如圖。
交集的面積
= (扇形 CAD) + (扇形 CBD) - 2(△CAB)
= ½ (r1)² (2θ1) + ½ (r2)² (2θ2) - 2[½ dr1 sin(θ1)] ﹝By 扇形及三角形的面積公式﹞
= (r1)² (θ1) + (r2)² (θ2) - dr1 sin(θ1)
= (r1)² cos-¹{[d² + (r1)² - (r2)²] / (2dr1)} ﹝By 餘弦定理﹞
+ (r2)² cos-¹{[d² + (r2)² - (r1)²] / (2dr2)}
- dr1 sqrt[1 - cos²(θ1)]
= (r1)² cos-¹{[d² + (r1)² - (r2)²] / (2dr1)}
+ (r2)² cos-¹{[d² + (r2)² - (r1)²] / (2dr2)}
- ½ sqrt{4d²(r1)² - [d² + (r1)² - (r2)²]²} ■