Area of Circle Formula and Every Other Area Formula You Need

Area of circle formula with radius and diameter

A circle is fixed by one length, its radius $r$: the distance from the centre to any point on the edge. The diameter $d$ runs right across through the centre, so $d = 2r$. The space enclosed by the boundary is the area, and it is measured in square units such as $\mathrm{cm^2}$ or $\mathrm{m^2}$.

When the radius is given, square it and multiply by pi:

When only the diameter is given, you can halve it first, or put $r = \tfrac{d}{2}$ straight into the formula:

Textbook questions usually tell you which value of $\pi$ to use: $\tfrac{22}{7}$ (handy when the radius is a multiple of 7) or 3.14. Follow whatever value the question names. If it says nothing, leave the answer in terms of $\pi$ or use 3.14.

Why the area of a circle is pi r squared

Here is the cutting proof that most teachers draw on the board. It is the simplest way to derive the formula of area of circle without calculus.

1. Cut the circle into many equal sectors, like slices of a pizza, say 16 or 32 of them.
2. Lay the slices in a row, point up, point down, point up, and so on.
3. The shape now looks like a wavy parallelogram. As you cut thinner slices, the wavy edges flatten and the shape becomes a rectangle.
4. The height of that rectangle is the radius $r$. Its length is half the circumference, because half the curved edges lie along the top and half along the bottom: $\tfrac{1}{2} \times 2\pi r = \pi r$.

In class 12 you can also prove it by integration: adding thin rings of radius $x$ and width $dx$ gives $\int_0^r 2\pi x\,dx = \pi r^2$. The integration formulas page covers that tool. The link between area and the boundary length of a circle is on the perimeter and circumference page.

Triangle area formula: base, height, three sides and equilateral

The basic area of triangle formula is half of base times height:

The height must be perpendicular to the base you chose. In a right triangle the two legs are already perpendicular, so one leg is the base and the other is the height. In an obtuse triangle the height may fall outside the triangle, and that is fine.

Area of triangle with 3 sides formula

When no height is given but all three sides are, use Heron's formula (class 9). For sides 13, 14 and 15 cm, $s = 21$, so

The proof and more examples are on the Heron's formula page.

Area of equilateral triangle formula

All sides are equal to $a$, and the height works out to $\tfrac{\sqrt{3}}{2}a$, so $A = \tfrac{\sqrt{3}}{4}a^2$. For a side of 6 cm, $A = 9\sqrt{3} \approx 15.59\ \mathrm{cm^2}$.

Isosceles triangle area formula

If the two equal sides are $a$ and the base is $b$, the height splits the base in half, and Pythagoras gives the height. The result is

With $a = 10$ and $b = 12$: the height is 8, and the area is $\tfrac{12}{4} \times \sqrt{256} = 3 \times 16 = 48$ square units.

Square, rectangle and parallelogram area formulas

Area of square formula

A square with side $a$ has area $a^2$. If you only know its diagonal $d$, the area is $\tfrac{d^2}{2}$, because the two diagonals of a square are equal and meet at right angles. A square of side 9 m covers 81 square metres; one with a diagonal of 10 cm covers 50 square centimetres. People sometimes search "area of square ka formula": it is the same, side into side.

Area of rectangle formula

Length times breadth: $A = l \times b$. A classroom floor 15 m long and 8 m wide has an area of 120 square metres. Both lengths must be in the same unit before you multiply.

Parallelogram area formula

Base times the perpendicular height, $A = b \times h$. Do not multiply the two slanting sides together: that gives a wrong, larger number. A parallelogram with base 11 cm and height 6 cm has area 66 square centimetres. Cut a triangle off one end and slide it to the other end, and you get a rectangle of the same base and height, which is why the formula matches the rectangle one.

Rhombus, trapezium and quadrilateral area formulas

Area of rhombus formula

The diagonals of a rhombus cut each other at right angles, so the area is half their product, $\tfrac{1}{2}d_1 d_2$. A rhombus is also a parallelogram, so base times height works too.

Area of trapezium formula

A trapezium (called a trapezoid in American books) has one pair of parallel sides, $a$ and $b$. Add them, halve the sum, and multiply by the distance $h$ between them: $A = \tfrac{1}{2}(a+b)h$. This trapezoid area formula is part of middle school mensuration.

Area of a general quadrilateral

Draw one diagonal $d$ and drop perpendiculars $h_1$ and $h_2$ on it from the other two corners. The diagonal splits the shape into two triangles, so $A = \tfrac{1}{2}d(h_1 + h_2)$. With $d = 18$, $h_1 = 5$ and $h_2 = 7$, the area is 108 square units. If only the four sides and one diagonal are known, find each triangle by Heron's formula and add.

Ellipse, sector, semicircle and segment

Area of an ellipse

An ellipse is a stretched circle with a semi-major axis $a$ and a semi-minor axis $b$. Its area is $\pi ab$. When $a = b = r$, this turns back into $\pi r^2$. For $a = 5$ and $b = 3$, the area is $15\pi \approx 47.12$ square units.

Area of sector formula

A sector is the slice between two radii. Its share of the full circle equals its angle $\theta$ out of 360 degrees:

Semicircle and quadrant

A semicircle is a sector of 180 degrees, so its area is $\tfrac{1}{2}\pi r^2$; radius 7 cm gives 77 square cm with $\pi = \tfrac{22}{7}$. A quadrant (90 degrees) is a quarter of the circle.

Area of segment formula

A minor segment is the part between a chord and its arc. Take the sector and remove the triangle formed by the two radii and the chord:

For $r = 14$ cm and $\theta = 90^\circ$, with $\pi = \tfrac{22}{7}$: the sector is 154, the right triangle is $\tfrac{1}{2} \times 14 \times 14 = 98$, so the segment is 56 square cm. Sectors and segments form the class 10 chapter Areas Related to Circles. For values like $\sin 60^\circ$, see the trigonometry formulas page.

Ring (annulus)

The region between two circles with the same centre has area $\pi(R^2 - r^2)$. For outer radius 14 and inner radius 7, with $\pi = \tfrac{22}{7}$, that is $\tfrac{22}{7} \times 147 = 462$ square units.

Mensuration formula summary table

Here are the mensuration formulas for plane figures in one table. Copy it into your notebook, or save the one-page PDF version offered below.

The hexagon area formula comes from six equilateral triangles of side $a$; a side of 4 cm gives $24\sqrt{3} \approx 41.57$ square cm. Solid shapes need a different set of formulas. The area of cylinder, cuboid and sphere surfaces (curved and total surface area) is covered with the volume and surface area formulas, where, for example, the sphere has surface area $4\pi r^2$.

Units of area and conversions

Area is always in square units. When a length unit changes by a factor, the area unit changes by the square of that factor. That is the root of most conversion mistakes.

• $1\ \mathrm{cm^2} = 100\ \mathrm{mm^2}$ (since 1 cm = 10 mm)
• $1\ \mathrm{m^2} = 10\,000\ \mathrm{cm^2}$ (since 1 m = 100 cm)
• 1 hectare $= 10\,000\ \mathrm{m^2}$, and 1 are $= 100\ \mathrm{m^2}$
• $1\ \mathrm{km^2} = 100$ hectares $= 10^6\ \mathrm{m^2}$

So a floor of 120 square metres is 1 200 000 square centimetres, not 12 000. Convert lengths first and then multiply. That is safer than converting the final area. Percentage questions on area, such as "each side grows by 10 percent, by what percent does the area grow?", use the successive change idea on the percentage formula page. Here the area grows by 21 percent, not 20, because $1.1^2 = 1.21$.

Mensuration formula in Hindi

Hindi medium students meet the same formulas with these names. Mensuration itself is क्षेत्रमिति, and area is क्षेत्रफल.

Which class teaches which area formula

Knowing where a formula sits in the syllabus helps you revise the right chapter. The rectangle, square, triangle, parallelogram and circle come first, in the middle school classes, followed by the trapezium, rhombus and general quadrilateral. Class 9 teaches Heron's formula for a triangle with three sides. Class 10 has the chapter on areas related to circles, which covers sectors and segments, plus a separate chapter on solids. The exact class for each middle school shape depends on your book edition, so check its contents page.

Competitive exams such as the ones after class 12 mix these shapes: a square with a circle cut out, a rectangle with semicircles on two ends, or a path around a garden. The rule is always the same: break the figure into standard shapes, find each area, then add or subtract. For the full list across topics, see all maths formulas.