Question

1. a patio includes a paved stone patio and a small lawn. the lawn is enclosed by an arc ab of a circle with centre o and radius 6m, such that $angle aob = 135^circ$. the straight border of the lawn is defined by segment ab.

the lawn is shown as the shaded region and the paved stone patio is shown as the unshaded region in the following diagram.
diagram not to scale
(a) find the area of the entire patio. 2
(b) find the area of the paved stone patio (the unshaded region). 2
(c) find the area of the lawn (the shaded region). 2

Explanation:

Step1: Calculate full circle area

The entire patio is a full circle with radius 6 m. The formula for the area of a circle is $A = \pi r^2$.
$A_{full} = \pi \times 6^2 = 36\pi$

Step2: Find unshaded area (part a)

The unshaded region is the fraction of the circle not in the 135° sector. First find the angle of the unshaded sector: $360^\circ - 135^\circ = 225^\circ$. The area is the fraction $\frac{225}{360}$ of the full circle.
$A_{unshaded} = \frac{225}{360} \times 36\pi = 22.5\pi$

Step3: Find shaded lawn area (part b)

The shaded region is the 135° sector minus the area of triangle AOB. First calculate the sector area:
$A_{sector} = \frac{135}{360} \times 36\pi = 13.5\pi$
Then calculate the area of triangle AOB using the formula $A_{\triangle} = \frac{1}{2}ab\sin\theta$ where $a=b=6$, $\theta=135^\circ$:
$A_{\triangle} = \frac{1}{2} \times 6 \times 6 \times \sin(135^\circ) = 18 \times \frac{\sqrt{2}}{2} = 9\sqrt{2}$
Subtract to get the shaded area:
$A_{shaded} = 13.5\pi - 9\sqrt{2}$

Answer:

(a) The area of the entire patio is $\boldsymbol{36\pi}$ m² (or approximately 113.10 m²)
(b) The area of the paved stone patio is $\boldsymbol{22.5\pi}$ m² (or approximately 70.69 m²)
(c) The area of the lawn is $\boldsymbol{13.5\pi - 9\sqrt{2}}$ m² (or approximately 30.17 m²)