The shadow area of a figure is a figure which is a combination of three equilateral triangles with 6 cm one side and three sectors 6 cm in radius.
When a circle 1 cm in radius takes one round touching the circumference of this figure, find the area of the portion along which this circle passes.
Pi is assumed to be 3.14.
When a circle 1 cm in radius takes one round touching the circumference of this figure, find the area of the portion along which this circle passes.
Pi is assumed to be 3.14.
Answer
Motion of the B portion becomes a sector which makes the diameter of a circle a radius, as shown in Fig. 2.
The area along which the circle passed for three kinds of every motions is determined.
① In the case of A
It is a form of the rectangle shown in Fig. 1, and there are three, the area is 6 × 2 × 3 = 36cm2.
② In the case of C
It becomes a form of C shown in Fig.1.
The sum total of the central angle of three sectors, 360° - (60° × 3) = 180°.
The area is (8 × 8 - 6 × 6) × 3.14 × 180/360 = 14 x 3.14cm2.
③ In the case of B
The central angle of one sector is 180° - 60° - 90° = 30° as shown in Fig. 2.
Since there are six sectors, total area is 2 × 2 × 3.14 × 30/360 × 6 = 2 x 3.14cm2.
86.24 cm2
Solution
When a circle takes one round the shadow area, as shown in Fig. 1, there are three kinds of motions, A, B, and C.
Motion of the B portion becomes a sector which makes the diameter of a circle a radius, as shown in Fig. 2.
This sector is made in six all.
The area along which the circle passed for three kinds of every motions is determined.
① In the case of A
It is a form of the rectangle shown in Fig. 1, and there are three, the area is 6 × 2 × 3 = 36cm2.
② In the case of C
It becomes a form of C shown in Fig.1.
The sum total of the central angle of three sectors, 360° - (60° × 3) = 180°.
The area is (8 × 8 - 6 × 6) × 3.14 × 180/360 = 14 x 3.14cm2.
③ In the case of B
The central angle of one sector is 180° - 60° - 90° = 30° as shown in Fig. 2.
Since there are six sectors, total area is 2 × 2 × 3.14 × 30/360 × 6 = 2 x 3.14cm2.
Total area overall is 36 + 14 × 3.14 + 2 × 3.14 = 86.24 cm2.
