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Zorluk: ZorPerimeter and Area of Plane Shapes

A sector of a circle of radius 6 cm6\text{ cm} has a total perimeter whose numerical value is equal to the numerical value of its area. What is the area of the sector, in cm2\text{cm}^2?

Cevap: 18 cm^2

Cevap

The area of the sector is 18 cm218\text{ cm}^2.
For a sector of radius r=6 cmr = 6\text{ cm}, its total perimeter is P=2(6)+s=12+sP = 2(6) + s = 12 + s, where ss is the arc length. Its area is A=12(6)s=3sA = \frac{1}{2}(6)s = 3s. Setting P=AP = A gives 12+s=3s  ⟹  2s=12  ⟹  s=6 cm12 + s = 3s \implies 2s = 12 \implies s = 6\text{ cm}. Substituting s=6s = 6 into the area expression gives A=3(6)=18 cm2A = 3(6) = 18\text{ cm}^2.

Adım Adım Çözüm

1
Express the perimeter and area of the sector in terms of the arc length ss
P=2r+s=12+sP = 2r + s = 12 + s and A=12rs=3sA = \frac{1}{2}rs = 3s
The total perimeter of a sector includes two straight radii plus the arc length, while its area is given by 12rs\frac{1}{2}rs.
2
Equate the numerical values of perimeter and area
12+s=3s12 + s = 3s
The question specifies that the numerical value of the total perimeter equals the numerical value of its area.
3
Solve for the arc length ss
2s=12  ⟹  s=6 cm2s = 12 \implies s = 6\text{ cm}
Subtracting ss from both sides yields 2s=122s = 12, so s=6 cms = 6\text{ cm}.
4
Calculate the sector area
A=3(6)=18 cm2A = 3(6) = 18\text{ cm}^2
Substituting s=6s = 6 into A=3sA = 3s yields an area of 18 cm218\text{ cm}^2.

Anahtar Kavram

Perimeter and Area of a Circular Sector
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Konular

Perimeter and Area of Plane Shapes

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