  ### Home > CCG > Chapter 12 > Lesson 12.2.3 > Problem12-92

12-92.

Find the area and perimeter of the shape at right. Assume that any non-straight portions of the shape are part of a circle. Show all work. Break up the shape into smaller, more manageable shapes.
There is more than one way to do this.

$125^\circ − 90^\circ = 35^\circ$

Fill in all known angles and lengths.

Use trigonometry to find the length of the triangle's unknown leg.

$\tan35^\circ = \frac{4}{x} \text { or } \frac{\sin 55^\circ}{x} = \frac {\sin 35^\circ}{4}$

Find the area of each the smaller shapes.

How does each of these smaller areas relate to the area of the larger shape?     $Area_{tri}=\frac{1}{2}\text{(base)(height)}$$Area_{tri}=\frac{1}{2}\text{(5.71)(4)}$$Area_{tri}=11.42 \text{ units} ^2$ $Area_{sec}=\frac{1}{4} r^2 \pi$$Area_{sec}=\frac{1}{4} 4^2 \pi$$Area_{sec}=12.566 \text{ units} ^2$ $Area_{rec}=\text{(base)(height)}$$Area_{rec}=(4)(5.71+4)$$Area_{rec}=38.84 \text{ units} ^2$

Find the lengths of the sides that form the perimeter
the triangle.

$x^2 + 4^2 = a^2$
$5.71.^2 + 4^2 = a^2$
$48.6041 = a^2$
$6.972 \approx a^2$

$b=\frac{1}{4}(2r\pi)$
$b=\frac{1}{4}(2 \: \cdot \pi)$
$b \approx 6.26$

Add these answers together to find the perimeter of the entire shape.

Perimeter = $6.972 + 6.26 + 4 + 9.71 + 4 = 30.942$ units

Area = $62.826$ units$^2$