Circles Lesson Test for ASVAB Math Knowledge

Answer: C

Correct answer: 9 inches. A diameter is two radii laid end to end, so

• A: This doubles the diameter, but the radius is half the diameter, not twice it.
• B: This is the diameter itself; the radius is only half of it.
• C (correct): Half of the 18-inch diameter is 9 inches, which is the radius.
• D: This halves the diameter twice; one halving already gives the radius.

Answer: A

Correct answer: 12π cm. Use

With d = 12 cm, the circumference is $12\pi$ cm.

• A (correct): Circumference equals pi times the diameter, so it is 12 pi cm.
• B: This treats the 12 cm diameter as the radius in 2 pi r, doubling the answer.
• C: This is the area formula pi r squared with r = 6, which gives square units, not a length.
• D: This multiplies pi by the radius only; circumference needs 2 pi r, or pi d.

Answer: C

Correct answer: 9π square meters. Use

Square the radius first:

so the area is $9\pi$ square meters.

• A: This is 2 pi r, the circumference formula, not the area.
• B: This multiplies pi by the radius without squaring the radius.
• C (correct): Squaring the radius gives 9, and multiplying by pi gives 9 pi square meters.
• D: This squares the diameter, 6, instead of the radius, making the area four times too big.

Answer: D

Correct answer: 16 units. The longest chord passes through the center, so it is a diameter. Its length is

units.

• A: This is the radius, which runs only from the center to the circle, not across it.
• B: This squares the radius; a chord is a length found by doubling the radius, not squaring it.
• C: This is the circumference, the distance around the circle, not a straight chord.
• D (correct): The longest chord is a diameter, which is twice the radius: 2 times 8 is 16.

Answer: A

Correct answer: 1/5. A central angle marks off $\frac{\theta }{360^{\circ }}$ of the circle. Here

• A (correct): Dividing 72 by 360 gives 1/5, since 5 times 72 is 360.
• B: This compares 72 with 180 instead of with the full 360 degrees of a circle.
• C: One fourth of a circle corresponds to a 90 degree central angle, not 72 degrees.
• D: One sixth of a circle corresponds to a 60 degree central angle, not 72 degrees.

Answer: D

Correct answer: 90 degrees. A diameter intercepts a 180 degree arc, and an inscribed angle is half its arc. So angle PQR is

• A: This assumes the triangle is equilateral, but a side that is a diameter forces a right angle.
• B: This is the arc cut off by the diameter; the inscribed angle is only half of that arc.
• C: This halves the right angle again; an angle standing on a diameter is exactly half of 180.
• D (correct): Angle PQR stands on a diameter, which cuts off a 180 degree arc, and half of 180 is 90.

Answer: A

Correct answer: 11 inches. Solve

for r:

• A (correct): Dividing the circumference by 2 pi gives 22 pi over 2 pi, which is 11 inches.
• B: This is the diameter, found by dividing by pi only; the radius is half of that.
• C: This multiplies by 2 instead of dividing by 2 pi.
• D: This divides the diameter by 2 twice; 22 pi divided by 2 pi is already the radius.

Answer: A

Correct answer: 16 feet. Divide the area by pi:

The diameter is

• A (correct): Dividing the area by pi gives r squared = 64, so r = 8 and the diameter is 2 times 8, or 16 feet.
• B: This halves 64 instead of taking its square root to find the radius.
• C: This doubles 64 without first taking the square root to get the radius.
• D: This is the radius, the square root of 64; the diameter is twice the radius.

Answer: D

Correct answer: 7π + 14 feet. The straight edge is the diameter, so r = 7 and the curved edge is

Adding the straight edge gives $7\pi +14$ feet.

• A: This counts only the curved edge and leaves out the 14-foot straight edge.
• B: This adds the radius instead of the full straight edge, which is the diameter.
• C: This uses a full circumference with diameter 14 for the curve instead of half of it.
• D (correct): The curve is half of 14 pi, or 7 pi, and adding the 14-foot straight edge gives 7 pi + 14 feet.

Answer: B

Correct answer: 16π square cm. The full circle has area

A quarter circle has

square centimeters.

• A: This takes half of the full area, which describes a semicircle rather than a quarter circle.
• B (correct): One fourth of the full area, 64 pi, is 16 pi square centimeters.
• C: This is the area of the whole circle; a quarter circle covers only one fourth of it.
• D: This is the quarter circle's arc length, one fourth of 16 pi, not its area.

Answer: B

Correct answer: 4π units. The fraction is

The circumference is

so the arc is

units.

• A: This uses the diameter 18 in place of the radius in 2 pi r, doubling the arc.
• B (correct): The fraction is 80/360 = 2/9, and 2/9 of the 18 pi circumference is 4 pi units.
• C: This is the sector area, 2/9 of 81 pi, which is in square units, not the arc length.
• D: This applies the 2/9 fraction to pi r, 9 pi, instead of to the full circumference 2 pi r.

Answer: D

Correct answer: 27π square cm. The fraction is

The full area is

so the sector is

square centimeters.

• A: This is the arc length, one third of 18 pi, rather than the sector area.
• B: This uses the diameter 18 as the radius, so the full area becomes 324 pi instead of 81 pi.
• C: This is the area of the remaining 240 degree part of the circle, not the 120 degree sector.
• D (correct): The fraction is 120/360 = 1/3, and one third of the 81 pi full area is 27 pi square centimeters.

Answer: B

Correct answer: 56π square mm. Ring area is $\pi (R^2-r^2)$. Here

square millimeters.

• A: This adds the hole's area to the outer disk instead of subtracting it.
• B (correct): The outer disk is 81 pi and the hole is 25 pi, so the ring is 81 pi minus 25 pi, or 56 pi.
• C: This is the area of the outer disk alone; the hole in the middle still has to be removed.
• D: This subtracts the radii first and squares the difference, which is the classic ring-area error.

Answer: A

Correct answer: 108π square inches. When the radius is multiplied by k = 3, the area is multiplied by

So the new area is

• A (correct): Tripling the radius multiplies the area by 3 squared, or 9, and 9 times 12 pi is 108 pi.
• B: This multiplies the area by 6, doubling the factor instead of squaring it.
• C: This squares the original area value, 12, instead of multiplying by the square of the scale factor.
• D: This multiplies the area by 3, but area scales by the square of the factor.

Answer: B

Correct answer: 80 + 20π meters. The straight sides add

The two semicircular ends together make one full circle with diameter 20 m, whose circumference is

The perimeter is $80+20\pi$ meters; the diameters are inside the shape and are not counted.

• A: This adds the two 20-meter diameters, but those are interior edges where the ends meet the straight part, so they are not on the outside.
• B (correct): The two straight sides give 80 m, and the two semicircles together form one circle of diameter 20, adding 20 pi m.
• C: This treats each semicircular end as a full circle of diameter 20, doubling the curved length.
• D: This counts only one of the two straight sides.

Answer: D

Correct answer: 4π + 20 cm. The angle fraction is

The full circumference is

so the arc is $4\pi$ cm. Add the two radii:

• A: This uses one fifth of the area, 100 pi, which gives the sector area 20 pi rather than the arc length.
• B: This adds only one radius, but a sector is bounded by two radii.
• C: This is only the arc length; the two straight radii along the sides of the slice were left out.
• D (correct): The arc is one fifth of the 20 pi cm circumference, 4 pi cm, and the two 10 cm radii add 20 cm.

Answer: C

Correct answer: 180π feet. One turn moves the wheel one circumference:

For 60 turns, the distance is

• A: This multiplies 60 by the diameter and leaves out pi, so it uses the distance across, not around.
• B: This treats the 3-foot diameter as the radius, giving 6 pi per turn instead of 3 pi.
• C (correct): Each turn covers one circumference, 3 pi feet, and 60 turns cover 60 times 3 pi, or 180 pi feet.
• D: This uses pi times the radius, 1.5 pi, as the distance per turn instead of the full circumference.

Answer: C

Correct answer: 15 cm. A tangent meets the radius at 90 degrees, so triangle OPT is a right triangle with hypotenuse OT = 17 and leg OP = 8.

So TP is 15 cm.

• A: This adds the squares, treating the 17 cm segment as a leg, but OT is the hypotenuse opposite the right angle.
• B: This adds the radius to the distance from the center, which does not follow the right-triangle relationship.
• C (correct): Radius OP meets the tangent at a right angle, so TP squared = 17 squared minus 8 squared = 225, and TP = 15 cm.
• D: This subtracts the lengths directly; in a right triangle the squares must be subtracted, not the sides.

Answer: B

Correct answer: 4. Set up the area formula:

• A: This sets 2x equal to 7 and forgets to account for the minus 1 in the radius.
• B (correct): Dividing by pi gives a radius squared of 49, so 2x minus 1 = 7, 2x = 8 and x = 4.
• C: This sets the radius equal to 49 without taking the square root of r squared.
• D: This sets the radius equal to half of 49, as if the area were 2 pi r instead of pi r squared.

Answer: B

Correct answer: 64 - 16π square inches. The square has area

The quarter circle has area

Subtracting gives $64-16\pi$, about 13.7 square inches.

• A: This takes one eighth of the full circle's area, as if the slice were 45 degrees instead of 90.
• B (correct): The square is 64 and the quarter circle is one fourth of 64 pi, or 16 pi, so the outside part is 64 minus 16 pi.
• C: This subtracts the arc length, 4 pi, which is a length rather than the quarter circle's area.
• D: This is the area of the quarter circle itself, the part inside the arc rather than outside it.

Answer: C

Correct answer: 16 + 2π feet. Count only the outside edges: two sides of 6 feet and the bottom of 4 feet give 16 feet. The semicircle has radius 2, so its curve is

The total trim is $16+2\pi$ feet.

• A: This uses a full circle's circumference with diameter 4 for the top instead of half of it.
• B: This also counts the rectangle's top edge, which is hidden inside the window where it meets the semicircle.
• C (correct): The outside edge is two 6-foot sides, the 4-foot bottom and a curved top of half of 4 pi, which totals 16 + 2 pi feet.
• D: This leaves out the 4-foot bottom edge, which is part of the outside of the window.

Answer: D

Correct answer: 62 degrees. The central angle shows that arc AB measures 124 degrees. Angle ACB has its vertex on the circle and intercepts that arc, so it measures

• A: This halves the 236 degree major arc, but C sits on that arc, so angle ACB cuts off the 124 degree arc.
• B: This subtracts 124 from 180, as if the angles were supplementary, instead of halving the arc.
• C: This sets the inscribed angle equal to the central angle, but an inscribed angle is half of its arc.
• D (correct): Angle ACB intercepts the same 124 degree arc as the central angle, and an inscribed angle is half its arc.

Answer: C

Correct answer: 100π square cm. The hypotenuse is

Because the right angle is inscribed, the hypotenuse is a diameter, so r = 10 and the area is

square centimeters.

• A: This adds the legs to get 28 as the diameter, but the diameter is the hypotenuse found with the Pythagorean theorem.
• B: This uses the 20 cm hypotenuse as the radius, but the hypotenuse is a diameter, so the radius is 10.
• C (correct): The right angle stands on a diameter, so the 20 cm hypotenuse is the diameter, the radius is 10 and the area is 100 pi.
• D: This is the circumference, pi times the 20 cm diameter, not the area.

Answer: A

Correct answer: 0.09π square meters. Halve the diameter and convert before squaring: r = 30 cm = 0.3 m. Then

• A (correct): The radius is 30 cm, or 0.3 m, and squaring it gives 0.09, so the area is 0.09 pi square meters.
• B: This divides 900 pi square centimeters by 100, but a square meter holds 10,000 square centimeters.
• C: This uses the full 0.6 m diameter as the radius instead of halving it to 0.3 m.
• D: This is the area in square centimeters with the units left unconverted.

Answer: B

Correct answer: 20π feet. Work backward from the area: r squared = 100π / π = 100, so r = 10 feet. The binding follows the edge, so find the circumference: C = 2π(10) = 20π feet.

• A: This is π times r; the circumference formula is 2πr, so the radius must be doubled.
• B (correct): The area gives r = 10, and the circumference is 2π times 10 = 20π feet.
• C: This halves the area; dividing by 2 does not turn an area into a circumference.
• D: This uses 20, the diameter, in place of the radius in 2πr.

Answer: D

Correct answer: 45π square feet. The outer radius is

The ring is the big disk minus the fountain.

The walkway covers 45π square feet.

• A: This squares the radius difference, π times 3 squared; the ring needs R squared minus r squared.
• B: This is the whole outer disk; the fountain's 36π still has to be removed.
• C: This adds the two disk areas instead of subtracting the fountain from the outer disk.
• D (correct): Outer radius 9, inner radius 6: π times (81 - 36) = 45π square feet.

Answer: A

Correct answer: 6π inches. The angle fraction is

The full circumference is 2π(8) = 16π inches, so the arc is 3/8 of 16π = 6π inches.

• A (correct): The fraction 3/8 of the 16π-inch circumference is 6π inches.
• B: This takes 3/8 of the area, 64π; arc length uses the circumference.
• C: This is the full circumference; only 3/8 of the circle is cut off.
• D: This takes 3/8 of 8π, using πr instead of the full circumference 2πr.

Answer: A

Correct answer: 10π square cm. The fraction is

The full area is π(4)^2 = 16π square cm, so the sector is 5/8 of 16π = 10π square cm.

• A (correct): The fraction 5/8 of the 16π full area gives 10π square cm.
• B: This is 5/8 of the circumference, 8π, which is the arc length, not the sector's area.
• C: This is the area of the entire circle, not the 225-degree slice.
• D: This is the area of the remaining 135-degree slice, 3/8 of 16π.

Answer: C

Correct answer: 2.25 times. The diameters are in the ratio

so the radii are too. Area scales by the square of that ratio: 1.5 squared = 2.25.

• A: This is the ratio of diameters; it tells how lengths compare, but areas scale by its square.
• B: This doubles the scale factor of 1.5 instead of squaring it.
• C (correct): The scale factor is 1.5, and area scales by 1.5 squared, which is 2.25.
• D: This cubes the scale factor, which is how volumes compare, not flat areas.

Answer: D

Correct answer: 18π square feet. The straight edge is the diameter, so r = 6 feet. A semicircle covers half the full disk: one half of π(6)^2 = one half of 36π = 18π square feet.

• A: This is the area of the whole circle; a semicircle is only half of it.
• B: This uses 12 as the radius, giving 144π, and then halves it; the 12-foot edge is the diameter.
• C: This is π times 6, the length of the curved edge, not the area inside.
• D (correct): The radius is 6, and half of 36π is 18π square feet.

Answer: D

Correct answer: 50. Each turn moves the bicycle one circumference: πd = 28π inches. Turns = distance divided by circumference.

• A: This uses 2π times 28 = 56π per turn, treating the diameter as the radius.
• B: This uses 14π per turn, multiplying π by the radius instead of the diameter.
• C: This divides 50 by π again; the π in the distance and the circumference already cancel.
• D (correct): One turn is 28π inches, and 1400π divided by 28π is 50.

Answer: A

Correct answer: 72 - 18π square inches. Each hole spans the 6-inch width, so its diameter is 6 and its radius is 3. Each hole is π(3)^2 = 9π, so two holes are 18π.

The remaining metal is 72 - 18π square inches.

• A (correct): Each hole has radius 3 and area 9π; the plate's 72 square inches minus 18π remains.
• B: This removes only one hole; the plate has two.
• C: This subtracts the two circumferences, 6π each, which are lengths, not areas.
• D: This computes the plate as 6 by 6; the plate is 12 by 6, or 72 square inches.

Answer: A

Correct answer: 28π inches. Circumference is C = 2πr = 2π(14) = 28π inches.

• A (correct): Two times π times 14 is 28π inches.
• B: This is π times the radius; the formula needs 2πr, or π times the diameter.
• C: This is π times 14 squared, which is the area, not the distance around.
• D: This uses 4πr, doubling the circumference formula by mistake.

Answer: C

Correct answer: 225π square cm. A radius drawn to the point of contact meets the tangent at a right angle, so the radius, the tangent segment and the 25 cm distance form a right triangle with hypotenuse 25.

The area is π(15)^2 = 225π square cm.

• A: This is the circumference, 2π times 15, not the area.
• B: This uses the 25 cm distance as the radius; that distance is the hypotenuse, from the center to an outside point.
• C (correct): The right triangle gives r = 15, and π times 15 squared is 225π square cm.
• D: This uses the 20 cm tangent segment as the radius; the radius is the other leg of the right triangle.

Answer: D

Correct answer: 25π square feet. Halve the diameter first: r = 5 feet. Then A = π(5)^2 = 25π square feet.

• A: This squares the diameter instead of the radius, making the area four times too big.
• B: This is π times the diameter, which is the circumference, not the area.
• C: This is 2π times 10, a circumference formula with the diameter used as the radius.
• D (correct): The radius is 5, and π times 5 squared is 25π square feet.

Answer: B

Correct answer: 20π cm. The square's sides are inside the shape now, so they do not count. The outer edge is four semicircles of diameter 10, which equal two full circles: 2 times 10π = 20π cm.

• A: This also adds the square's four sides, but they are covered by the semicircles and are not on the outside.
• B (correct): Four semicircular arcs of diameter 10 make two full circumferences, 2 times 10π.
• C: This counts only two semicircles, one full circle; there is a semicircle on all four sides.
• D: This counts each attachment as a full circle instead of a half circle.

Answer: B

Correct answer: 55 degrees. An inscribed angle is half of its intercepted arc:

• A: This is the measure a central angle on the same arc would have; inscribed angles are half as large.
• B (correct): An inscribed angle is half its arc, and half of 110 is 55.
• C: This doubles the arc instead of halving it.
• D: This subtracts the arc from 180; the inscribed angle is simply half its arc.

Answer: C

Correct answer: 78.5 meters. The line follows the boundary, so use C = πd = 3.14 times 25 = 78.5 meters.

• A: This doubles π times the diameter; 2πr already equals πd, so doubling again is too much.
• B: This multiplies 3.14 by the radius, 12.5, which gives only half the circumference.
• C (correct): Pi times the diameter, 3.14 times 25, is 78.5 meters.
• D: This is π times 12.5 squared, the pad's area, which is not a length.

Answer: D

Correct answer: 90 degrees. A radius drawn to the point where a tangent touches the circle always meets the tangent at a right angle, 90 degrees.

• A: The radius does not cut the tangent at a slant; the angle is always a right angle.
• B: A 180-degree angle would mean the radius lies along the tangent line, which cannot happen.
• C: No angle other than 90 degrees is fixed by the tangent rule, whatever the size of the circle.
• D (correct): A tangent is always perpendicular to the radius drawn to the point of contact.

Answer: B

Correct answer: 30 inches. Pi is the circumference divided by the diameter, so d = C / π =

• A: This is the radius, half of the diameter.
• B (correct): Diameter is circumference divided by π: 94.2 / 3.14 = 30 inches.
• C: This doubles the diameter, as if C / π gave the radius instead of the diameter.
• D: This multiplies the circumference by π instead of dividing.

Answer: C

Correct answer: 34π cm. A diameter is two radii, so

Then 3x = 24 and x = 8, so the radius is

• A: This is π times 17 squared, the area, not the circumference.
• B: This is π times the radius; the circumference is 2πr.
• C (correct): Setting the diameter equal to twice the radius gives x = 8, r = 17 and C = 34π cm.
• D: This uses 2π times the diameter, 34, which doubles the circumference.

Answer: B

Correct answer: 1/9. The slice's share equals its central angle over the full 360 degrees: 40/360, which simplifies to 1/9.

• A: This compares 40 to 180 degrees, which is only half a circle.
• B (correct): The fraction is 40/360, which reduces to 1/9.
• C: This puts the angle in the denominator; the fraction is angle over 360.
• D: This compares 40 to 90 degrees, which is only a quarter turn.

Answer: A

Correct answer: 76 degrees. An inscribed angle is half its intercepted arc, so the arc is twice the angle: 2 times 38 = 76 degrees.

• A (correct): The arc is twice the inscribed angle, 2 times 38.
• B: This halves the angle; it is the angle that is half the arc, so the arc is double.
• C: This treats the inscribed angle as equal to its arc, which holds only for a central angle.
• D: This subtracts 38 from 180; there is no supplement relationship between an inscribed angle and its arc.

Answer: A

Correct answer: 96 square cm. An angle that stands on a diameter is a right angle, so angle Y is 90 degrees and XZ is the hypotenuse. The third side is the square root of

The legs are 16 and 12, so the area is one half times 16 times 12 = 96 square cm.

• A (correct): Angle Y is 90 degrees, the missing leg is 12, and half of 16 times 12 is 96.
• B: This uses the diameter, 20, as a leg; the diameter is the hypotenuse, so it is not a base-height pair with XY.
• C: This multiplies the legs 16 and 12 but forgets the one half in the triangle area.
• D: This assumes the two legs are equal at 16 cm; the Pythagorean relationship makes the other leg 12.

Answer: D

Correct answer: 324π square inches. Convert the radius first: 1.5 feet = 18 inches. Then A = π(18)^2 = 324π square inches.

• A: This is the area in square feet, π times 1.5 squared, with the units never converted.
• B: This converts 2.25 square feet by multiplying by 12; a square foot is 144 square inches.
• C: This is 2π times 18, the circumference in inches, not the area.
• D (correct): Convert 1.5 feet to 18 inches before squaring: π times 324.

Answer: B

Correct answer: 3π + 12 feet. The curved part is one fourth of 2π(6) = 12π, which is 3π feet. Add the two straight radii,

The total is 3π + 12 feet.

• A: This is just the curved edge; the outline also includes the two straight radii.
• B (correct): The arc is one fourth of 12π, or 3π, plus two 6-foot radii.
• C: This uses the quarter-circle's area, 9π, in place of the arc length.
• D: This uses half the circumference, which is a semicircle's arc, not a quarter circle's.

Answer: C

Correct answer: 81π square inches. Find the radius from the circumference: r = 18π / 2π = 9 inches. Then A = π(9)^2 = 81π square inches.

• A: This squares 18, which is the diameter, so the area comes out four times too large.
• B: This reuses the circumference value, a length, as if it were the area.
• C (correct): The circumference gives r = 9, and π times 9 squared is 81π.
• D: This uses 2πr squared, mixing the circumference formula into the area formula.

Answer: C

Correct answer: 20π cm. The perpendicular from the center cuts the chord into two 8 cm halves. The radius to an end of the chord is the hypotenuse of a right triangle with legs 8 and 6.

The circumference is 2π(10) = 20π cm.

• A: This adds 8 and 6 to get a radius of 14; the radius is the hypotenuse, not the sum of the legs.
• B: This is the area, π times 10 squared, not the circumference.
• C (correct): Half the chord, 8, and the distance, 6, give a radius of 10, so C = 20π.
• D: This treats the chord as a diameter, but a chord 6 cm from the center does not pass through it.