Find the area of a sector determined by a 315degrees° angle in a circle of radius 4.

1. Break Down the Problem

To find the area of a sector, we need to identify the formula used for calculating it. A sector of a circle is defined by a central angle, and we will use the radius given.

2. Relevant Concepts

The formula for the area $A$ of a sector of a circle is given by:

$A = \frac{\theta}{360^\circ} \times \pi r^2$

where:

• $\theta$ is the angle in degrees,
• $r$ is the radius of the circle.

3. Analysis and Detail

Given:

• $\theta = 315^\circ$
• $r = 4$

Using the formula, we can substitute the values into the equation:

$A = \frac{315}{360} \times \pi \times (4)^2$

Calculating $(4)^2$:

$(4)^2 = 16$

Substituting back, we have:

$A = \frac{315}{360} \times \pi \times 16$

Calculating $\frac{315}{360}$:

$\frac{315}{360} = \frac{7}{8}$

Now substituting this value:

$A = \frac{7}{8} \times \pi \times 16$

Calculating further:

$A = \frac{7 \times 16 \pi}{8} = 14 \pi$

4. Verify and Summarize

The calculations seem accurate as all steps were followed methodically, and the values substituted where necessary.

Final Answer

The area of the sector is $14\pi$ square units, which is approximately $43.98$ square units when calculated numerically.