The perimeter of a sector is 172m and radiusis 42cm. Find the \$\theta\$?

Perimeter of a sector.

\$P = 2r + l \$

Substitute 172 for P and 42 for r.

\$172 = 2(42) + l\$

Solve for l.

\$ l = 88 \$

Arc length of a sector.

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

Substitute \$22/7\$ for \$pi\$, 42 for r and 88 for l.

\$ 88 = \frac{\theta}{360°} \times 2 \times \frac{22}{7} \times 42 \$

Cancel out common factor.

\$ \cancel{88}^8 = \frac{\theta}{360°} \times 2 \times \frac{\cancel{22}^2}{7} \times 42\$

Cancel out common factor.

\$ 8 = \frac{\theta}{360°} \times 2 \times \frac{2}{\cancel{7}^1} \times \cancel{42}^6 \$

Cancel out common factor.

\$ \cancel{8}^4 = \frac{\theta}{360°} \times 2 \times \frac{\cancel{2}^1}{1} \times 6 \$

Cancel out common factor.

\$ \cancel{4}^2 = \frac{\theta}{360°} \times \cancel{2}^1 \times \frac{1}{1} \times 6 \$

Cancel out common factor.

\$ 2 = \frac{\theta}{\cancel{360°} ^{60°}} \times 1 \times \frac{1}{1} \times \cancel{6}^1 \$

Multiply.

\$\theta = 120° \$

Answer

The θ is \$120°\$.