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To find the angle \(\theta\), given the area and the lengths of the sides, we can use the formula for the area of a triangle:

\[
\text{Area} = \frac{1}{2}ab \sin(\theta)
\]

where \( a \) and \( b \) are the sides adjacent to the angle \(\theta\). In this problem:

- The area is \( 17.8 \ \text{cm}^2 \).
- \( a = 5.2 \ \text{cm} \).
- \( b = 11.3 \ \text{cm} \).

Substitute the given values into the formula:

\[
17.8 = \frac{1}{2} \times 5.2 \times 11.3 \times \sin(\theta)
\]

Simplifying this gives:

\[
17.8 = 29.38 \times \sin(\theta)
\]

\[
\sin(\theta) = \frac{17.8}{29.38}
\]

\[
\sin(\theta) \approx 0.6059
\]

Now, use the inverse sine function to find \(\theta\):

\[
\theta = \arcsin(0.6059)
\]

\[
\theta \approx 37.3^\circ
\]

So, the angle \(\theta\) is approximately \( 37.3^\circ \) to 3 significant figures.

Would you like further details or have any questions? Here are some related questions:

1. How would the angle change if the area of the triangle were different?
2. What is the effect of changing the lengths of the sides on the angle \(\theta\)?
3. Can you derive the formula for the area of a triangle using trigonometry?
4. How do you find the remaining angles of a triangle using the sine rule?
5. What happens if \(\theta\) is obtuse instead of acute?

**Tip:** Always double-check your unit consistency when solving geometry problems to avoid mistakes.

Work out the size of angle θ, given that it is acute, using the area of 17.8 cm² and sides of lengths 5.2 cm and 11.3 cm.

Learn how to calculate the acute angle θ in a triangle given the area of 17.8 cm² and the sides of 5.2 cm and 11.3 cm. This solution uses trigonometry and the sine rule to solve for θ, providing a step-by-step guide for students in Grades 9-12.

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