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Circles

25 questions

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Question 1Question →

A circular swimming pool has a circumference of 20π20\pi feet. What is the diameter of the pool in feet?

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Answer: 20

Answer

20
The formula for the circumference of a circle is C=πdC = \pi d. By substituting the given circumference of 20π20\pi into the formula, we get 20π=πd20\pi = \pi d. Dividing both sides by π\pi yields a diameter of 2020 feet.

Step-by-Step Solution

1
Identify the formula for the circumference of a circle.
C=πdC = \pi d, where CC is circumference and dd is diameter.
This formula directly relates the given circumference to the unknown diameter.
2
Substitute the given value into the formula.
20π=πd20\pi = \pi d
The problem states the circumference is 20π20\pi feet.
3
Solve for the diameter (dd).
d=20d = 20
Dividing both sides of the equation by π\pi isolates the value of dd.

Key Concept

Circumference of a circle
Estimated Time:45s
Question 2Question →

An automated lawn sprinkler rotates through an angle of 120∘120^\circ and sprays water outward to a maximum distance of 1818 feet. Which of the following represents the total area of the lawn covered by this sprinkler? (Leave your answer in terms of π\pi.)

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Answer: 108π sq ft108\pi \text{ sq ft}

Answer

108π sq ft108\pi \text{ sq ft}
The sprinkler covers a sector of a circle. The radius of this circle is the maximum spray distance, which is 1818 feet. The full area of a circle with this radius would be π(18)2=324π\pi(18)^2 = 324\pi square feet. Since the sprinkler rotates 120∘120^\circ, which is 120360\frac{120}{360} or 13\frac{1}{3} of a full circle, the covered area is 13\frac{1}{3} of 324π324\pi, resulting in 108π108\pi square feet.

Step-by-Step Solution

1
Identify the geometric shape and the relevant area formula.
The area covered by the sprinkler is a sector of a circle. The formula for the area of a sector is Area=θ360∘×πr2Area = \frac{\theta}{360^\circ} \times \pi r^2.
The sprinkler covers a fractional portion of a full circle defined by its rotation angle.
2
Determine the fraction of the circle covered.
120∘360∘=13\frac{120^\circ}{360^\circ} = \frac{1}{3}
The sprinkler rotates 120∘120^\circ out of a total possible 360∘360^\circ for a complete circle.
3
Substitute the radius into the formula and solve.
13×π×(18)2=13×π×324=108π\frac{1}{3} \times \pi \times (18)^2 = \frac{1}{3} \times \pi \times 324 = 108\pi
The maximum spray distance (1818 feet) is the radius. Squaring 1818 gives 324324, and finding one-third of this yields the sector's area.

Key Concept

Area of a Circular Sector
Question 3Question →

A circular target has a diameter of 1010 inches. What is the area of the target in square inches?

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Answer: 25π25\pi

Answer

The correct answer is 25π25\pi square inches.
The correct area is found by first dividing the diameter of 1010 inches by 22 to get a radius of 55 inches. Squaring the radius gives 2525, which is then multiplied by π\pi to yield 25π25\pi square inches.

Step-by-Step Solution

1
Find the radius of the circular target.
r=102=5r = \frac{10}{2} = 5 inches.
The area formula requires the radius, which is exactly half of the given diameter.
2
Apply the formula for the area of a circle.
A=πr2=π(5)2=25πA = \pi r^2 = \pi (5)^2 = 25\pi.
Substitute the calculated radius into the standard area formula to find the final square footage.

Key Concept

Area of a Circle
Question 4Question →

Match each fundamental geometric term related to a circle with its correct definition.

Click a left item, then click its matching right item

Items

Radius
Diameter
Circumference
Chord

Matches

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Answer

Radius matches with a line segment from the center to the edge; Diameter matches with a line segment passing through the center; Circumference matches with the total distance around the circle; Chord matches with a line segment connecting any two points on the circle.
Each geometric term possesses a precise mathematical definition that distinguishes its function and location within a circle. Correctly identifying these fundamental components is an essential prerequisite for performing geometric calculations such as finding area or circumference.

Step-by-Step Solution

1
Identify the term that represents the perimeter or boundary distance of the circle.
Circumference matches with 'The total distance around the outer boundary of a circle.'
Circumference is the only term listed that measures a 1D distance around a shape rather than a straight interior line segment.
2
Differentiate between the line segments based on their relationship to the circle's center point.
Radius connects the center to the edge. Diameter passes all the way through the center from edge to edge.
This establishes the mathematical relationship where the diameter is exactly twice the length of the radius.
3
Determine the general term for a line segment connecting any two points on the circle's edge.
Chord matches with 'A line segment whose endpoints lie on the circle, but which does not necessarily pass through the center.'
A diameter is a special type of chord, but 'chord' is the broader category that does not require passing through the center.

Key Concept

Fundamental vocabulary and structural properties of circles.
Question 5Question →

A landscaper is placing a decorative border around a circular flower bed. The radius of the flower bed is 1212 feet. What is the total length of the border needed in terms of π\pi?

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Answer: 24π24\pi feet

Answer

The total length of the border is 24π24\pi feet, which is found by calculating the circumference of the circle.
To find the total length of the border, you must calculate the circumference of the circular flower bed. The formula for circumference is C=2πrC = 2\pi r. Substituting the given radius of 1212 feet into the formula gives C=2⋅π⋅12=24πC = 2 \cdot \pi \cdot 12 = 24\pi feet.

Step-by-Step Solution

1
Identify the mathematical concept required to find the length of a border around a circle.
The length of the border corresponds to the circumference of the circle.
Perimeter (circumference) measures the distance around the outside of a 2D shape.
2
Select the correct formula for the circumference of a circle.
Formula: C=2πrC = 2\pi r
The problem provides the radius (rr), so this is the most direct formula to use.
3
Substitute the given radius value into the formula and solve.
C=2⋅π⋅12=24πC = 2 \cdot \pi \cdot 12 = 24\pi feet.
Replacing rr with 1212 computes the exact circumference in terms of π\pi.

Key Concept

Circumference of a circle
Question 6Question →

A circular helipad has a diameter of 2020 meters. What is the area of the helipad in square meters?

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Answer: 100π100\pi

Answer

The correct area is 100π100\pi square meters.
The area of a circle is found using the formula A=πr2A = \pi r^2. Since the given diameter is 2020 meters, the radius is half of that amount, which is 1010 meters. Squaring the radius gives 100100, so the area is 100π100\pi square meters.

Step-by-Step Solution

1
Identify the given information and what needs to be found.
The diameter of the helipad is 2020 meters, and the area needs to be calculated.
Understanding the starting values is required to choose the correct geometric formulas.
2
Calculate the radius of the helipad.
The radius is half of the diameter: r=20÷2=10r = 20 \div 2 = 10 meters.
The standard formula for the area of a circle requires the radius, not the diameter.
3
Apply the area formula for a circle.
A=πr2=π(10)2=100πA = \pi r^2 = \pi (10)^2 = 100\pi square meters.
Squaring the radius and multiplying by π\pi yields the total area.

Key Concept

Calculating the area of a circle from its diameter.
Question 7Question →

An engineer is evaluating the specifications of a circular pipe and recording various measurements. Match each measurement description with the correct geometric term.

Click a left item, then click its matching right item

Items

The total length of the continuous boundary enclosing the pipe's cross-section
The straight-line measurement from the exact center of the pipe to its outer edge
The longest straight distance across the pipe, passing directly through its center
The total amount of two-dimensional flat space enclosed within the pipe's boundary

Matches

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Answer

The boundary length matches to Circumference, the center-to-edge measurement to Radius, the edge-to-edge measurement through the center to Diameter, and the enclosed space to Area.
Each practical measurement perfectly aligns with a fundamental geometric property of a circle. The continuous outer boundary corresponds to the circumference. The distance from the center point to the edge is the radius. The line segment completely crossing the circle through its center is the diameter. Finally, the total two-dimensional region contained inside is the area.

Step-by-Step Solution

1
Analyze the description for the boundary of the circular pipe.
Matches with Circumference.
By definition, the perimeter or outer boundary of a circle is called its circumference.
2
Analyze the description for the measurement from the center to the outer edge.
Matches with Radius.
The radius is the line segment connecting the center point to any point on the outer edge of the circle.
3
Analyze the description for the full width passing through the center.
Matches with Diameter.
The diameter is the longest chord of a circle, spanning from one edge to the other directly through the center.
4
Analyze the description for the enclosed two-dimensional space.
Matches with Area.
Area is the mathematical term for the amount of surface space enclosed by a two-dimensional shape.

Key Concept

Basic terminology and properties of circles
Question 8Question →

A circular crop field has a diameter of 3030 yards. A dirt road that is 55 yards wide is built along the entire outer edge of the field. What is the area of the dirt road in square yards?

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Answer: 175π175\pi

Answer

The area of the dirt road is 175π175\pi square yards.
The dirt road forms a ring (an annulus) around the circular field. To find this area, calculate the area of the total circular space (field plus road) and subtract the area of the field itself. The field's radius is 1515 yards (30÷230 \div 2), making the total outer radius 2020 yards (15+515 + 5). Subtracting the inner area (225π225\pi) from the outer area (400π400\pi) gives the road's area: 175π175\pi square yards.

Step-by-Step Solution

1
Find the radius of the inner circular field.
rinner=302=15r_{inner} = \frac{30}{2} = 15 yards
The area formula for a circle requires the radius, but the diameter of the field is given.
2
Determine the total radius of the entire area, including the road.
router=15+5=20r_{outer} = 15 + 5 = 20 yards
The dirt road adds 55 yards to the outer edge of the field's radius.
3
Calculate the total area enclosed by the outer edge of the road.
Areaouter=π(20)2=400πArea_{outer} = \pi (20)^2 = 400\pi square yards
This represents the total area of the field and the road combined.
4
Calculate the area of just the inner crop field.
Areainner=π(15)2=225πArea_{inner} = \pi (15)^2 = 225\pi square yards
This area must be subtracted to isolate the area of the road.
5
Subtract the inner area from the total outer area.
400π−225π=175π400\pi - 225\pi = 175\pi square yards
The area of the ring (annulus) is the difference between the two concentric circular areas.

Key Concept

Area of an annulus (a ring formed by two concentric circles)
Question 9Question →

A circular hatch on a maritime vessel has a diameter of 1414 inches. What is the area of the hatch in square inches?

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Answer: 49π49\pi

Answer

The correct area is 49π49\pi square inches.
The correct answer is found by first determining the radius, which is half of the given diameter (14÷2=714 \div 2 = 7 inches). Applying the area formula A=πr2A = \pi r^2 yields π×72=49π\pi \times 7^2 = 49\pi square inches.

Step-by-Step Solution

1
Determine the radius of the circular hatch.
Radius = 14÷2=714 \div 2 = 7 inches.
The formula for the area of a circle requires the radius, which is exactly half of the diameter.
2
Apply the area formula for a circle.
Area = π×72=49π\pi \times 7^2 = 49\pi square inches.
The area of a circle is found using the formula A=πr2A = \pi r^2.

Key Concept

Calculating the area of a circle by first deriving the radius from the diameter.
Question 10Question →

Match each circular geometry scenario with its corresponding simplified algebraic expression.

Click a left item, then click its matching right item

Items

The area of a circular pathway of uniform width xx surrounding a pool with radius rr
The total perimeter of a semicircular window with radius rr
The number of full revolutions a tire with diameter dd makes to travel a distance DD
The radius of a circle whose area is exactly half the area of a circle with radius RR

Matches

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Answer

The correct pairings connect each geometric scenario to its simplified algebraic derivation by applying circle formulas, expanding binomials, and rationalizing denominators.
Each expression is correctly derived by applying the fundamental formulas for circle area and circumference, followed by rigorous algebraic simplification such as factoring, binomial expansion, or radical rationalization.

Step-by-Step Solution

1
Analyze the annular pathway area scenario.
πx(2r+x)\pi x(2r + x)
Requires subtracting the inner pool area from the total outer area and expanding the squared binomial (r+x)2(r+x)^2.
2
Determine the semicircular perimeter expression.
r(π+2)r(\pi + 2)
Requires adding the semicircular arc length to the straight diameter base and factoring out the radius.
3
Calculate the algebraic expression for tire revolutions.
Dπd\frac{D}{\pi d}
Requires dividing the total linear distance by the circumference of the tire, utilizing the diameter.
4
Solve for the half-area radius.
R22\frac{R\sqrt{2}}{2}
Requires setting up an area ratio equation, solving for the new radius, and rationalizing the denominator of the resulting square root.

Key Concept

Algebraic representation and derivation of advanced circular measurements
Question 11Question →

Match each advanced circular geometry scenario with its exact calculated numerical value.

Click a left item, then click its matching right item

Items

A wheel with a diameter of 2424 inches rolls forward for a total distance of 120π120\pi inches. The number of complete revolutions the wheel makes.
The measure of the central angle (in degrees) of a circular sector that has a radius of 66 inches and an area of 15π15\pi square inches.
The outer circumference (in yards) of a circular track that has an inner circumference of 40π40\pi yards and a uniform width of 55 yards.
The area (in square inches) of a square perfectly inscribed inside a circular base that has an area of 64π64\pi square inches.

Matches

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Answer

Match the scenarios to their respective values: the wheel makes 55 revolutions, the sector angle is 150∘150^{\circ}, the track's outer circumference is 50π50\pi yards, and the inscribed square's area is 128128 square inches.
Each scenario relies on accurately applying distinct circular properties. 1) The distance traveled by a wheel uses the circumference formula (C=πdC=\pi d). 2) The sector area requires using the proportion of the circle θ360πr2\frac{\theta}{360}\pi r^2. 3) Concentric track perimeters require adding the width to the inner radius before recalculating circumference. 4) An inscribed square has a diagonal equal to the circle's diameter, allowing for area calculation via the diagonal.

Step-by-Step Solution

1
Calculate the number of wheel revolutions.
The circumference is 24π24\pi. Dividing the total distance 120π120\pi by 24π24\pi yields 55 revolutions.
One full revolution covers a linear distance equal to exactly one circumference.
2
Determine the sector's central angle.
Using the area formula A=θ360πr2A = \frac{\theta}{360}\pi r^2, setting 15π=θ360π(36)15\pi = \frac{\theta}{360}\pi(36) simplifies to θ=150∘\theta = 150^{\circ}.
The area of a sector is a proportional fraction of the full circle's area based on its central angle.
3
Find the outer circumference of the circular track.
The inner radius is 2020 (from 40π=2πr40\pi = 2\pi r). The outer radius is 20+5=2520 + 5 = 25. The outer circumference is 2π(25)=50π2\pi(25) = 50\pi.
Concentric circles share the same center; the outer radius equals the inner radius plus the track's width.
4
Calculate the area of the inscribed square.
The circle's radius is 88, so the diameter (and square's diagonal) is 1616. The square's area is 12(16)2=128\frac{1}{2}(16)^2 = 128.
An inscribed square's diagonal is strictly equal to the diameter of the circumscribing circle.

Key Concept

Advanced applications of circle formulas including circumference, sector area, and relationships with inscribed shapes.
Question 12Question →

A windshield wiper assembly has a total length of 2020 inches from its pivot point to its tip. The innermost 44 inches of the assembly is a bare metal arm, and the outer 1616 inches is the rubber wiper blade. When turned on, the assembly sweeps back and forth through an angle of 135∘135^\circ. What is the total area of the windshield that is wiped by the rubber blade during one sweep? (Leave your answer in terms of π\pi)

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Answer: 144π144\pi sq in.

Answer

The correct answer is 144π144\pi sq in.
The region wiped by the blade forms a shape called an annulus sector. To find its area, we calculate the area of the larger sector created by the full assembly (radius = 20) and subtract the area of the smaller un-wiped sector created by the metal arm (radius = 4). Both sectors cover a 135∘135^\circ angle, which is 38\frac{3}{8} of a full circle. Large sector area is 38π(202)=150π\frac{3}{8}\pi(20^2) = 150\pi, and small sector area is 38π(42)=6π\frac{3}{8}\pi(4^2) = 6\pi. The difference is 144π144\pi.

Step-by-Step Solution

1
Identify the geometric shapes involved in the problem.
The total area swept is a large circular sector with a radius of 2020 inches. The inner un-wiped area is a smaller circular sector with a radius of 44 inches. The wiped area is the difference between them.
The rubber blade starts 44 inches away from the pivot, leaving a small sector un-wiped.
2
Determine the fraction of a full circle the wiper sweeps.
135∘360∘=38\frac{135^\circ}{360^\circ} = \frac{3}{8} of a circle.
Sector area requires knowing what fraction of the total 360∘360^\circ circle is being used.
3
Calculate the area of the large sector (total sweep).
Area=38×π×(20)2=38×400π=150π sq in.\text{Area} = \frac{3}{8} \times \pi \times (20)^2 = \frac{3}{8} \times 400\pi = 150\pi \text{ sq in.}
This finds the entire area enclosed by the 20-inch assembly.
4
Calculate the area of the small un-wiped sector.
Area=38×π×(4)2=38×16π=6π sq in.\text{Area} = \frac{3}{8} \times \pi \times (4)^2 = \frac{3}{8} \times 16\pi = 6\pi \text{ sq in.}
This finds the area swept only by the bare metal arm.
5
Subtract the un-wiped area from the total area.
150π−6π=144π sq in.150\pi - 6\pi = 144\pi \text{ sq in.}
Subtracting gives the area exclusively wiped by the rubber blade.

Key Concept

Sector Area of an Annulus
Question 13Question →

A flat circular roof of a silo has a circumference of 4444 meters. A sealant must be applied to the entire top surface. If one bucket of sealant covers exactly 3030 square meters, what is the minimum number of buckets that must be purchased to seal the roof? (Use π=227\pi = \frac{22}{7})

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Answer: 6

Answer

6
To find the correct number of buckets, we first derived the radius (77 meters) from the given circumference (4444 meters). Next, we calculated the total area (154154 square meters). Finally, dividing the total area by the coverage per bucket (3030 square meters) gives a value of 5.1335.133. Because we cannot purchase a fraction of a bucket and need to cover the entire roof, we must round up to 66 full buckets.

Step-by-Step Solution

1
Set up the circumference formula C=2πrC = 2\pi r using the given values to solve for the radius.
44=2×227×r44 = 2 \times \frac{22}{7} \times r, which simplifies to 44=447×r44 = \frac{44}{7} \times r. Multiplying both sides by 744\frac{7}{44} gives r=7r = 7 meters.
The radius is required to calculate the surface area of the circular roof.
2
Calculate the area of the circular roof using the formula A=πr2A = \pi r^2.
A=227×(7)2=227×49=154A = \frac{22}{7} \times (7)^2 = \frac{22}{7} \times 49 = 154 square meters.
The total area determines the total amount of sealant that is needed.
3
Determine the number of buckets needed by dividing the total area by the coverage per bucket.
154÷30=5154 \div 30 = 5 with a remainder of 44 (or 5.133...5.133...). Since partial buckets cannot be purchased, you must round up to 66 buckets.
To seal the entire surface area without leaving any sections exposed, you must purchase enough full buckets to cover the total area.

Key Concept

Calculating the area of a circle from its circumference and applying the result to a real-world scenario requiring discrete quantities.
Question 14Question →

A radar system on a ship sweeps a 45∘45^\circ sector of the surrounding water. If the total area scanned by this single sweep is 32π32\pi square nautical miles, what is the maximum detection range (radius) of the radar in nautical miles?

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Answer: 16

Answer

The maximum detection range (radius) of the radar is 16 nautical miles.
The area of a sector is found using the formula A=θ360πr2A = \frac{\theta}{360} \pi r^2. Substituting the given values gives 32π=45360πr232\pi = \frac{45}{360} \pi r^2. Simplifying the fraction 45360\frac{45}{360} to 18\frac{1}{8} and dividing both sides by π\pi yields 32=18r232 = \frac{1}{8} r^2. Multiplying both sides by 8 isolates r2r^2, resulting in 256=r2256 = r^2. Taking the square root gives a radius of 16.

Step-by-Step Solution

1
Identify the mathematical formula for the area of a circle's sector.
Area = (θ / 360) * π * r^2
This formula relates the known area and central angle to the unknown radius.
2
Substitute the known values from the word problem into the formula.
32π = (45 / 360) * π * r^2
The scanned area is 32π and the sweep angle is 45 degrees.
3
Simplify the fraction and divide both sides by π.
32 = (1/8) * r^2
The fraction 45/360 simplifies to 1/8, and removing π from both sides simplifies the algebraic equation.
4
Multiply both sides of the equation by 8 to isolate r^2.
256 = r^2
Applying the correct inverse operation (multiplying by the reciprocal of 1/8) isolates the squared variable.
5
Take the square root of both sides to solve for r.
r = 16
Since a physical distance must be positive, the principal square root gives the correct radius.

Key Concept

Calculating the radius of a circle given the area of a sector.
Question 15Question →

Match each geometric term related to circles with its correct mathematical definition.

Click a left item, then click its matching right item

Items

Chord
Secant
Tangent
Diameter

Matches

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Answer

Chord matches with a line segment having endpoints on the circle. Secant matches with a straight line intersecting at two points. Tangent matches with a straight line touching at exactly one point. Diameter matches with a line segment passing through the center.
Each geometric term possesses a precise mathematical definition based on whether it is a finite segment or an infinite line, and how many distinct points it shares with the circle. A chord is a finite segment with two points on the circle, a diameter is a specific chord that passes through the center, a secant is an infinite line intersecting the circle at two points, and a tangent is an infinite line intersecting at exactly one point.

Step-by-Step Solution

1
Differentiate between lines and line segments in the definitions.
Identify that Chord and Diameter are finite line segments, while Secant and Tangent are straight lines extending infinitely.
This fundamental distinction splits the matching options into two distinct groups, making the problem easier to solve.
2
Distinguish between the two line segments (Chord and Diameter).
A diameter must pass through the center point, making it a specific type of chord. Therefore, 'Diameter' matches the definition mentioning the center, and 'Chord' matches the broader segment definition.
Both have endpoints on the circle, but only the diameter requires passing through the center point.
3
Distinguish between the two straight lines (Secant and Tangent).
A secant cuts through the circle at two points, while a tangent only touches the outer boundary at one point.
Counting the exact number of intersection points (two for a secant, one for a tangent) determines the correct match.

Key Concept

Geometric properties and terminology of lines and segments associated with circles
Question 16Question →

The circumference of a circular stained glass window is 18π18\pi inches. What is the area of the window in square inches?

(Enter only the numerical value that multiplies π\pi. For example, if the area is 10π10\pi, enter 10.)

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Answer: 81

Answer

81
The correct answer is 81. First, the radius is found by dividing the circumference (18π18\pi) by 2π2\pi, which gives a radius of 9 inches. Then, the area is calculated by squaring this radius and multiplying by π\pi (92=819^2 = 81), resulting in a total area of 81π81\pi. Since the prompt asks for just the numerical value multiplying π\pi, the final response is 81.

Step-by-Step Solution

1
Find the radius of the circular window using the circumference formula C=2πrC = 2\pi r.
18π=2πr⇒r=918\pi = 2\pi r \Rightarrow r = 9 inches.
The area formula requires the radius, so we must work backward from the given circumference to find it.
2
Calculate the area of the window using the area formula A=πr2A = \pi r^2.
A=π(9)2=81πA = \pi(9)^2 = 81\pi square inches.
Squaring the radius and multiplying by π\pi yields the total area of the circle.
3
Identify the requested numerical value.
81
The question specifically asks for the numerical coefficient that multiplies π\pi.

Key Concept

Calculating the area of a circle by first determining the radius from the circumference.
Question 17Question →

A circular track has an inner circumference of 440440 meters. The track has a uniform width of 77 meters. What is the circumference of the outer edge of the track? (Use π≈227\pi \approx \frac{22}{7})

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Answer: 484 meters

Answer

484 meters
The inner circumference is 440440 meters. Using the formula C=2πrC = 2\pi r, the inner radius is 7070 meters (440=2×227×r440 = 2 \times \frac{22}{7} \times r). Because the track has a uniform width of 77 meters, the outer radius is 70+7=7770 + 7 = 77 meters. Using the circumference formula again for the outer edge, C=2×227×77=484C = 2 \times \frac{22}{7} \times 77 = 484 meters.

Step-by-Step Solution

1
Find the inner radius using the given circumference.
rinner=70r_{inner} = 70 meters
Using the formula C=2πrC = 2\pi r, substitute 440440 for CC and 227\frac{22}{7} for π\pi: 440=2×227×r440 = 2 \times \frac{22}{7} \times r, which simplifies to r=70r = 70.
2
Calculate the outer radius by adding the track's width.
router=77r_{outer} = 77 meters
The outer edge is 77 meters further from the center than the inner edge, so you must add the width to the inner radius (70+7=7770 + 7 = 77).
3
Calculate the outer circumference.
484484 meters
Use the formula C=2πrC = 2\pi r with the new outer radius: C=2×227×77=44×11=484C = 2 \times \frac{22}{7} \times 77 = 44 \times 11 = 484.

Key Concept

Calculating the circumference of concentric circles by determining and adjusting their respective radii.

Alternative Method

You can also solve this using the diameter. The inner diameter is 140140 meters (440=πd440 = \pi d). Since the track is 77 meters wide on all sides, the total width added to the overall diameter is 1414 meters (7×27 \times 2). The outer diameter is 140+14=154140 + 14 = 154 meters. The outer circumference is 227×154=484\frac{22}{7} \times 154 = 484 meters.
Estimated Time:1m 30s
Question 18Question →

Match each circle-related calculation on the left to its correct numerical value on the right.

Click a left item, then click its matching right item

Items

Area of a circle with a radius of 55
Circumference of a circle with a diameter of 1010
Radius of a circle with an area of 100π100\pi
Diameter of a circle with a circumference of 12π12\pi

Matches

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Answer

Area with radius 5 matches to 25π; Circumference with diameter 10 matches to 10π; Radius of area 100π matches to 10; Diameter of circumference 12π matches to 12.
Each match is verified by accurately applying the formulas A=πr2A = \pi r^2 and C=πdC = \pi d to either solve for the resulting property or to work backwards to find the missing dimension.

Step-by-Step Solution

1
Calculate the area for the first description.
Area = 25π25\pi
The formula for the area of a circle is A=πr2A = \pi r^2. Substituting r=5r = 5 yields 25π25\pi.
2
Calculate the circumference for the second description.
Circumference = 10π10\pi
The formula for circumference is C=πdC = \pi d. Substituting d=10d = 10 yields 10π10\pi.
3
Solve for the radius in the third description.
Radius = 1010
Given area A=100πA = 100\pi, use 100π=πr2100\pi = \pi r^2 to find that r2=100r^2 = 100, so r=10r = 10.
4
Solve for the diameter in the fourth description.
Diameter = 1212
Given circumference C=12πC = 12\pi, use 12π=πd12\pi = \pi d to determine that d=12d = 12.

Key Concept

Calculating area, circumference, radius, and diameter of a circle using fundamental geometry formulas.
Question 19Question →

A drone is tethered to a fixed landing pad and can fly within a maximum circular boundary with an area of 400π400\pi square meters. If a safety protocol requires the tether to be shortened by 88 meters, what is the maximum area of the new circular flight boundary?

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Answer: 144π square meters144\pi \text{ square meters}

Answer

The new flight boundary has an area of 144π square meters144\pi \text{ square meters}.
The initial area of 400π400\pi square meters indicates the original tether length (radius) was 2020 meters, because πr2=400π\pi r^2 = 400\pi. Shortening the tether by 88 meters results in a new radius of 1212 meters. The new area is calculated by squaring the new radius and multiplying by π\pi, yielding π(12)2=144π\pi(12)^2 = 144\pi square meters.

Step-by-Step Solution

1
Find the original radius using the area formula A=πr2A = \pi r^2.
400π=πr2400\pi = \pi r^2, so r2=400r^2 = 400, which means r=20r = 20 meters.
The original tether length represents the radius of the circular boundary.
2
Calculate the new radius by subtracting 88 meters.
20−8=1220 - 8 = 12 meters.
The safety protocol shortens the tether, directly reducing the flight radius.
3
Calculate the area of the new circular boundary.
A=π(12)2=144π square metersA = \pi(12)^2 = 144\pi \text{ square meters}.
The problem asks for the maximum area of the new boundary.

Key Concept

Area of a circle and its relationship to the radius.
Question 20Question →

A flat metal washer is manufactured by stamping a circular hole with a diameter of 4 mm4 \text{ mm} out of the exact center of a solid circular disk with a diameter of 12 mm12 \text{ mm}. What is the area of the remaining flat face of the washer in square millimeters? (Enter only the numerical coefficient of π\pi in your answer. For example, if the area is 50π50\pi, enter 50).

Show answer & explanation

Answer: 32

Answer

32
The area of the remaining metal washer face is found by calculating the area of the large outer circle (36π36\pi) and subtracting the area of the smaller inner circle hole (4π4\pi), which leaves a remaining area of 32π32\pi. Since the question asks for the coefficient of π\pi, the correct numerical entry is 32.

Step-by-Step Solution

1
Calculate the radius of the outer disk.
R=12÷2=6 mmR = 12 \div 2 = 6 \text{ mm}
The area formula for a circle requires the radius, which is half of the diameter.
2
Calculate the area of the outer disk.
Areaouter=π×62=36π mm2\text{Area}_{\text{outer}} = \pi \times 6^2 = 36\pi \text{ mm}^2
This represents the total surface area before the center hole is removed.
3
Calculate the radius of the inner hole.
r=4÷2=2 mmr = 4 \div 2 = 2 \text{ mm}
The radius of the removed section is needed to determine how much area to subtract.
4
Calculate the area of the inner hole.
Areainner=π×22=4π mm2\text{Area}_{\text{inner}} = \pi \times 2^2 = 4\pi \text{ mm}^2
This is the area of the metal that is stamped out.
5
Subtract the inner area from the outer area.
36π−4π=32π mm236\pi - 4\pi = 32\pi \text{ mm}^2
The remaining face area of the washer is the difference between the full disk and the removed hole.
6
Identify the coefficient of π\pi.
32
The stem specifically asks for the numerical coefficient only.

Key Concept

Calculating the area of an annulus (the region between two concentric circles) by subtracting the area of the inner circle from the area of the outer circle.
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