Let
r
1 = radius of circle O
1 = 15 cm
r
2 = radius of circle O
2 = 20 cm
r
3 = radius of circle O
3 = 25 cm
α = ∠BAC = arctan (40/30) = 53.13°
β = ∠ACB = arctan (30/40) = 36.87°
Solving for A1
$\theta_1 = 180^\circ - 2\alpha = 180^\circ - 2(53.13^\circ)$
$\theta_1 = 73.74^\circ$
A1 = semicircle of radius r1 - circular segment of radius r3 and central angle θ1
$A_1 = \frac{1}{2}\pi{r_1}^2 - \frac{1}{2}{r_3}^2(\theta_{1rad} - \sin \theta_{1deg})$ → Calculator if DEG mode
$A_1 = \frac{1}{2}\pi(15^2) - \frac{1}{2}(25^2)~ \left[ 73.74^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 73.74^\circ \right]$
$A_1 = 353.43 - 102.19$
$A_1 = 251.24 \, \text{ cm}^2$ answer
Solving for A2
$\theta_2 = 180^\circ - 2\beta = 180^\circ - 2(36.87^\circ)$
$\theta_2 = 106.26^\circ$
A2 = semicircle of radius r2 – circular segment segment of radius r3 and central angle θ2
$A_2 = \frac{1}{2}\pi{r_2}^2 - \frac{1}{2}{r_3}^2(\theta_{2rad} - \sin \theta_{2deg})$ → Calculator if DEG mode
$A_2 = \frac{1}{2}\pi(20^2) - \frac{1}{2}(25^2)~ \left[ 106.26^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 106.26^\circ \right]$
$A_2 = 628.32 – 279.56$
$A_2 = 348.76 \, \text{ cm}^2$ answer
Solving for A3
ABC and O1BO2 are both 3-4-5 triangles, thus,
$\theta_3 = 2\alpha = 2(53.13^\circ)$
$\theta_3 = 106.26^\circ$
$\theta_4 = 2\beta = 2(36.87^\circ)$
$\theta_4 = 73.74^\circ$
A3 = circular segment of radius r1 and central angle θ3 + circular segment of radius r2 and central angle θ4
$A_3 = \frac{1}{2}{r_1}^2(\theta_{3rad} - \sin \theta_{3deg}) + \frac{1}{2}{r_2}^2(\theta_{4rad} - \sin \theta_{4deg})$ → Calculator if DEG mode
$A_3 = \frac{1}{2}(15^2)~ \left[ 106.26^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 106.26^\circ \right] + \frac{1}{2}(20^2)~ \left[ 73.74^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 73.74^\circ \right]$
$A_3 = 100.64 + 65.40$
$A_3 = 166.04 \, \text{ cm}^2$ answer
Solving for A4
$\theta_5 = 180^\circ - 2\alpha = 180^\circ - 2(53.13^\circ)$
$\theta_5 = 73.74^\circ$
$\theta_6 = 180^\circ - 2\beta = 180^\circ - 2(36.87^\circ)$
$\theta_6 = 106.26^\circ$
A4 = semicircle of radius r3 – circular segment of radius r1 and central angle θ5 – circular segment of radius r2 and central angle θ6
$A_4 = \frac{1}{2}\pi{r_3}^2 - \frac{1}{2}{r_1}^2(\theta_{5rad} - \sin \theta_{5deg}) - \frac{1}{2}{r_2}^2(\theta_{6rad} - \sin \theta_{6deg})$ → Calculator if DEG mode
$A_4 = \frac{1}{2}\pi(25^2) - \frac{1}{2}(15^2)~ \left[ 73.74^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 73.74^\circ \right] - \frac{1}{2}(20^2)~ \left[ 106.26^\circ \left( \dfrac{\pi}{180^\circ} \right) - \sin 106.26^\circ \right]$
$A_4 = 981.75 – 36.79 – 178.92$
$A_4 = 766.04 \, \text{ cm}^2$ answer
Checking
$A_1 + A_2 + A_3 = 251.24 + 348.76 + 166.04$
$A_1 + A_2 + A_3 = 766.04 \, \text{ cm}^2$
$A_1 + A_2 + A_3 = A_4 = 766.04 \, \text{ cm}^2$ (okay!)
