Question

Please provide all the important formulas and tips for the Chapter, "Areas related to circles".

⇒ Class 10

⇒ Mathematics

⇒ Chapter 12

Thanks for helping me guys!!

Answer 1

Answer:

What are the formulas for circles?

Formulas Related to Circles

Diameter of a Circle D = 2 × r

Circumference of a Circle C = 2 × π × r

Area of a Circle A = π × r2

Answer 2

• For a circle C (O,r) :

$\leadsto \sf {\pink{Radius:r=OA,OB,OC}}$

$\leadsto \sf {\pink{Diameter:2r=AB}}$

$\leadsto \sf {\pink{Circumference:2\pi r}}$

$\leadsto \sf {\pink{Area~ of ~circle=\pi r^2}}$

• For a sector of a circle of radius r with central angle $\sf \theta$ :

$\leadsto \sf {\green{Length~ of ~arc=\dfrac{\theta}{360\degree}(2\pi r)}}$

$\leadsto \sf {\green{Area=\dfrac{\theta}{360\degree}(\pi r^2)}}$

• Area of segment = Area of sector - Area of triangle
• Area of major sector (segment) = Area of circle - Area of minor sector (segment)

• For a semicircle of radius r :

$\leadsto \sf {\blue{Perimeter=\dfrac{1}{2}(2\pi r)(2r)=\pi r +2r}}$

$\leadsto \sf {\blue {Area=\dfrac{\pi r^2}{2}}}$

• For a quadrant of radius r :

$\leadsto \sf {\orange{Perimeter=\dfrac{1}{4}(2\pi r)(2r)=\dfrac{\pi r}{2}+2r}}$

$\leadsto \sf {\orange {Area =\dfrac{\pi r^2}{4}}}$

☞ QUADRANT : A quadrant is a one-fourth of a circle.

• Area of a ring = Area of outer ring - Area of inner ring

$\leadsto \sf {\purple{\pi R^2-\pi r^2}}$

$\leadsto \sf {\purple{\pi (R+r)(R-r)~units^2}}$

• What is an arc of a circle?

A part of the circumference of the circle is called it's arc. It's length is denoted by $l$.

☞ MINOR ARC : An arc whose length is less than half the circumference of a circle is called a minor arc.

☞ MAJOR ARC : An arc whose length is more than half the circumference of a circle is called a major arc.

☞ Circumference = Length of minor arc + Length of major arc.

• What is a sector of a circle?

The portion of a circular region enclosed by two radii and the corresponding arc is called a sector of the circle.

☞ MINOR SECTOR : A sector is called a minor sector if it's bonding arc is a minor arc of the circle.

☞ MAJOR SECTOR : A sector is called a major sector if it's bonding arc is the major arc of the circle.

☞ In the figure, the shaded region OAPB is the minor sector and the unshaded region is the major sector OAQB.

☞ ANGLE OF A SECTOR : An angle which the bounding arc of a sector subtends at the centre of the circle is called the angle of the sector.

☞ Angle of major sector = 360° - Angle of minor sector : $\sf {\green{360\degree-\theta}}$.

☞ Length of an arc AB of a sector of a circle with radius r angle with degree $\sf {\green{\theta : \dfrac{\theta}{360\degree}(2\pi r)}}$

☞ Perimeter of sector OAPB = OA + OB + AB

$: \red{\sf {r + r + l}}$

$:\sf {\red{2r + l}}$

$:\sf {\red{2r+\dfrac{\theta}{360\degree}(2\pi r)}}$

☞ Area of sector OAPB of radius r angle $\sf{\red{ \theta=\dfrac{\theta}{360\degree}(\pi r^2)~units^2}}$

• What is a segment of a circle?

☞ The portion of a circular region enclosed between a chord and the corresponding arc is called a segment of the circle.

☞ MINOR SEGMENT : A segment is called a minor segment if it is less than a semicircle.

☞ MAJOR SEGMENT : A segment is called a major segment if it's greater than semicircle.

☞ Area of the circle = Area of major sector + Area of minor sector

☞ Area of segment of a circle = Area of the sector - Area of the triangle

$:\sf{\blue{ \dfrac{\pi r^2\theta}{360\degree}-\dfrac{1}{2}r^2(sin\theta)}}$

$:\sf {\blue{\dfrac{r^2}{2}[\dfrac{\pi \theta}{180\degree}-sin\theta]~units^2}}$

___________________

Hey! This is the most important stuff of all time from the chapter - Areas related to circles.

Seems like you are having ya' exams...All the best! :D