**Circles**

### Area and circumference

**Example 1**

The circumference of a circle is 12. Find the area of the circle.

**Example 2**

The diameter of a circle is 20. Find the area of the circle.

**Example 3**

One metre is added to the circumference of the circle. How much does the radius of the circle increase?

Let the radius at the beginning be* r*. Let* r + x* be a new radius and we add *1* to the circumference.

The radius increases by 0,16 meters, which is 16 cm.

### Sector and segment

**Example 4**

Find the area of the sector and the length of the arc.

The radius of the circle is *2,5* and the central angle is *80°*

**Example 5**

Find the area and perimeter of the segment.

The area of the segment is obtained by calculating the area of the sector and then subtracting the area of the isosceles triangle. Calculating the area of a triangle using two sides and the angle between them is covered in the Area section.

The perimeter of the segment consists of the base of an isosceles triangle and the arc of the sector. We find the base by using a right triangle. We mark the half of the base with *x*

Now the perimeter of the segment.

### The tangent of a circle

The tangent of a circle only touches the circle at one point and is always perpendicular to the radius.

**Example 6**

How far is point *A *from the circle?

From point* A*, draw a line segment to the centre of the circle. This line segment is the hypotenuse of the formed right triangle. Subtracting the radius from it gives the distance of point *A* from the circle.

**Example 7**

Liisa-Petter's balloon escaped. The diameter of the balloon was 10 meters. How far was the balloon when it was seen at an angle of 18°?

The radius of the balloon is 5 and the magnitude of the angle opposite the radius of the resulting right triangle is 9 °. The distance from the balloon is the hypotenuse of a right triangle minus the radius of the balloon.

### The central angle and inscribed angles

An inscribed angle is half of the central angle. All inscribed angles sharing the same arc of a circle are equal.

In this circle, angle 𝛂 is also* 35 °*. The central angle corresponding to the inscribed angles is 𝛽 *= 70 °*

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