Area Calculator

About the Area Calculator

Area tells you how much flat surface a shape covers: how much paint a wall needs, how many tiles fit on a floor, how much fabric a pattern uses or how large a plot of land is. Each common shape has its own formula, and most of them grow out of the rectangle's simple rule of length times width.

This calculator collects the shapes students meet most often in one place. It keeps fractions and decimals exact where the arithmetic allows, writes circular answers as a multiple of π with a decimal alongside, and shows the formula with your numbers substituted so you can copy the working into your notes.

What is area?

Area is the number of unit squares needed to cover a region. A rectangle 8 cm long and 3 cm wide holds 8 columns of 3 one-centimetre squares, so its area is 24 cm². Because area counts squares, it is always measured in squared units such as cm², m² or ft².

Other formulas follow from the rectangle. A parallelogram can be cut and rearranged into a rectangle with the same base and height. A triangle is half of a parallelogram. A trapezoid is the average of its two parallel sides times its height. Circles and ellipses need π, and regular polygons split into identical triangles around the centre.

How to use the Area Calculator

Choose the shape from the menu.
Enter the dimensions that appear for that shape, all in the same unit.
For triangles, parallelograms and trapezoids, use the perpendicular height, not a slanted side.
Press Calculate area.
Read the area in square units and, where available, the perimeter.

Formulas

Rectangle and square

A = lw,\quad A = s^2

Triangle and parallelogram

A = \tfrac12 bh,\quad A = bh

Trapezoid

A = \tfrac12 (a + b)h

Circle, ellipse and sector

A = \pi r^2,\quad A = \pi ab,\quad A = \tfrac{\theta}{360^\circ}\pi r^2

Regular polygon

A = \frac{n s^2}{4\tan(\pi/n)}

Worked example: Rectangle 8 × 3.5

The worked example is a rectangle 8 units long and 3.5 units wide. Its area is $8 \times 3.5 = 28$ square units and its perimeter is $2(8 + 3.5) = 23$ units.

A trapezoid with parallel sides 6 and 10 and height 4 has area $\frac{6 + 10}{2} \times 4 = 32$ . A circle of radius 3 has area $9\pi \approx 28.274$ , and a regular hexagon with side 2 has area about 10.392.

The solution below is generated by the same engine that powers the calculator above.

Final answer

A = 28

How we read your problem

Compute the area from the shape's dimensions, all in the same length unit; the area is in that unit squared.

A = lw

Step-by-step solution

Length times width
$A = 8 \cdot \frac{7}{2} = 28$

Formulas used

Area
$A = lw$

Area measures how many unit squares fit inside a flat shape, so it is always in squared units such as cm² or m².

Verification

The area 28 is positive and in square units.

Detailed explanation

Height means perpendicular height

In the triangle, parallelogram and trapezoid formulas, the height is the perpendicular distance between the base and the opposite vertex or side. It is not the length of a slanted side. For a parallelogram with base 10 and slanted sides of 6 that meet the base at 60°, the height is $6\sin 60° \approx 5.196$ , not 6.

If you only know side lengths and angles of a triangle, use the triangle calculator, which finds the area with $\frac12 ab\sin C$ without needing the height.

Circles, ellipses and sectors

A circle's area is π times the radius squared. An ellipse with semi-axes a and b stretches a circle in two directions, so its area is $\pi ab$ ; when $a = b = r$ this becomes $\pi r^2$ . An ellipse with semi-axes 5 and 3 has area $15\pi \approx 47.124$ .

A sector is a slice of a circle. Its area is the fraction of a full turn the central angle covers, times the circle's area. A quarter circle of radius 4 has area $\frac{90}{360} \times 16\pi = 4\pi \approx 12.566$ .

Regular polygons

A regular polygon with n equal sides of length s can be split into n congruent isosceles triangles meeting at the centre. Each has base s and height equal to the apothem $a = \frac{s}{2\tan(\pi/n)}$ , so the total area is half the perimeter times the apothem.

For a regular hexagon with side 2, the apothem is $\sqrt3 \approx 1.732$ and the area is $\frac12 \times 12 \times 1.732 \approx 10.392$ , which equals six equilateral triangles of side 2. As n grows, a regular polygon approaches a circle, and its area formula approaches $\pi r^2$ .

Common mistakes

Using a slanted side instead of the perpendicular height.
Forgetting the ½ in the triangle and trapezoid formulas.
Using the diameter as the radius in $\pi r^2$ .
Mixing units, such as a length in metres and a width in centimetres.
Reporting area in plain units instead of square units.

Tips

Convert all measurements to one unit before calculating.
Split irregular shapes into rectangles and triangles, then add their areas.
To convert square units, square the length conversion: 1 m² = 10,000 cm².
Doubling every length multiplies the area by four.
Use the volume calculator when the shape has depth as well.

Frequently asked questions

How do you calculate the area of a rectangle?

Multiply the length by the width. A rectangle 8 m by 3.5 m has an area of 28 m².

How do I find the area of a trapezoid?

Add the two parallel sides, divide by 2 and multiply by the perpendicular height: A = ½(a + b)h.

What is the area of an ellipse?

It is π times the product of the two semi-axes, A = πab. For semi-axes 5 and 3 the area is 15π ≈ 47.12.

How do I find the area of a regular hexagon?

Use A = (3√3 / 2)s², which is the regular polygon formula with n = 6. A hexagon with side 2 has an area of about 10.39.

Why is area measured in square units?

Area counts how many unit squares cover a shape, so multiplying two lengths gives a unit squared, such as cm × cm = cm².

How do I convert square feet to square metres?

Multiply by 0.092903, because one foot is 0.3048 m and 0.3048² ≈ 0.092903. So 100 ft² is about 9.29 m².

How do I find the area of an irregular shape?

Divide it into rectangles, triangles and other simple shapes, find each area separately and add them, subtracting any cut-out parts.

Calculate an area

Choose a shape above and enter its dimensions to get the area in square units with the formula shown.

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