ASVAB Study Guide
Geometry · Lesson 4 of 8

Circles

Key rules and formulas, worked examples and 15 practice problems with step-by-step solutions.

1.Area of a Circle

Calculating the total flat space enclosed within the boundary of a circle.

$$A = \pi \times r^2$$

r is the radius. pi is a mathematical constant (approximately equal to 3.14 or $$\frac{22}{7}$$)

Example

Find the area of a circle with a radius of 5 cm.

Show solution

Step 1: Square the radius: $$5 \times 5 = 25$$

Step 2: Multiply the result by $$\pi$$: $$25 \times 3.14$$

Step 3: Get the final area: 78.5 cm²

Practice problems

  1. Problem 1

    A circular rug has a radius of 4 meters. What is the total area of the rug?

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    Answer: 50.24 square meters

    Step 1: Identify the radius: $$r = 4\text{ m}$$

    Step 2: Apply the area formula: $$A = 3.14 \times 4^2 = 3.14 \times 16$$

    Step 3: Perform the multiplication: $$3.14 \times 16 = 50.24$$ square meters

  2. Problem 2

    Find the area of a circular metal plate with a diameter of 20 inches.

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    Answer: 314 square inches

    Step 1: Convert diameter to radius: $$r = \frac{20}{2} = 10$$ inches

    Step 2: Set up the area formula: $$A = 3.14 \times 10^2 = 3.14 \times 100$$

    Step 3: Calculate the final area: $$3.14 \times 100 = 314$$ square inches

  3. Problem 3

    The area of a circular target zone is 154 square feet. Using $$\pi = \frac{22}{7}$$, find the square of its radius $$(r^2)$$

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    Answer: 49

    Step 1: Write the area formula: $$A = \pi \times r^2$$

    Step 2: Substitute the values: $$154 = \frac{22}{7} \times r^2$$

    Step 3: Solve for the square of the radius: $$r^2 = 154 \times \frac{7}{22} = 49$$

Remember

Always square the radius first (r times r), then multiply by pi. If given the diameter, divide it by 2 to find the radius.

2.Circumference

Calculating the linear distance around the outside boundary of a circle.

$$C = 2 \times \pi \times r$$ (or $$C = \pi \times d$$)

r is the radius, d is the diameter $$(d = 2 \times r)$$, and pi is approximately 3.14.

Example

Find the circumference of a circular pool with a diameter of 10 meters.

Show solution

Step 1: Multiply the diameter directly by $$\pi$$: $$10 \times 3.14$$

Step 2: Calculate and state the result: 31.4 meters

Practice problems

  1. Problem 4

    A bicycle wheel has a radius of 14 inches. What is its circumference?

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    Answer: 88 inches

    Step 1: Identify the given radius parameter: $$r = 14$$ inches

    Step 2: Use the radius-based formula: $$C = 2 \times (\frac{22}{7}) \times 14$$

    Step 3: Simplify and calculate: $$2 \times 22 \times 2 = 88$$ inches

  2. Problem 5

    A circular racing track has a diameter of 100 meters. Find the distance around the track.

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    Answer: 314 meters

    Step 1: Identify the parameter: $$d = 100\text{ m}$$

    Step 2: Apply the formula: $$C = \pi \times d = 3.14 \times 100$$

    Step 3: Compute the distance value: 314 meters

  3. Problem 6

    If the circumference of a circular table is 18.84 feet, what is its radius?

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    Answer: 3 feet

    Step 1: Set up the equation: $$18.84 = 2 \times 3.14 \times r$$-> $$18.84 = 6.28 \times r$$

    Step 2: Isolate the radius variable: $$r = \frac{18.84}{6.28}$$

    Step 3: Divide to solve: $$r = 3$$ feet

Remember

Circumference is a linear measurement of distance, so its unit is never squared.

3.Diameter & Radius

Converting back and forth between the two foundational dimensions of a circle.

$$d = 2 \times r$$ (or $$r = \frac{d}{2}$$)

r is the radius (center to edge). d is the diameter (straight line passing through the center connecting two edges).

Example

The radius of a gear is 7.5 cm. Find its diameter.

Show solution

Step 1: Multiply the radius length by 2: $$7.5 \times 2$$

Step 2: Get the diameter result: 15 cm

Practice problems

  1. Problem 7

    A large circular clock has a diameter of 36 inches. What is the radius of this clock face?

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    Answer: 18 inches

    Step 1: Identify the given total diameter dimension: $$d = 36$$ in

    Step 2: Apply the radius translation formula: $$r = \frac{d}{2}$$

    Step 3: Divide the value: $$\frac{36}{2} = 18$$ inches

  2. Problem 8

    A circular pipe opening has a radius of 4.25 inches. Find the complete diameter of the pipe.

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    Answer: 8.5 inches

    Step 1: Identify the given radius value: $$r = 4.25$$ in

    Step 2: Multiply by 2 to convert to diameter: $$d = 2 \times 4.25$$

    Step 3: Calculate the product: 8.5 inches

  3. Problem 9

    If the distance from the center of a circular pond to its outermost edge is 11 yards, what is the length of its diameter?

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    Answer: 22 yards

    Step 1: Recognize that the distance from the center to the edge is the radius: $$r = 11$$ yards

    Step 2: Apply the conversion rule: $$d = 2 \times r$$

    Step 3: Double the value: $$2 \times 11 = 22$$ yards

Remember

In the same circle, the diameter is always exactly twice as long as the radius.

4.Arc Length

Calculating the length of a portion of the circle's boundary (arc) defined by a central angle.

$$\text{Arc Length} = (\text{Angle} / 360) \times 2 \times \pi \times r$$

Angle is the degree measurement of the central angle. $$(2 \times \pi \times r)$$ is the total circumference of the entire circle.

Example

Find the length of an arc with a central angle of 90 degrees and a radius of 7 cm.

Show solution

Step 1: Calculate the fractional slice ratio: $$\frac{90}{360} = \frac{1}{4}$$ (which is one-fourth of a circle)

Step 2: Calculate the total circumference: $$2 \times (\frac{22}{7}) \times 7 = 44$$ cm

Step 3: Multiply the fraction by the circumference to find the arc length: $$\frac{1}{4} \times 44 = 11$$ cm

Practice problems

  1. Problem 10

    A circle has a radius of 6 inches. Find the arc length intercepted by a central angle of 60 degrees.

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    Answer: 6.28 inches

    Step 1: Create the fractional ratio: $$\frac{60}{360} = \frac{1}{6}$$

    Step 2: Calculate the full circumference: $$C = 2 \times 3.14 \times 6 = 37.68$$

    Step 3: Multiply the fraction by total circumference: $$(\frac{1}{6}) \times 37.68 = 6.28$$ inches

  2. Problem 11

    An engineer tracks a rotation arc of 180 degrees on a wheel with a radius of 10 cm. What is the length of this arc segment?

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    Answer: 31.4 cm

    Step 1: Simplify the rotation fraction: $$\frac{180}{360} = \frac{1}{2}$$ (a semicircle)

    Step 2: Determine total circumference value: $$C = 2 \times 3.14 \times 10 = 62.8$$

    Step 3: Take half of the total boundary: $$0.5 \times 62.8 = 31.4$$ cm

  3. Problem 12

    Find the central angle in degrees if the arc length is exactly $$\frac{1}{3}$$ of the full circumference of the circle.

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    Answer: 120 degrees

    Step 1: Set up the fractional ratio relationship: $$\text{Angle} / 360 = \frac{1}{3}$$

    Step 2: Isolate the angle variable by cross-multiplying: Angle $$= \frac{360}{3}$$

    Step 3: Evaluate the expression: Angle = 120 degrees

Remember

Think of it as taking a specific percentage/fraction of the angle (Angle / 360) and multiplying it by the total circumference.

5.Sector Area

Calculating the area of a pie-shaped slice (like a pizza slice) bounded by a central angle.

$$\text{Sector Area} = (\text{Angle} / 360) \times \pi \times r^2$$

Angle is the degree measurement of the central angle. $$(\pi \times r^2)$$ is the total area of the entire circle.

Example

Find the area of a sector with a central angle of 90 degrees and a radius of 4 cm. (Use $$\pi = 3.14$$)

Show solution

Step 1: Simplify the angle fraction: $$\frac{90}{360} = \frac{1}{4}$$ of the circle

Step 2: Calculate the total area of the circle: $$3.14 \times 4^2 = 50.24$$ cm²

Step 3: Multiply the fraction by the total area to find the sector area: $$\frac{1}{4} \times 50.24 = 12.56$$ cm²

Practice problems

  1. Problem 13

    A pizza with a radius of 6 inches is cut into a slice with a central angle of 60 degrees. What is the area of this slice?

    Show solution

    Answer: 18.84 square inches

    Step 1: Reduce the slice angle fraction: $$\frac{60}{360} = \frac{1}{6}$$

    Step 2: Compute the full area value: $$A = 3.14 \times 6^2 = 3.14 \times 36 = 113.04$$

    Step 3: Take one-sixth of the full area value: $$\frac{113.04}{6} = 18.84$$ square inches

  2. Problem 14

    Calculate the area of a semicircular sector (central angle of 180 degrees) that has a radius of 10 feet.

    Show solution

    Answer: 157 square feet

    Step 1: Set up the area fraction ratio: $$\frac{180}{360} = \frac{1}{2}$$

    Step 2: Calculate the absolute full area parameter: $$A = 3.14 \times 10^2 = 314$$

    Step 3: Multiply the fraction by the total area: $$0.5 \times 314 = 157$$ square feet

  3. Problem 15

    A circular patch has a total area of 72 square centimeters. A sector slice of this circle has a central angle of 45 degrees. Find its sector area.

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    Answer: 9 square centimeters

    Step 1: Convert the sector angle into a fraction: $$\frac{45}{360} = \frac{1}{8}$$

    Step 2: Identify the given full circle area value directly: Total Area = 72

    Step 3: Calculate the area of the slice segment: $$(\frac{1}{8}) \times 72 = 9$$ square centimeters

Remember

Simplify the angle fraction first, then multiply it by the total area of the circle to avoid confusion.