https://file.aiquickdraw.com/mathbot/file/1727343244301fqdi6ohb.jpg

To solve this problem, we need to use the formula that relates the arc length of a sector to the radius and the angle subtended at the center (in radians).

The formula for the length of an arc \( L \) of a sector is:
\[
L = r\theta
\]
Where:
- \( L \) is the arc length,
- \( r \) is the radius of the sector,
- \( \theta \) is the angle subtended by the arc at the center, measured in radians.

From the problem:
- The total length of the wire is 70 cm, which includes both the arc length and the two radii of the sector.
- The radius \( r \) is given as 20 cm.

So the arc length \( L \) is:
\[
L = 70 - 2 \times 20 = 70 - 40 = 30 \, \text{cm}
\]

Now, using the formula for arc length:
\[
L = r\theta \quad \Rightarrow \quad 30 = 20 \times \theta
\]

Solving for \( \theta \):
\[
\theta = \frac{30}{20} = 1.5 \, \text{radians}
\]

Thus, the angle \( \angle AOB \) is \( 1.5 \) radians.

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Would you like more details or clarification? Here are some related questions to deepen your understanding:

1. How is the arc length related to the radius and angle of a sector?
2. What would happen if the radius of the sector were different?
3. How can you convert the angle from radians to degrees?
4. What if the wire length was given as a variable instead of a specific value?
5. How do you calculate the area of the sector based on the angle?

**Tip:** Always make sure to subtract the lengths of the radii from the total wire length when the problem involves arcs and radii combined!

A piece of wire of 70 cm long is bent to form a sector OAB. Identify the angle of AOB in radians.

This page explains how to calculate the angle of a sector formed by a wire bent into shape. Given a wire of 70 cm and a radius of 20 cm, learn how to use the arc length formula to determine the angle in radians. Suitable for students in Grades 9-11.

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