The area of a circle is found using the formula πr², where r denotes the radius. This formula accurately provides the measure of the space enclosed within the circle.

A full circle encompasses a complete rotation of 360°. This basic property is fundamental in understanding the geometry of circles.

Since 90° is one fourth of 360°, a sector with this angle represents one quarter of the circle's area. This directly uses the proportional relationship between the central angle and the area.

The area enclosed by two radii and the intercepted arc is known as a sector. This is a common term used in circle geometry.

The circumference of a circle is calculated using the formula 2πr, where r represents the radius. This measures the total distance around the circle.

Using the area formula πr² with r = 5 gives π(5²) = 25π. This straightforward computation emphasizes the relationship between radius and area.

The area of the sector is computed using the formula (θ/360)πr². Substituting 60° and r = 4 gives (60/360)×π×16 = 8π/3.

Given the circle's area πr² equals 49π, r² must be 49, which means the radius is 7. This uses the inverse process of finding the radius from the area.

The arc length is obtained by taking the fraction of the circle's circumference corresponding to the central angle. This fraction is θ/360, multiplied by the full circumference 2πr.

Using the arc length formula: 5π = (θ/360)*2π×5, solving for θ gives 180°. This shows how the arc length relates to its central angle.

The area of the sector is (θ/360)×π×8². Setting this equal to 16π leads to (θ/360)×64 = 16, resulting in θ = 90°.

To convert an angle from degrees to radians, you multiply by π/180. This factor is essential for converting between the two measurement systems.

When the angle is provided in radians, the arc length is the product of the radius and the angle. This simplifies the calculation considerably.

A diameter of 12 implies a radius of 6. Therefore, the area is calculated as π(6²) = 36π using the standard area formula.

Using the sector area formula, (150/360)×π×10² simplifies to (5/12)×100π, which reduces to (125π)/3. This demonstrates careful application of the formula.

The central angle in radians is found using the formula angle = arc length / radius. Thus, 15.7 divided by 5 is approximately 3.14 radians.

Since the sector occupies one-fifth of the circle's area, its central angle is one-fifth of the full angle 2π radians, resulting in 2π/5 radians. This directly uses proportionality between area and angle.

The area of a sector is directly proportional to its central angle. An increase of 50% in the area implies that the central angle also increases by 50%, maintaining the proportional relationship.

The area of the triangle is given by 0.5r²sinθ and the area of the sector by 0.5r²θ. Their ratio simplifies to sinθ/θ, illustrating the relationship between the two areas.

Using the chord length formula c = 2r*sin(θ/2), solving 6 = 2×5*sin(θ/2) gives sin(θ/2) = 0.6, leading to θ/2 ≈ 36.87°. Doubling this value provides an approximate central angle of 73.74°.

Given the arc length (rθ) and the area ((1/2)r²θ) of a sector, the ratio simplifies to [rθ] / [0.5r²θ] = (rθ) / (0.5r²θ) = 2/r. However, since the question asks for a ratio in terms of θ for a fixed r, this reveals that r is constant and the role of θ cancels out; therefore, the intended ratio expression emphasizes the invariant nature for fixed r. (Note: This question challenges the student to address nuances in variable separation.)