Circle’s Circumference And Area Of Sectors And Segments

Circle

We can calculate area and perimeter (circumference) of a circle because it is a closed curve by the following formulas.

Circumference of circle

Circumference is the length of the boundary of the circle. Its units are same as that of length i.e meter, centimeter, millimeter etc.

The circumference of a circle with radius r can be calculated by multiplying diameter with the constant π.

As diameter of a circle is double of its radius. So, the formula used to find circumference is 2 π r.

where, r is radius of circle and π is a constant value.

The value of π = $\frac{22}{7}$

or = 3.14

Circumference of circle

In the figure below the boundary of the circle is marked with "Circumference".

Area of circle

Area of circle is a measure of space of the surface of circle. Its units are same as that of area i.e meter², centimeter², millimeter² etc.

In the figure below the shaded part inside the circle denotes the area of the circle.

Area of circle

The area of a circle with radius r can be calculated using the formula π r². where, π is the constant value and r is radius of the circle.

Semi circle

Circumference

Semi circle is a half part of a circle divided into two equal parts by the diameter. The circumference of semi circle is the length of its curved path. This length becomes equal to the half of circumference of the circle.

Circumference of semi circle

The circumference of a semi circle with radius r can be calculated by doing half of the Circumference of circle $\frac{2\pi r}{2}$ , which becomes π r.

Perimeter

Perimeter of semi circle is the total length of it boundary, which is equal to sum of its circumference and diameter.

Area

The Area of semi circle is half of the area of the circle.

Area of semi circle

The area of a semi circle with radius r can be calculated by doing half of the area of circle as $\frac{\pi r^2}{2}$

Area of minor sector

Area of sector of a circle is the region of the circle that is bounded by an arc and two radii of the circle.

In the figure below the shaded part inside the circle denotes the area of the sector of the circle, formed by two radii which make an angle θ at the centre of circle (called as central angle) and the opposite minor arc.

Area of sector is always assumed as the area of minor sector area if "minor" is not mentioned.

Area of minor sector of circle

In degrees

The area of circle π r² covers the full circle. This whole area of circle is covered by a sector whose two radii coincides each other and make the central angle θ = 360°.

Central angle of 360° covers the area of circle = π r²

Central angle of 1° covers the area of circle = $\frac{\pi r^2}{{360}^{\mathrm{o}}}$

Central angle of θ covers the area of circle = $\frac{\pi r^2}{{360}^{\mathrm{o}}}\times \theta$

∴ area of sector of circle with central angle θ = $\frac{\pi r^2}{{360}^{\mathrm{o}}}\times \theta$

= $\frac{\theta }{360}\times \pi r^2$

In radians

360° in radians can be written as 2 π radians

∴ the area of sector in radians = $\frac{\theta }{\mathrm{2\pi }}\times \pi r^2$

With length of arc

The whole area of circle is covered by an arc whose length is equal to the circumference of the circle, which is 2 π r.

∴ Length of arc 2 π r covers the area of circle = π r²

Length of arc 1 covers the area of circle = $\frac{\pi r^2}{2^{\pi }}$

Length of arc l covers the area of circle = $\frac{\pi r^2}{2^{\pi }}\times l=\frac{\mathrm{rl}}{2}$

∴ area of sector of circle with arc of length l = $\frac{\mathrm{rl}}{2}$

The minor sectors in the circle have their names depending upon the size of the central angle as listed in the following table.

Area of major sector

Area of major sector is the area left other than the area of minor sector of circle.

The figure below shows the area of major sector with the shaded part formed by two radii which make a central angle of θ in minor sector and the opposite major arc.

Area of major sector of circle

It is calculated by subtracting the area of minor sector from the total area of the circle.

Area of minor segment

Area of minor segment is the region enclosed between a minor arc and the chord. The figure below shows the area of minor segment with the shaded part.

Minor segment of circle

Area of minor segment of circle

In the above diagram, the shaded part below the chord AB is the area of the minor segment.

OA and OB are the two radii of length r. θ is the central angle.

So, Area of minor segment = area of minor sector - area of ΔAOB

Area of minor sector = $\frac{\theta }{{360}^o}\times \pi r^2$

To calculate the area of ΔAOB, follow the steps below.

Area of ΔAOB = $\frac{1}{2}\times \mathrm{AB}\times \mathrm{OL}$

In ΔAOL $\frac{\mathrm{AL}}{\mathrm{OA}}=\mathrm{Sin}\frac{\theta }{2}$

$\frac{\mathrm{AL}}{r}=\mathrm{Sin}\frac{\theta }{2}$

$\mathrm{AL}=r\mathrm{Sin}\frac{\theta }{2}$

AB = 2 × AL

∴ $\mathrm{AB}=2\times r\mathrm{Sin}\frac{\theta }{2}=2r\mathrm{Sin}\frac{\theta }{2}$

Also, $\frac{\mathrm{OL}}{\mathrm{OB}}=\mathrm{Cos}\frac{\theta }{2}$

$\frac{\mathrm{OL}}{r}=\mathrm{Cos}\frac{\theta }{2}$

$\mathrm{OL}=r\mathrm{Cos}\frac{\theta }{2}$

∴ Area of ΔAOB = $\frac{1}{2}\times 2\times r\mathrm{Sin}\frac{\theta }{2}\times r\mathrm{Cos}\frac{\theta }{2}$

= $r^2\mathrm{Sin}\frac{\theta }{2}\times \mathrm{Cos}\frac{\theta }{2}$

Again, Area of minor segment = area of minor sector - area of ΔAOB

= $\frac{\theta }{{360}^{\mathrm{o}}}\times \pi r^2-r^2\mathrm{Sin}\frac{\theta }{2}\times \mathrm{Cos}\frac{\theta }{2}$

= r² $(\frac{\theta }{{360}^{\mathrm{o}}}\times \pi -\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2})$

Area of major segment

Area of major segment is the region enclosed between a major arc and the chord. The figure below shows the area of major segment with the shaded part.

Area of major segment of circle

Area of major segment = π r² - Area of minor segment

= π r² - r² $(\frac{\theta }{{360}^{\mathrm{o}}}\times \pi -\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2})$

= π r² - $\frac{\theta }{{360}^{\mathrm{o}}}\times \pi r^2-r^2\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2}$

= r² $(\pi -\frac{\theta }{{360}^{\mathrm{o}}}\times \pi -\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2})$

= r² $\left[\pi \right(1-\frac{\theta }{{360}^{\mathrm{o}}})-\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2}]$

= r² $\left[\pi \right(\frac{{360}^{\mathrm{o}}-\theta }{{360}^{\mathrm{o}}})-\mathrm{Sin}\frac{\theta }{2}\mathrm{Cos}\frac{\theta }{2}]$

Length of arc

The length of arc is the distance between two points on the boundary of a circle.

Length of arc of circle

Formulas list of areas & perimeters