This area calculator lets you calculate the area for several different planar shapes. The shapes currently supported are triangles, squares, rectangles, parallelograms, rhombuses, and circles.

For most shapes, you have the option to choose what combination of inputs you want to provide. And to top it all, the calculator will give you not just the answer, but also the step by step solution.

Here's a quick overview of the concept of volume and its formula for a few important shapes.

The area of any two-dimensional or plane figure is a measure of the region it cobers.

For example, the area of the circle above refers to the shaded region.

Alright, here are formulas for the areas of some of the most popular shapes.

### Area of a Circle

For a circle with a radius $\hspace{0.2em} r \hspace{0.2em}$, the area is given by —

$A = \frac{\pi \cdot r^2}{2}$

### Area of a Parallelogram

The formula for the area of a parallelogram with a base $\hspace{0.2em} b \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$ is —

$A = b \cdot h$

Rectangles, rhombuses, and squares are just parallelograms with certain additional properties. So the formulas for the area of a parallelogram also work for those other shapes.

However, because of their additional properties, the relevant formulas can look a bit different, often simpler.

### Area of a Rectangle

For a rectangle with a length $\hspace{0.2em} l \hspace{0.2em}$ and width $\hspace{0.2em} w \hspace{0.2em}$, the area would be —

$A \hspace{0.25em} = \hspace{0.25em} l \cdot w$

### Area of a Rhombus

The most popular formula for the area of a rhombus is —

$A \hspace{0.25em} = \hspace{0.25em} \frac{d_1 \cdot d_2}{2}$

Here, $\hspace{0.2em} d_1 \hspace{0.2em}$ and $\hspace{0.2em} d_2 \hspace{0.2em}$ are the diagonal-lengths of the rhombus.

Alternatively, if you know the length of the rhombus' sides $\hspace{0.2em} a \hspace{0.2em}$ and one internal angle $\hspace{0.2em} \theta \hspace{0.2em}$, you can use the formula —

$A = a^2 \cdot \sin \theta$

### Area of a Square

The area of a square with sides of length $\hspace{0.2em} a \hspace{0.2em}$ is given by the formula —

$A = a^2$

### Area of a Triangle

Here are the two most commly used formulas for the area of a triangle.

1. The area of a triangle with a base $\hspace{0.2em} b \hspace{0.2em}$ and height $\hspace{0.2em} h \hspace{0.2em}$ would be —

$A = \frac{b \cdot h}{2}$

2. (Heron's formula) For a triangle with sides $\hspace{0.2em} a \hspace{0.2em}$, $\hspace{0.2em} b \hspace{0.2em}$, and $\hspace{0.2em} c \hspace{0.2em}$, the area is given by —

$A = \sqrt{s (s - a) (s - b) (s - c)}$

Here, $\hspace{0.2em} s \hspace{0.2em}$ is the semi-perimeter of the triangle.

$s = \frac{a + b + c}{2}$

## Area Calculations — Examples

Example

Calculate the area of a circle with a radius of $\hspace{0.2em} 4 \hspace{0.2em}$ inches.

Solution

The area of a circle is given by the formula —

$A \hspace{0.25em} = \hspace{0.25em} \pi r^2$

Substituting the value of $\hspace{0.2em} r \hspace{0.2em}$, we get

$\begin{align*} A \hspace{0.25em} &= \hspace{0.25em} \pi \cdot 4^2 \\[1em] &= \hspace{0.25em} 50.26 \end{align*}$

So the area of the circle is $\hspace{0.2em} 50.26 \text{ cm}^2 \hspace{0.2em}$.

Example

The sides of a triangle measure $\hspace{0.2em} 4 \hspace{0.2em}$ cm, $\hspace{0.2em} 5 \hspace{0.2em}$ cm, and $\hspace{0.2em} 7 \hspace{0.2em}$ cm. Calculate its area.

Solution

When we have the length of each side of a triangle, we can find it's area using the Heron's formula.

$A = \sqrt{s (s - a) (s - b) (s - c)}$

So, let's start by finding the semi-perimeter $\hspace{0.2em} (s) \hspace{0.2em}$.

$\begin{align*} s \hspace{0.25em} &= \hspace{0.25em} \frac{a + b + c}{2} \\[1em] &= \hspace{0.25em} \frac{4 + 5 + 7}{2} \\[1em] &= \hspace{0.25em} 8 \end{align*}$

Now, substituting the values of $\hspace{0.2em} a \hspace{0.2em}$, $\hspace{0.2em} b \hspace{0.2em}$, $\hspace{0.2em} c \hspace{0.2em}$, and $\hspace{0.2em} s \hspace{0.2em}$ into the Heron's formula, we get —

$\begin{align*} A \hspace{0.25em} &= \hspace{0.25em} \sqrt{s (s - a) (s - b) (s - c)} \\[1em] &= \hspace{0.25em} \sqrt{8 \cdot (8 - 4) \cdot (8 - 5) \cdot (8 - 7)} \\[1em] &= \hspace{0.25em} \sqrt{72} \\[1em] &= \hspace{0.25em} \sqrt{8.48} \end{align*}$

The area of the triangle is $\hspace{0.2em} 8.48 \text{ cm}^2 \hspace{0.2em}$.

Example

Calculate the area of a rectangular farm that is $\hspace{0.2em} 75 \text{ m} \hspace{0.2em}$ long and $\hspace{0.2em} 60 \text{ m} \hspace{0.2em}$ wide.

Solution

Using the formula for the area of a rectangle —

$\begin{align*} A \hspace{0.25em} &= \hspace{0.25em} l \cdot w \\[1em] &= \hspace{0.25em} 75 \cdot 60 \\[1em] &= \hspace{0.25em} 4500 \end{align*}$

So, the area of the farm is $\hspace{0.2em} 4500 \text{ m}^2 \hspace{0.2em}$.

Example

If the area of a rhombus is $\hspace{0.2em} 60 \text{ in}^2 \hspace{0.2em}$ and one diagonal measures $\hspace{0.2em} 6 \text{ in} \hspace{0.2em}$, what is the length of the other diagonal.

Solution

We know that for a rhombus with diagonals $\hspace{0.2em} d_1 \hspace{0.2em}$ and $\hspace{0.2em} d_2 \hspace{0.2em}$, the area is

$A \hspace{0.25em} = \hspace{0.25em} \frac{d_1 \cdot d_2}{2}$

Substituting the values of $\hspace{0.2em} A \hspace{0.2em}$ and $\hspace{0.2em} d_1 \hspace{0.2em}$ and solving for $\hspace{0.2em} d_2 \hspace{0.2em}$, we have

$\begin{align*} A \hspace{0.25em} &= \hspace{0.25em} \frac{d_1 \cdot d_2}{2} \\[1.75em] 60 \hspace{0.25em} &= \hspace{0.25em} \frac{6 \cdot d_2}{2} \\[1.75em] d_2 \hspace{0.25em} &= \hspace{0.25em} 20 \end{align*}$

So the other diagonal is $\hspace{0.2em} 20 \text{ in} \hspace{0.2em}$ long.