Perimeter of 3

Geometry Level 2

If a circular sector of radius 1 has perimeter 3, what is the central angle?

3 0 ∘ 30 ^ \circ 3 π \frac{3}{ \pi } radians 1 radian 6 0 ∘ 60 ^ \circ

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Chung Kevin by Chung Kevin , 45, Australia

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7 solutions

Evan Harahap
Apr 24, 2015

If the perimeter of a circular sector with radius 1 is 3, that means radius + radius + arc = 3. So the length of arc is 1. 1 radians is the angle of a circular sector when the arc length is equal to the radius. So the answer of this question is 1 radians.

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I found this surprising. I was thinking that an equilateral triangle with perimeter 3 had a 6 0 ∘ 60 ^ \circ angle, and so wondered what happened with a sector of perimeter 3.

Chung Kevin - 6 years, 1 month ago

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One radian is ~57 degrees, so this is quite a useful way to think about it - just remember that the arc is length 1 but its horizontal component is slightly less!

Tom Benn - 6 years, 1 month ago
Ashraful Islam
Apr 26, 2015

It's obvious that arc length is 1. For perimeter of 2 * pi * radius, angle is 2 * pi radians. So, for perimeter of 1, angle is 1 radians (when radius=1).

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Moderator note:

Good. What would the perimeter be if the answer is 2 radians? Or 3 radians?

That's right!

Chung Kevin - 6 years, 1 month ago

2 * pi radians forms a perimeter equal to 2* pi * radius

so, 1 radians forms a perimeter equal to radius

and so, 2 radians will form a perimeter of 2*radius ... so on!

Ashraful Islam - 6 years ago
Niraj Agrawal
May 3, 2015

Arc length S = Radius*Angle in radian. So angle = S/r = 1/1 = 1 Radian

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If the total perimeter is 3 and the radius is 1, then it means that the length of the arc is 1. Therefore, the angle is 1 radian, as 1 radian equals the angle subtended by an arc of same length as the radius!

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Good solution!

Chung Kevin - 6 years, 1 month ago
Vishnu Bhagyanath
Apr 27, 2015

Radius = 1 unit,

Perimeter = 2(radius) + arc

Arc = 3-2 = 1 unit.

1 radian is the angle formed by an arc of radius 'x' with length 'x'.

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2r+rθ=3 Since r=1, 2+θ=3 θ=1 rad (ans)

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la figura tiene un arco y dos radios, cada radio mide uno, si son dos radios, tenemos 2 unidades de las 3 que nos pide el perímetro, entonces la medida de arco equivale a un radio, 1 unidad, por lo tanto el angulo del arco comprendido entre los dos radios es de 1 radian.

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