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Step 12: Geometry

Circles, arcs, and chords

Step 12 · Geometry topics

Use central and inscribed angles, find arc lengths and sector areas, and use the facts about chords and tangents to find lengths.

Central and inscribed angles

A central angle has its vertex at the center of the circle; the arc it cuts off has the same measure. An inscribed angle has its vertex on the circle; it is half of the arc it cuts off.

Center O, with A and B on the circle. The central angle AOB is 90°, and the inscribed angle ACB, cutting off the same arc, is 45°.

Example: An inscribed angle

An inscribed angle cuts off an arc of 110°. What is the angle?

  1. It is half its arc: 110 ÷ 2 = 55.

Answer55°

Example: A semicircle

An inscribed angle cuts off a semicircle. What is it?

  1. A semicircle is a 180° arc.
  2. Half of 180 is 90.

Answer90°: an angle in a semicircle is a right angle

Arc length and sectors

An arc is a fraction of the whole circumference: an arc of θ° is θ ÷ 360 of it. A sector, a slice of the circle, is the same fraction of the area.

arc length=θ360×2πrsector area=θ360×πr2\text{arc length} = \frac{\theta}{360} \times 2\pi r \qquad \text{sector area} = \frac{\theta}{360} \times \pi r^{2}

Example: Arc length

Find the length of a 60° arc in a circle of radius 9.

  1. 60360=16\frac{60}{360} = \frac{1}{6}.
  2. The circumference is 18π, and a sixth of it is 3π.

Answer3π, about 9.42

Example: Sector area

Find the area of a 90° sector of a circle of radius 10.

  1. 90° is a quarter of the circle.
  2. The area is 100π, and a quarter of it is 25π.

Answer25π, about 78.5

Chords and tangents

A chord joins two points of a circle. A tangent touches the circle at exactly one point, and it is perpendicular to the radius drawn to that point.

The tangent PT meets the radius OT at a right angle, so OTP is a right triangle with hypotenuse OP.

Example: A tangent

A tangent from P touches a circle of radius 5 at T, and P is 13 from the center O. How long is PT?

  1. OT is perpendicular to PT, so OTP is a right triangle.
  2. PT² = 13² − 5² = 169 − 25 = 144.
  3. PT = 12.

Answer12

Example: Two tangents

Two tangents from the same point P touch a circle at A and B. PA = 8. What is PB?

  1. Two tangents from the same point are equal in length.

Answer8

Example: A chord and the center

A chord of length 16 is 6 from the center. What is the radius?

  1. The perpendicular from the center cuts the chord in half: 8 each side.
  2. r² = 8² + 6² = 64 + 36 = 100.

Answer10

Try it yourself

Work each one on paper first, then open the answer to check.

  1. A central angle is 84°. What is its arc?

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    84°

  2. An inscribed angle cuts off a 140° arc. What is the angle?

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    70°

  3. Find the length of a 45° arc of a circle with radius 8, in terms of π.

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    2π

  4. Find the area of a 120° sector of a circle with radius 6, in terms of π.

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    12π

  5. A tangent from P touches a circle of radius 8 at T, and OP = 17. How long is PT?

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    15

  6. A chord of a circle with radius 13 is 5 from the center. How long is the chord?

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    24

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