return 1;
  const Point h = projection(l, c.p);
  if (dot(l.a - h, l.b - h) < 0)
    return 2;
  return 0;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_A&lang=jp
int intersect(Circle c1, Circle c2) {
  if (c1.r < c2.r)
    swap(c1, c2);
  double d = abs(c1.p - c2.p);
  if (c1.r + c2.r < d)
    return 4;
  if (eq(c1.r + c2.r, d))
    return 3;
  if (c1.r - c2.r < d)
    return 2;
  if (eq(c1.r - c2.r, d))
    return 1;
  return 0;
}

double distance(const Point &a, const Point &b) { return abs(a - b); }

double distance(const Line &l, const Point &p) {
  return abs(p - projection(l, p));
}

double distance(const Line &l, const Line &m) {
  return intersect(l, m) ? 0 : distance(l, m.a);
}

double distance(const Segment &s, const Point &p) {
  Point r = projection(s, p);
  if (intersect(s, r))
    return abs(r - p);
  return min(abs(s.a - p), abs(s.b - p));
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_D
double distance(const Segment &a, const Segment &b) {
  if (intersect(a, b))
    return 0;
  return min(
      {distance(a, b.a), distance(a, b.b), distance(b, a.a), distance(b, a.b)});
}

double distance(const Line &l, const Segment &s) {
  if (intersect(l, s))
    return 0;
  return min(distance(l, s.a), distance(l, s.b));
}

Point crosspoint(const Line &l, const Line &m) {
  double A = cross(l.b - l.a, m.b - m.a);
  double B = cross(l.b - l.a, l.b - m.a);
  if (abs(A) < EPS && abs(B) < EPS)
    return m.a;
  return m.a + (m.b - m.a) * B / A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_2_C
Point crosspoint(const Segment &l, const Segment &m) {
  double A = cross(l.b - l.a, m.b - m.a);
  double B = cross(l.b - l.a, l.b - m.a);
  if (abs(A) < EPS && abs(B) < EPS)
    return m.a;
  return m.a + (m.b - m.a) * B / A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_D
pair<Point, Point> crosspoint(const Circle &c, const Line l) {
  Point pr = projection(l, c.p);
  Point e = (l.b - l.a) / abs(l.b - l.a);
  if (eq(distance(l, c.p), c.r))
    return {pr, pr};
  double base = sqrt(c.r * c.r - norm(pr - c.p));
  return {pr - e * base, pr + e * base};
}

pair<Point, Point> crosspoint(const Circle &c, const Segment &l) {
  Line aa = Line(l.a, l.b);
  if (intersect(c, l) == 2)
    return crosspoint(c, aa);
  auto ret = crosspoint(c, aa);
  if (dot(l.a - ret.first, l.b - ret.first) < 0)
    ret.second = ret.first;
  else
    ret.first = ret.second;
  return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_E
pair<Point, Point> crosspoint(const Circle &c1, const Circle &c2) {
  double d = abs(c1.p - c2.p);
  double a = acos((c1.r * c1.r + d * d - c2.r * c2.r) / (2 * c1.r * d));
  double t = atan2(c2.p.imag() - c1.p.imag(), c2.p.real() - c1.p.real());
  Point p1 = c1.p + Point(cos(t + a) * c1.r, sin(t + a) * c1.r);
  Point p2 = c1.p + Point(cos(t - a) * c1.r, sin(t - a) * c1.r);
  return {p1, p2};
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_F
pair<Point, Point> tangent(const Circle &c1, const Point &p2) {
  return crosspoint(c1, Circle(p2, sqrt(norm(c1.p - p2) - c1.r * c1.r)));
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_7_G
Lines tangent(Circle c1, Circle c2) {
  Lines ret;
  if (c1.r < c2.r)
    swap(c1, c2);
  double g = norm(c1.p - c2.p);
  if (eq(g, 0))
    return ret;
  Point u = (c2.p - c1.p) / sqrt(g);
  Point v = rotate(PI * 0.5, u);
  for (int s : {-1, 1}) {
    double h = (c1.r + s * c2.r) / sqrt(g);
    if (eq(1 - h * h, 0)) {
      ret.emplace_back(c1.p + u * c1.r, c1.p + (u + v) * c1.r);
    } else if (1 - h * h > 0) {
      Point uu = u * h, vv = v * sqrt(1 - h * h);
      ret.emplace_back(c1.p + (uu + vv) * c1.r, c2.p - (uu + vv) * c2.r * s);
      ret.emplace_back(c1.p + (uu - vv) * c1.r, c2.p - (uu - vv) * c2.r * s);
    }
  }
  return ret;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_B
bool is_convex(const Polygon &p) {
  int n = (int)p.size();
  for (int i = 0; i < n; i++) {
    if (ccw(p[(i + n - 1) % n], p[i], p[(i + 1) % n]) == -1)
      return false;
  }
  return true;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_A
Polygon convex_hull(Polygon &p) {
  int n = (int)p.size(), k = 0;
  if (n <= 2)
    return p;
  sort(p.begin(), p.end());
  vector<Point> ch(2 * n);
  for (int i = 0; i < n; ch[k++] = p[i++]) {
    while (k >= 2 && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < 0)
      --k;
  }
  for (int i = n - 2, t = k + 1; i >= 0; ch[k++] = p[i--]) {
    while (k >= t && cross(ch[k - 1] - ch[k - 2], p[i] - ch[k - 1]) < 0)
      --k;
  }
  ch.resize(k - 1);
  return ch;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_C
enum { OUT, ON, IN };

int contains(const Polygon &Q, const Point &p) {
  bool in = false;
  for (int i = 0; i < Q.size(); i++) {
    Point a = Q[i] - p, b = Q[(i + 1) % Q.size()] - p;
    if (a.imag() > b.imag())
      swap(a, b);
    if (a.imag() <= 0 && 0 < b.imag() && cross(a, b) < 0)
      in = !in;
    if (cross(a, b) == 0 && dot(a, b) <= 0)
      return ON;
  }
  return in ? IN : OUT;
}

bool merge_if_able(Segment &s1, Segment s2) {
  if (abs(cross(s1.b - s1.a, s2.b - s2.a)) > EPS)
    return false;
  if (ccw(s1.a, s2.a, s1.b) == 1 || ccw(s1.a, s2.a, s1.b) == -1)
    return false;
  if (ccw(s1.a, s1.b, s2.a) == -2 || ccw(s2.a, s2.b, s1.a) == -2)
    return false;
  s1 = Segment(min(s1.a, s2.a), max(s1.b, s2.b));
  return true;
}

void merge_segments(vector<Segment> &segs) {
  for (int i = 0; i < segs.size(); i++) {
    if (segs[i].b < segs[i].a)
      swap(segs[i].a, segs[i].b);
  }
  for (int i = 0; i < segs.size(); i++) {
    for (int j = i + 1; j < segs.size(); j++) {
      if (merge_if_able(segs[i], segs[j])) {
        segs[j--] = segs.back(), segs.pop_back();
      }
    }
  }
}

vector<vector<int>> segment_arrangement(vector<Segment> &segs,
                                        vector<Point> &ps) {
  vector<vector<int>> g;
  int N = (int)segs.size();
  for (int i = 0; i < N; i++) {
    ps.emplace_back(segs[i].a);
    ps.emplace_back(segs[i].b);
    for (int j = i + 1; j < N; j++) {
      const Point p1 = segs[i].b - segs[i].a;
      const Point p2 = segs[j].b - segs[j].a;
      if (cross(p1, p2) == 0)
        continue;
      if (intersect(segs[i], segs[j])) {
        ps.emplace_back(crosspoint(segs[i], segs[j]));
      }
    }
  }
  sort(begin(ps), end(ps));
  ps.erase(unique(begin(ps), end(ps)), end(ps));

  int M = (int)ps.size();
  g.resize(M);
  for (int i = 0; i < N; i++) {
    vector<int> vec;
    for (int j = 0; j < M; j++) {
      if (intersect(segs[i], ps[j])) {
        vec.emplace_back(j);
      }
    }
    for (int j = 1; j < vec.size(); j++) {
      g[vec[j - 1]].push_back(vec[j]);
      g[vec[j]].push_back(vec[j - 1]);
    }
  }
  return (g);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_C
Polygon convex_cut(const Polygon &U, Line l) {
  Polygon ret;
  for (int i = 0; i < U.size(); i++) {
    Point now = U[i], nxt = U[(i + 1) % U.size()];
    if (ccw(l.a, l.b, now) != -1)
      ret.push_back(now);
    if (ccw(l.a, l.b, now) * ccw(l.a, l.b, nxt) < 0) {
      ret.push_back(crosspoint(Line(now, nxt), l));
    }
  }
  return (ret);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_3_A
double area2(const Polygon &p) {
  double A = 0;
  for (int i = 0; i < p.size(); ++i) {
    A += cross(p[i], p[(i + 1) % p.size()]);
  }
  return A;
}

double area2(const Polygon &p, const Circle &c) {
  if (p.size() < 3)
    return 0.0;
  function<double(Circle, Point, Point)> cross_area =
      [&](const Circle &c, const Point &a, const Point &b) {
        Point va = c.p - a, vb = c.p - b;
        auto f = cross(va, vb), ret = 0.0;
        if (eq(f, 0.0))
          return ret;
        if (max(abs(va), abs(vb)) < c.r + EPS)
          return f;
        if (distance(Segment(a, b), c.p) > c.r - EPS)
          return c.r * c.r * arg(vb * conj(va));
        auto u = crosspoint(c, Segment(a, b));
        vector<Point> tot{a, u.first, u.second, b};
        for (int i = 0; i + 1 < tot.size(); i++) {
          ret += cross_area(c, tot[i], tot[i + 1]);
        }
        return ret;
      };
  double A = 0;
  for (int i = 0; i < p.size(); i++) {
    A += cross_area(c, p[i], p[(i + 1) % p.size()]);
  }
  return A;
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_4_B
double convex_diameter(const Polygon &p) {
  int N = (int)p.size();
  int is = 0, js = 0;
  for (int i = 1; i < N; i++) {
    if (p[i].imag() > p[is].imag())
      is = i;
    if (p[i].imag() < p[js].imag())
      js = i;
  }
  double maxdis = norm(p[is] - p[js]);

  int maxi, maxj, i, j;
  i = maxi = is;
  j = maxj = js;
  do {
    if (cross(p[(i + 1) % N] - p[i], p[(j + 1) % N] - p[j]) >= 0) {
      j = (j + 1) % N;
    } else {
      i = (i + 1) % N;
    }
    if (norm(p[i] - p[j]) > maxdis) {
      maxdis = norm(p[i] - p[j]);
      maxi = i;
      maxj = j;
    }
  } while (i != is || j != js);
  return sqrt(maxdis);
}

// http://judge.u-aizu.ac.jp/onlinejudge/description.jsp?id=CGL_5_A
double closest_pair(Points ps) {
  if (ps.size() <= 1)
    throw(0);

  auto compare_y = [&](const Point &a, const Point &b) {
    return imag(a) < imag(b);
  };
  vector<Point> beet(ps.size());

  function<double(int, int)> rec = [&](int left, int right) {
    if (right - left <= 1)
      return 1e18;
    int mid = (left + right) >> 1;
    auto x = real(ps[mid]);
    auto ret = min(rec(left, mid), rec(mid, right));
    inplace_merge(begin(ps) + left, begin(ps) + mid, begin(ps) + right,
                  compare_y);
    int ptr = 0;
    for (int i = left; i < right; i++) {
      if (abs(real(ps[i]) - x) >= ret)
        continue;
      for (int j = 0; j < ptr; j++) {
        auto luz = ps[i] - beet[ptr - j - 1];
        if (imag(luz) >= ret)
          break;
        ret = min(ret, abs(luz));
      }
      beet[ptr++] = ps[i];
    }
    return ret;
  };
  return rec(0, (int)ps.size());
}

int main() {
  int N;
  double R;
  cin >> N >> R;

  Circle c(Point(0, 0), R);
  Polygon p(N);
  for (auto &s : p)
    cin >> s;
  cout << fixed << setprecision(10) << area2(p, c) * 0.5 << endl;
}