Area of a Composite Figure – Triangle and Two Semicircles

88.40 in²

Explanation

The shaded figure is a combination of a triangular region and two semicircular regions. Its area is found by adding the three parts:

\[A_{\text{total}} = A_{\text{triangle}} + A_{\text{smaller semicircle}} + A_{\text{larger semicircle}}\]

Area of the triangle (base = 6 in, height = 8 in):

\[A_{\text{triangle}} = \frac{1}{2}bh = \frac{1}{2}(6)(8) = \frac{1}{2}(48) = 24 \text{ in}^2\]

Area of the smaller semicircle (radius = 4 in, diameter = 8 in = one leg of the triangle):

Since the formula for the area of a semicircle is \(A = \frac{1}{2}\pi r^2\):

\[A_{\text{smaller}} = \frac{1}{2}\pi(4)^2 = \frac{1}{2}\pi(16) = 8\pi\]

Area of the larger semicircle (radius = 5 in, diameter = 10 in = hypotenuse of the triangle):

\[A_{\text{larger}} = \frac{1}{2}\pi(5)^2 = \frac{1}{2}\pi(25) = 12.5\pi\]

Total area:

\[A_{\text{total}} = 24 + 8\pi + 12.5\pi \approx 88.4026494\]

To the nearest hundredth, the area of the shaded figure is 88.40 in².