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- 1. CIRCLES Made by :- Amit choube Class :- 10th ‘ B ’
- 2. Introduction In this power point presentation we will discuss about • Circle and its related terms . • Concepts of perimeter and area of a circle . • Finding the areas of two special parts of a circular region known as sector and segment . • Finding the areas of some combinations of plane figures involving circles or their parts .
- 3. Contents Circle and its related terms . Area of a circle . Areas related to circle . Perimeter of a circle . Sector of a circle and its area . Segment of a circle and its area Areas of combinations of plane figures .
- 4. Circle – Definition The collection of all the points in a plane which are at a fixed distance from in the plane is called a circle . or A circle is a locus of a point which moves in a plane in such a way that its distance from a fixed point always remains same.
- 5. 1. Radius – The line segment joining the centre and any point on the circle is called a radius of the circle . O P Here , in fig. OP is radius of the circle with centre ‘O’ . Related terms of circle
- 6. 2. A circle divides the plane on which it lies into three parts . They are • The Interior of the circle . • The circle . Exterior • The exterior of the circle . Interior circle Here , in the given fig . We can see that a circle divides the plane on which it lies into three parts .
- 7. 3. Chord – if you take two points P and Q on a circle , then the line segment PQ is called a chord of the circle . 4. Diameter – the chord which passes through the centre of the circle is called a diameter of the circle . O P R Here in the given fig. OR is the diameter of the circle and PR is the chord of the circle . Note :- A diameter of a circle is the longest chord of the circle .
- 8. 1. Arc – the piece of circle between two points is called an arc of the circle . Q . Major Arc PQR P . . R Minor Arc PR Here in the given fig. PQR is the major arc because it is the longer one whereas PR is the minor arc of the given circle . When P and Q are ends of a diameter , then both arcs are equal and each is called a semicircle
- 9. Segment – the region between a chord and either of its arc is called a segment of the circle . Major segment Minor segment Here , in the given fig. We can clearly see major and minor segment .
- 10. Sector – the region between two radii , joining the centre to the end points of the arc is called a sector . A B Here in the given fig. you find that minor arc corresponds to minor sector and major arc correspondence to major sector .
- 11. Perimeter of a circle • The distanced covered by travelling around a circle is its perimeter , usually called its circumference . We know that circumference of a circle bears a constant ratio with its diameter . → 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 = 𝜋 → 𝑐𝑖𝑟𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋 × 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 → 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋 × 2𝑟 (diameter = 2r) → 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋r
- 12. Area of a circle Area of a circle is 𝜋𝑟2 , where is the radius of the circle . We have verified it in class 7 , by cutting a circle into a number of sectors and rearranging them as shown in fig. 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 = 𝜋𝑟2
- 13. Area and circumference of semicircle Area of circle = 𝜋𝑟2 Area of semi – circle = 1 2 (Area of circle) Area of semicircle = 1 2 𝜋𝑟2 and Perimeter of circle = 2𝜋𝑟 Perimeter of semi circle = 1 2 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 + 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 Perimeter of semi circle = 𝜋𝑟 +2𝑟 = 𝜋 + 2 𝑟
- 14. Area of a sector . Following are some important points to remember 1. A minor sector has an angle 𝜃 , (say) , subtended at the centre of the circle , whereas a major sector has no angle . 2. The sum of arcs of major and minor sectors of a circle is equal to the circumference of the circle . 3. The sum of the areas of major and minor sectors of a circle is equal to the areas of the circle . 4. The boundary of a sector consists of an arc of the circle and the two radii .
- 15. If an arc subtends an angle of 180° at the centre , then its arc length is 𝜋r . If the arc subtends an angle of θ at the centre , then its arc length is → 𝑙 = 𝜃 180 × 𝜋𝑟 → 𝑙 = 𝜃 360 × 2𝜋𝑟 If the arc subtends an angle θ , then the area of the corresponding sector is → 𝜋𝑟2 𝜃 360 = 𝜃 180 × 1 2 𝜋𝑟2 Thus the area A of a sector of angle θ then area of the corresponding sector is → 𝐴 = 𝜃 360 × 𝜋𝑟2 Now, → 𝐴 = 1 2 𝜃 180 × 𝜋𝑟 𝑟 → 𝐴 = 1 2 𝑙𝑟 Area of a sector.
- 16. Area of a sector.
- 17. Some useful results to remember . 1. Angle described by one minute hand in 60 minute =360 ͦ → Angle described by minute hand in one minute = 360 60 = 6 Thus , minute hand rotates through an angle of 6 in one minute . 2. Angle described by hour hand in 12 hours = 360 ͦ → Angle described hour hand in one minute = 360 12 = 30 Thus , hour hand rotates through 30 ͦ in one minute .
- 18. Area of a segment of a circle
- 19. Thank you

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