Multiple Choice
Given a triangle with an included angle of and a side of length feet adjacent to the angle, if the area of the triangle is square feet, what is the length of the base adjacent to the angle?
7. Non-Right Triangles
Area of SAS & ASA Triangles
Multiple Choice
Given a circle with radius and a central angle measured in radians, what is the area of the shaded sector formed by this angle?
A
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Verified step by step guidance1
Recall that the area of a full circle is given by the formula \(\text{Area} = \pi \times r^{2}\), where \(r\) is the radius of the circle.
Understand that the central angle \(\theta\) (in radians) represents a fraction of the full circle. Since a full circle corresponds to an angle of \(2\pi\) radians, the fraction of the circle covered by the sector is \(\frac{\theta}{2\pi}\).
To find the area of the sector, multiply the total area of the circle by the fraction of the circle represented by the angle \(\theta\). This gives \(\text{Sector Area} = \left( \frac{\theta}{2\pi} \right) \times \pi r^{2}\).
Simplify the expression by canceling \(\pi\) in numerator and denominator, resulting in \(\text{Sector Area} = \frac{\theta \times r^{2}}{2}\).
Thus, the formula for the area of a sector with radius \(r\) and central angle \(\theta\) (in radians) is \(\boxed{\frac{\theta \times r^{2}}{2}}\).
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