Тарифы

Калькулятор длины дуги и площади сектора

Этот калькулятор находит длину дуги, площадь сектора и длину хорды по радиусу и центральному углу. Углы можно задавать в градусах или радианах; в градусах ответы даются точно через π и в десятичном виде.

Результат

Введите значения и нажмите «Вычислить», чтобы увидеть результат.

Формула

s=θ360∘×2πr=rθ (θ in radians)s = \frac{\theta}{360^\circ} \times 2\pi r = r\theta \ (\theta \text{ in radians})

The arc is the same fraction of the circumference as the angle is of a full turn.

A=θ360∘×πr2=12r2θA = \frac{\theta}{360^\circ} \times \pi r^2 = \tfrac{1}{2} r^2 \theta

The sector is the same fraction of the whole circle’s area.

c=2rsin⁡θ2c = 2r\sin\frac{\theta}{2}

The chord is the straight line joining the ends of the arc.

Как пользоваться этим калькулятором

  1. Enter the radius and the central angle, and choose degrees or radians.
  2. Press Calculate to see the arc length, sector area and chord length.

Разобранные примеры

Radius 6, angle 60°

60° is 1/6 of a full turn. Arc = 1/6 × 2π × 6 = 2π ≈ 6.283185.

Sector area = 1/6 × π × 36 = 6π ≈ 18.849556.

Chord = 2 × 6 × sin 30° = 6.

Radius 4, angle 1.5 radians

Arc = rθ = 4 × 1.5 = 6.

Area = ½ × 16 × 1.5 = 12.

Chord = 2 × 4 × sin 0.75 ≈ 5.45311.

A semicircle: radius 10, angle 180°

Arc = half the circumference = 10π.

Area = half the circle = 50π. The chord is the diameter, 20.

Частые ошибки

  • Using s = rθ with θ in degrees; that formula needs radians.
  • Mixing up the diameter and the radius.
  • Confusing the arc (curved) with the chord (straight).

Частые вопросы

How do you find the arc length?

Arc length = (θ/360°) × 2πr for θ in degrees, or s = rθ for θ in radians. For r = 6 and 60°: (60/360) × 12π = 2π.

How do you find the area of a sector?

Area = (θ/360°) × πr², or ½r²θ in radians. A 90° sector of radius 4 has area ¼ × 16π = 4π.

What is the difference between an arc and a chord?

The arc is the curved part of the circle between two points; the chord is the straight line joining them.