This exercise involves the formula for the area of a circular sector.

Question

This exercise involves the formula for the area of a circular sector.

The area of a sector of a circle with a central angle of 5𝜋/12 rad is 19 m². Find the radius of the circle. (Round your answer to one decimal place.)

 m

Answer 1

A full circle is 2𝜋 rad.

A sector with a central angle of 5𝜋/12 rad is a slice of the circle that is

(5𝜋/12)/2𝜋 = 5/24 the area of the whole circle:

A_sector = (5/24)A_circle which means that A_circle = (24/5)A_sector = 4.8A_sector

Since we know A_sector = 19 m², we can calculate

A_circle = 4.8A_sector =(4.8)(19) = 91.2

Since A_circle = 𝜋r², where r is the radius, we can calculate r as as follows:

Start with 𝜋r²=91.2

Divide both sides by 𝜋 to get r²=91.2/𝜋

Take the square root of both sides to get r = √(91.2/𝜋).

Answer (rounded to one decimal place)

r = 5.4 m

Answer 2

The area of a sector is a fraction of the area of an entire circle. This fraction is the ratio of the angle of the sector to the angle swept out by a full circle, measured in most cases by either 360 degrees or 2π radians.

The area of a complete circle is π × the radius squared, or π r² = area of a circle.

Therefore, the area of a sector is [ (sector width angle in radians) / (2×π radians) ] × (π r²).

This can be simplified to [ (sector width angle ) / (2) ] × (r²).

Or

(sector width angle in radians) × r² / 2 = sector area.

In this case, we are given the sector area, 19 m² and the sector width angle, 5π/12 radians.

That leads to this equation:

5π/12 × r² / 2 = 19 square meters.

Multiply both sides of the above equation by 2:

5π/12 radians × r² = 2 × 19 = 38 square meters

Now, divide by 5π and multiply by 12 on both sides:

r² = (38×12/(5π)) square meters

And taking the square root of both sides:

r = 5.4 meters, rounded to 1 decimal place.

Note: you might be wondering what happened to the "radians" unit. Recall that this is a fraction with the sector width angle being divided by the number of radians in an entire circle. These units cancel out and leave the equation.