Question

В круг вписаний правильний десятикут- ник, сторона якого дорівнює 4 см. Знайти площу сегмента, який відсікає від круга сторона десятикутника.

Answer 1

центральный угол равен 360°/10=36°; тогда радиус окружности равен R=4/(2*sin(180°/10))=2/sin(18°)≈6.5

cторона отсекает сегмент, площадь которого πR²n/360-sΔ, где sΔ- это площадь треугольника со сторонами, равными радиусам 6.5 и углом между радиусами в 36°.

π*(6.5)²*10/360-6.5²*(sin36°)/2≈132.665-42.25*0.58778525/2≈

132.665-12.414=120.236≈120/см²/

Answer 2

Finding the Area of the Segment Cut Off by the Decagon

To find the area of the segment cut off by the decagon from the circle, we can use the following steps:

1. Find the Area of the Sector 2. Subtract the Area of the Isosceles Triangle

Let's calculate the area of the segment step by step.

Step 1: Find the Area of the Sector

The area of the sector can be calculated using the formula: \[ \text{Sector Area} = \frac{1}{2} r^2 \theta \] where \( r \) is the radius of the circle and \( \theta \) is the central angle of the sector.

Given that the radius of the circle is 4 cm and the decagon is inscribed in the circle, we can calculate the central angle of the sector using the formula: \[ \theta = \frac{360^\circ}{n} \] where \( n \) is the number of sides of the regular polygon (in this case, the decagon).

The central angle \( \theta \) for a regular decagon is \( 360^\circ / 10 = 36^\circ \).

Using this information, we can calculate the area of the sector.

Step 2: Subtract the Area of the Isosceles Triangle

The area of the isosceles triangle formed by two radii and a side of the decagon can be calculated using the formula: \[ \text{Triangle Area} = \frac{1}{2} r^2 \sin \theta \] where \( r \) is the radius of the circle and \( \theta \) is the central angle of the sector.

We can then subtract the area of the isosceles triangle from the area of the sector to find the area of the segment cut off by the decagon from the circle.

Let's proceed with the calculations.