The area of a full circle is given by the formula π r². Since a semicircle is half of a circle, its area is half of the full circle's area.

Now, we substitute the calculated radius r = 5 cm into the area formula. We square the radius and then multiply by (1)/(2)π to get the final area.

Now regarding the shaded region. It definitely looks like a semicircle with PQ as the base. If PQ were a diameter, then $O$ would be the midpoint of PQ. But $O$ is the center of the circle, and PQ is a chord. The shaded region in the figure seems to be the area of the semicircle. Let's assume the shaded region is the semicircle whose diameter is the line segment from P to a point directly opposite on the circumference, passing through $O$ . The shading is from P to Q, then along arc Q to another. (semicircle, diameter, chord, circumference, arc)

Answer: The area of the shaded region is $12.5\pi \text{ cm}^2$ .

Step-by-step solution

Step 1: Identify the shape and given information

The problem states that the shaded region is a semicircle and that $PQ$ is its diameter. We are given the length of the diameter, $PQ = 10\text{ cm}$ .

Step 2: Calculate the radius

The radius ( $r$ ) of a circle or semicircle is half of its diameter ( $d$ ). By dividing the given diameter by 2, we find the radius.

Step 3: Recall the formula for the area of a semicircle

The area of a full circle is given by the formula $\pi r^2$ . Since a semicircle is half of a circle, its area is half of the full circle's area.

Step 4: Substitute the radius and calculate the area

Now, we substitute the calculated radius $r = 5\text{ cm}$ into the area formula. We square the radius and then multiply by $\frac{1}{2}\pi$ to get the final area.

More from this chapter

Solve your own maths question free →