The diagram shows the top view of a building WXYZ. OWX and OZY are sectors of a circle with centre $O$ . Find the area of the top surface of the building.

Full solution

Q. The diagram shows the top view of a building WXYZ. OWX and OZY are sectors of a circle with centre

O

. Find the area of the top surface of the building.

Identify Shapes: Identify the shapes that make up the top surface of the building. The sectors $OWX$ and $OZY$ are parts of circles, and $WXYZ$ is a rectangle.
Calculate Rectangle Area: Calculate the area of the rectangle WXYZ. Assume the length is $l$ and the width is $w$ . The area of a rectangle is length times width, $A = l \times w$ .
Calculate Sector OWX Area: Calculate the area of the sector OWX. Assume the radius of the circle is $r$ and the angle of the sector is $\theta$ degrees. The area of a sector is $(\theta/360) \cdot \pi \cdot r^2$ .
Calculate Sector OZY Area: Calculate the area of the sector OZY. It should be the same as the area of sector OWX since they are sectors of the same circle with the same angle. So, the area of OZY is also $(\theta/360) * \pi * r^2$ .
Calculate Total Area: Add the area of the rectangle $WXYZ$ to the areas of the two sectors $OWX$ and $OZY$ to find the total area of the top surface of the building. Total area = $A + 2 \times \left(\frac{\theta}{360} \times \pi \times r^2\right)$ .
Calculate Final Answer: Plug in the known values for $l$ , $w$ , $r$ , and $\theta$ into the total area formula and calculate the final answer.