Arc Length and Area of Sector

Front

What is the formula for the length of an arc in a circle?

Back

The length of an arc (L) can be calculated using the formula: L = r * θ, where r is the radius and θ is the central angle in radians.

Front

How do you convert degrees to radians?

Back

To convert degrees to radians, use the formula: radians = degrees * (π/180).

Front

What is the formula for the area of a sector in a circle?

Back

The area of a sector (A) can be calculated using the formula: A = (1/2) * r^2 * θ, where r is the radius and θ is the central angle in radians.

Front

What is the relationship between the radius, arc length, and central angle?

Back

The arc length is directly proportional to the radius and the central angle. As the radius or the angle increases, the arc length increases.

Front

If the radius of a circle is 10 cm and the central angle is 60 degrees, what is the arc length?

Back

First, convert 60 degrees to radians: 60 * (π/180) = π/3. Then, use the formula: L = r * θ = 10 * (π/3) = 10π/3 cm.

Front

What is the area of a sector with a radius of 5 cm and a central angle of 90 degrees?

Back

Convert 90 degrees to radians: 90 * (π/180) = π/2. Then, use the formula: A = (1/2) * r^2 * θ = (1/2) * 5^2 * (π/2) = 12.5π/2 = 6.25π cm².

Front

What is the measure of a central angle in radians if the arc length is equal to the radius?

Back

If the arc length (L) is equal to the radius (r), then the central angle (θ) is 1 radian, since L = r * θ.

Front

How do you find the central angle if you know the arc length and radius?

Back

Use the formula: θ = L/r, where L is the arc length and r is the radius.

Front

What is the area of a circle with a radius of 7 cm?

Back

The area (A) of a circle is calculated using the formula: A = π * r^2. For a radius of 7 cm, A = π * 7^2 = 49π cm².

Front

If the area of a sector is 20 cm² and the radius is 4 cm, what is the central angle in degrees?

Back

Use the formula: A = (1/2) * r^2 * θ. Rearranging gives θ = (2A)/(r^2). Substituting gives θ = (2*20)/(4^2) = 10/4 = 2.5 radians. Convert to degrees: 2.5 * (180/π) ≈ 143.24 degrees.