Question

do all work on your own paper and show complete work for every problem.
hint: area of a triangle is $\frac{1}{2}b\cdot h$, area of a circle is $\pi r^{2}$
calculate the area of the figure shown using the information given. the figure is composed of a triangle and a semicircle. round the area to 3 decimal places and remember units (use ^ to raise to a power in the answerbox below).
area =

Explanation:

Step1: Calculate the area of the triangle

The formula for the area of a triangle is \(A_{\text{triangle}}=\frac{1}{2}bh\). Here, \(b = 12\space\text{cm}\) and \(h=12\space\text{cm}\).

$$ A_{\text{triangle}}=\frac{1}{2}\times12\times12 = 72\space\text{cm}^2 $$

Step2: Calculate the radius of the semicircle

The diameter of the semicircle is equal to the hypotenuse of the right - angled triangle. Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\) (where \(a = b=12\)), \(c=\sqrt{12^{2}+12^{2}}=\sqrt{144 + 144}=\sqrt{288}=12\sqrt{2}\space\text{cm}\). The radius \(r=\frac{12\sqrt{2}}{2}=6\sqrt{2}\space\text{cm}\)

Step3: Calculate the area of the semicircle

The formula for the area of a circle is \(A=\pi r^{2}\), so the area of a semicircle is \(A_{\text{semicircle}}=\frac{1}{2}\pi r^{2}\). Substituting \(r = 6\sqrt{2}\), we get \(A_{\text{semicircle}}=\frac{1}{2}\pi(6\sqrt{2})^{2}=\frac{1}{2}\pi\times72 = 36\pi\space\text{cm}^2\approx36\times3.14159 = 113.097\space\text{cm}^2\)

Step4: Calculate the total area

The total area \(A=A_{\text{triangle}}+A_{\text{semicircle}}\)

$$ A=72 + 113.097=185.097\space\text{cm}^2 $$

Answer:

\(185.097\space\text{cm}^2\)