Circles

Circles have three formulas and one trap, and the trap catches people who know all three formulas.

The parts

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• Radius: center to the edge.
• Diameter: all the way across, through the center. Twice the radius.
• Chord: any segment joining two points on the circle. A diameter is the longest possible chord.
• Arc: a portion of the circle's edge.
• Sector: a wedge, bounded by two radii and an arc.
• Tangent: a line touching the circle at exactly one point, always perpendicular to the radius at that point.

The three formulas

The area formula squares the radius, never the diameter. Given a circle 14 inches across, the radius is 7 and the area is pi times 49, which is about 154. Using the 14 gives pi times 196, which is about 616 - exactly four times too large, and it is always a choice.

Four times, because squaring doubles the error: if you use twice the right number and then square it, you are out by a factor of four.

Working with pi

Use 3.14 when the choices are decimals. Use 22/7 if the numbers are multiples of 7 and the choices are fractions - a radius of 7 makes 22/7 cancel neatly.

When every answer choice contains the symbol for pi, do not multiply it out. A circle of radius 6 has area 36 pi, and that is the final answer. Converting to 113.1 and then hunting for it among choices written in pi wastes time and invites a rounding mismatch.

Worked examples

A circular pool is 20 feet across. How much fencing goes around it?

Fencing is circumference. The diameter is 20, so circumference is pi times 20, which is about 62.8 feet.

Note that this is the one formula where being handed the diameter is convenient.

The same pool: how much cover is needed?

Cover is area, which needs the radius. Radius is 10, so area is pi times 100, which is about 314 square feet.

Same circle, two questions, and the number you use differs. That is the lesson.

A circular garden has an area of 78.5 square feet. What is its radius?

Work backward. 78.5 divided by 3.14 is 25, and 25 is the radius squared, so the radius is 5 feet.

Divide by pi first, then take the root. Taking the root of 78.5 gives 8.86, which is meaningless here and is a choice.

Arcs and sectors

Both are fractions of the whole circle, and the fraction is the central angle over 360.

A circle has a radius of 12. What is the length of an arc with a central angle of 60 degrees?

• Whole circumference: 2 times pi times 12, which is 24 pi.
• Fraction: 60 / 360, which is 1/6.
• Arc: 24 pi / 6, which is 4 pi, or about 12.6.

The same fraction gives the sector's area:

• Whole area: pi times 144, which is 144 pi.
• Sector: 144 pi / 6, which is 24 pi.

One fraction, applied to whichever whole the question asks about. A quarter circle is 90/360, a semicircle is 180/360, and those two cover most of what the test asks.

Inscribed angles

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An inscribed angle has its vertex on the circle itself. It is half the central angle, or half the arc, that it cuts off.

An inscribed angle subtends an arc of 100 degrees. It measures 50 degrees.

An inscribed angle measures 40 degrees. The central angle on the same arc is 80 degrees.

The special case: an angle inscribed in a semicircle is always 90 degrees, because the semicircle's arc is 180.

A circle inside a square

A circle inscribed in a square touches all four sides, so its diameter equals the square's side. A square of side 10 holds a circle of radius 5 and area 25 pi, about 78.5. The region inside the square but outside the circle is the square's area minus the circle's: for a side of 12, that is 144 minus 36 pi. A sphere inscribed in a cube works the same way: its diameter is the cube's edge.

What you can skip

Across the 43 questions on this topic:

• The equation of a circle in coordinates never appears.
• Radians never appear. Angles are in degrees throughout.
• Tangent and secant theorems. Tangent lines come up twice, and no question uses the power-of-a-point relationships.

Where people lose points

Using the diameter in the area formula. The single biggest error, and it gives four times too much.

Using the radius where the circumference formula expects the diameter, which halves the answer. The two-pi-r form avoids this: if you have the radius, use 2 pi r and there is nothing to convert.

Multiplying pi out when the choices keep it.

Taking the root before dividing by pi when working backward from an area.

Forgetting that a semicircle's perimeter includes the straight edge, not just the curve.

Work one in under a minute

A circular tank has a diameter of 8 feet and stands 5 feet tall. How many cubic feet does it hold? Leave your answer in terms of pi.

Radius is 4, so the base area is 16 pi.

Volume is base area times height: 16 pi times 5 is 80 pi cubic feet.

Leaving pi in place kept this to two multiplications. Converting would have meant 50.27 times 5, and a rounding decision that was never needed.

Where this leads

The base of a cylinder is a circle, so every cylinder volume question starts here, and the pi-in-the-answer habit saves time across the whole geometry block.