If a circle has an area of 12π cm2 and a diameter \(\overline{AB}\), what is the length of arc AB?
-
Solution
The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, the radius of this circle is \(\sqrt{12}\) cm or 23 cm. The circumference of a circle is 2πr, so the circumference of this circle is 2π(2√3) = 43π cm. A diameter divides a circle into two 180° arcs. Therefore, the measure of arc AB is 180°. The length of the arc is equal to \(\frac{180}{360}\)(4√3π) = 1⁄2(4√3π) = 2√3π cm.
If the measure of angle COB below is 3x, what is the measure of arc CA?
-
Solution
Angles COB and COA form a line; they are supplementary angles. Therefore, the measure of angle COA is equal to 180 – 3x. Since angle COA is a central angle and CA is its intercepted arc, the measure of CA is also 180 – 3x.
The measure of a central angle is 18°. If the length of its intercepted arc is xπ units, what is the circumference of the circle?
-
Solution
The length of an arc is equal to the circumference of the circle multiplied by the fraction of the circle that the arc covers. Therefore, if c is the circumference of the circle, then xπ = \(\frac{18}{360}\)c. Multiply both sides of the equation by \(\frac{360}{18}\), or 20, to isolate c: (\(\frac{360}{18}\))xπ = (\(\frac{360}{18}\))\(\frac{18}{360}\)c, c = 20xπ square units.
If the central angle of a sector is x° and the radius of the circle is x units, then the area of the sector is equal to
-
Solution
The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, the area of this circle is πx2. The area of a sector is equal to the area of the circle multiplied by the fraction of the circle that the sector covers. That fraction is equal to \(\frac{x}{360}\), so the area of the sector is equal to \(\frac{x}{360}\)(πx2) = (\(\frac{x^{3}}{360}\))π square units.
If the area of a circle is (4x2 + 20x + 25)π, what is the diameter of the circle?
-
Solution
The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, the radius of this circle is equal to the square root of (4x2 + 20x + 25). Factor this trinomial into 2 identical factors; (2x)(2x) = 4x2, (5)(5) = 25, and (2x)(5) + (2x)(5) = 20x. Therefore, (4x2 + 20x + 25) = (2x + 5)(2x + 5), and the square root of (4x2 + 20x + 25) is 2x + 5. The diameter of a circle is twice its radius, so the diameter of this circle is 2(2x + 5) = 4x + 10.
The radius of Carly’s circle is 2x – 7, and the area of her circle is (16x + 9)π. Which of the following could be the value of x?
-
Solution
The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, π(2x – 7)2 = (16x + 9)π, 4x2 – 14x – 14x + 49 = 16x + 9, 4x2 – 44x + 40 = 0, x2 – 11x + 10 = 0, (x – 1)(x – 10) = 0, x = 10. x cannot be equal to 1, since that would make the radius equal to –5, and a radius cannot have a negative length.
If the diameter of a circle is doubled, the circumference of the new circle is
-
Solution
The circumference of a circle is equal to 2πr. Since the radius of a circle is half the diameter of a circle, 2r is equal to the diameter, d, of a circle. Therefore, the circumference of a circle is equal to πd. If the diameter is doubled, the circumference becomes π2d, or two times its original size.
If the diameter of a circle is 8x + 6, what is the area of the circle?
-
Solution
The radius of a circle is equal to half the diameter of a circle. The radius of this circle is equal to \(\frac{(8x+ 6)}{2}\) = 4x + 3. The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, the area of this circle is equal to π(4x + 3)2= π(16x2 + 12x + 12x + 9) = (16x2 + 24x + 9)π square units.
If the area of a circle is (121x)π square units, what is the circumference of the circle?
-
Solution
The area of a circle is equal to πr2, where r is the radius of the circle. Therefore, the radius of this circle is equal to the square root of 121x, or 11√x units. The circumference of a circle is equal to 2πr, so the circumference of this circle is equal to 2π(11√x) = (22√x)π units.
If the circumference of a circle triples, the area of the circle becomes
-
Solution
The circumference of a circle is 2πr, where r is the radius of the circle. If the circumference of a circle triples, that means the radius of the circle has tripled. The area of the circle has gone from πr2 to π (3r)2 = π9r2. The area of the circle is now 9 times bigger.