In this guide
To find area, identify the shape, choose its formula, substitute measurements in the same unit, and give your answer in square units. For a rectangle, multiply length by width. For a triangle, halve base × perpendicular height. For a circle, multiply π by the radius squared.
Calculate the area of a shape
The calculator covers 13 common shapes. It starts with an 8 cm by 5 cm rectangle, so you can see how the formula becomes a result. Change the shape or any measurement to try your own problem.
Area of a shape calculator
Choose a shape, enter its measurements, and follow the working.
Measure two adjacent sides that meet at a right angle.
Use the same unit for every length. Changing this label does not convert your numbers.
Measurement guide · not to scale
Area
Decimal approximation, displayed to up to 4 decimal places (6 significant digits in scientific notation).
Your working
A = l × w
8 × 5
≈ 40 cm²
Area formulas for all common shapes
Here, A means area. The letters are labels for measurements, not extra numbers to find. Height is always perpendicular to the chosen base, and r is a radius, not a diameter.
| Shape | Area formula | Measurements you need |
|---|---|---|
| Rectangle | l × w | Length (l); Width (w) |
| Square | s² | Side length (s) |
| Triangle | ½ × b × h | Base (b); Perpendicular height (h) |
| Parallelogram | b × h | Base (b); Perpendicular height (h) |
| Trapezoid / trapezium | ½ × (a + b) × h | Parallel side (a); Parallel side (b); Perpendicular height (h) |
| Rhombus | ½ × d₁ × d₂ | Full diagonal (d₁); Full diagonal (d₂) |
| Kite | ½ × d₁ × d₂ | Full diagonal (d₁); Full diagonal (d₂) |
| Circle | πr² | Radius (r) |
| Semicircle | ½πr² | Radius (r) |
| Sector | (θ / 360) × πr² | Radius (r); Central angle (θ) |
| Ellipse | πab | Semi-major axis (a); Semi-minor axis (b) |
| Annulus / ring | π(R² − r²) | Outer radius (R); Inner radius (r) |
| Regular polygon | ns² / (4 tan(π/n)) | Number of sides (n); Side length (s) |
How to read this table: Every formula is exact for the stated geometry. A decimal result involving π or a square root is usually rounded. The regular-polygon tangent uses radians.
Use exact formulas for correctly identified shapes, add/subtract methods for compound regions, and estimates for an irregular outline measured on a grid. These describe the method’s limits; they are not research ratings. Formula references: [1] [2] [5] [6] [7] [8]
How to find area, step by step
- Name the region. Is it one triangle, a whole circle, or a rectangle with a piece missing? Trace the boundary of the area you actually need.
- Label the useful measurements. Mark the perpendicular height, radius, diagonals, or semi-axes. If a length is missing, derive it from the other information before substituting.
- Make the units match. Convert metres and centimetres into one length unit. Do this before multiplying.
- Write the formula, then the working. For a triangle with b = 10 cm and h = 6 cm: A = ½bh = ½ × 10 × 6 = 30 cm².
- Check the size and the units. That triangle has half the area of a 10 cm by 6 cm rectangle. A result of 60 cm² would mean you forgot to halve.
Area counts how much flat space is covered. Length × length gives square units: cm × cm = cm². NIST identifies the square metre, m², as the SI unit of area. [3]
Area of rectangles, triangles and quadrilaterals
Rectangle and square: count the rows
A rectangle with 8 columns and 5 rows of unit squares holds 40 squares. That is why A = l × w. A square is the same idea with equal sides, so its formula shortens to A = s². A square with side 6 cm has area 6 × 6 = 36 cm². These are special cases of the same row-by-column method. [1]
Square: 6² = 36 cm²
Triangle: halve base × height
Two copies of a triangle fit together to make a parallelogram with the same base and height. One triangle therefore has half that area. The rule works for right, acute and obtuse triangles. For an obtuse triangle, the perpendicular height may meet an extension of the base outside the shape.
The height meets the base at a right angle
Labelled triangle schematic. Use the dashed height h and the base b. Do not substitute either sloping side for h.
How to read this: Use the dashed height h and the base b. Do not substitute either sloping side for h.
Original schematic, not to scale. Geometry: OpenStax, Prealgebra 2e, §9.4.
If all three sides are known, select “Three sides” in the calculator. Heron’s formula uses the semiperimeter p, which is half the sum of the sides. [4]
p = (5 + 6 + 7) / 2 = 9 cm
A = √[9 × 4 × 3 × 2] = √216 ≈ 14.70 cm²
Check that the sides form a triangle first: the two shortest must add up to more than the longest. Lengths 2, 3 and 5 lie flat, so they do not enclose a triangle. With two sides and their included angle C instead, use A = ½ab sin C; make sure your scientific calculator uses the same angle unit as the question.
Parallelogram: use the height, not the lean
Cut the triangular wedge from one end and move it to the other. The parallelogram becomes a rectangle without changing its area. That gives A = bh. With base 9 cm and perpendicular height 4 cm, the area is 36 cm². A sloping side of a different length does not change which height belongs in the formula.
Leaning the sides does not change base × height
Labelled parallelogram schematic. The dashed line spans the gap between the parallel bases. The sloping edge is a different measurement.
How to read this: The dashed line spans the gap between the parallel bases. The sloping edge is a different measurement.
Original rearrangement schematic, not to scale. The same base and perpendicular height define a rectangle of equal area.
Trapezoid or trapezium: average the parallel sides
Add the two parallel side lengths, halve that sum, and multiply by their perpendicular separation. Two matching trapezoids can form a parallelogram with base a + b, which explains the factor of ½. The shape is usually called a trapezoid in US school maths and a trapezium in UK school maths. [1]
Check your result against rectangles: it should lie between 6 × 4 = 24 cm² and 10 × 4 = 40 cm². Do not add the non-parallel sides into the formula.
Rhombus and kite: multiply the full diagonals, then halve
A rhombus has four equal sides. A convex kite has two pairs of adjacent equal sides. Their diagonals cross at right angles, letting you split them into right triangles whose areas sum to ½d₁d₂. The formula is shared; the shapes need not look the same. [8]
Kite: ½ × 10 × 6 = 30 cm²
Use both full diagonals, not the shorter segments from the crossing point to a corner. For an arbitrary quadrilateral whose diagonals are not perpendicular, the half-product rule does not apply.
Area of circles, sectors, rings and ellipses
Circle and semicircle: start with the radius
A circle’s area is πr². If the problem gives diameter d, first calculate r = d / 2. A semicircle takes half the area of the full circle, including only the region on one side of a diameter. [2]
Circle: π × 5² = 25π ≈ 78.54 cm²
Semicircle: ½ × 25π ≈ 39.27 cm²
Leave π in an exact answer if the question allows it. Otherwise use your calculator’s π key and round at the end. Using the diameter as r makes the result four times too large, because (2r)² = 4r².
Sector: take a fraction of the circle
A sector is the region between two radii and their connecting arc. A 60° sector takes 60/360 of the whole circle. If the angle is in radians, the equivalent formula is A = ½r²θ, because a full turn is 2π radians. The calculator supports both angle units and converts the angle when you switch. [5]
A = (60 / 360) × π × 6² = 6π ≈ 18.85 cm²
A sector’s angle tells you the fraction to keep
Labelled sector schematic. θ is measured at the centre between the two radii. Use θ/360 for degrees, or θ/(2π) for radians.
How to read this: θ is measured at the centre between the two radii. Use θ/360 for degrees, or θ/(2π) for radians.
Original schematic, not to scale. Formula: MathWorld, Circular Sector.
A circular segment is different: it is bounded by a chord and an arc. For a minor segment, subtract the triangle between the radii from its sector. The sector option alone does not calculate a segment.
Annulus: subtract the smaller circle
A ring between concentric circles has area π(R² − r²). Square the outer and inner radii separately, then subtract. Squaring the ring’s width, (R − r)², answers a different question.
A = π(25 − 9) = 16π ≈ 50.27 cm²
Ellipse: use the two half-axes
An ellipse is a circle stretched by different amounts along two perpendicular axes. Stretching the unit circle by factors a and b multiplies its area by ab, giving A = πab. These lengths run from the centre to the boundary, not across the whole ellipse. [6]
The ellipse formula needs half of each full axis
Labelled ellipse schematic. a is the semi-major axis and b is the semi-minor axis. Halve the full width and full height before entering them.
How to read this: a is the semi-major axis and b is the semi-minor axis. Halve the full width and full height before entering them.
Original schematic, not to scale. Formula and scaling derivation: MathWorld, Ellipse.
A = π × 6 × 3 = 18π ≈ 56.55 cm²
Area of regular pentagons, hexagons and other polygons
“Regular” means every side has the same length and every interior angle is equal. Join the centre to each vertex to make equal triangles. Each has a base equal to one side and a height called the apothem, the perpendicular distance from the centre to a side.
Or, with n sides of length s: A = ns² / [4 tan(π/n)]
The second form follows by finding the apothem of those triangles. It is what the calculator uses, so you only need the number of sides and one side length. The tangent is evaluated in radians. [7]
A = (6 × 4²) / [4 tan(π/6)] = 24√3 ≈ 41.57 cm²
For an equilateral triangle, n = 3 gives A = (√3/4)s². For a square, n = 4 gives A = s². A polygon with unequal sides or angles needs another method; entering its average side length does not repair the regular-polygon formula.
How to find the area of compound and irregular shapes
Look for a way to draw straight lines that split the region into shapes you recognise. Calculate each piece once, then add them. Alternatively, imagine an enclosing rectangle and subtract the missing pieces. OpenStax uses this decomposition approach for irregular figures. [2]
An L-shape can be one rectangle minus a cutout
Labelled compound schematic. Start with 60 m², then subtract the 12 m² corner. The dashed boundary shows the piece that is missing.
How to read this: Start with 60 m², then subtract the 12 m² corner. The dashed boundary shows the piece that is missing.
Original worked example and schematic, not to scale. A 10 m by 6 m rectangle has a 4 m by 3 m corner removed.
Missing corner: 4 × 3 = 12 m²
L-shaped region: 60 − 12 = 48 m²
You can check by splitting the same L into a 6 m by 6 m rectangle and a 4 m by 3 m rectangle: 36 + 12 = 48 m². Both approaches count exactly the same region. If two pieces overlap, adding their full areas counts that overlap twice.
For example, if a region contains 18 full cells and touches 8 more boundary cells, each of area 1 cm², its area lies between 18 and 26 cm². Taking half the boundary cells gives an estimate of 22 cm², but this is a rough convention, not a guarantee. Precisely specified curved boundaries may instead require integration.
Square units and the mistakes to avoid
| Mistake | What to do instead |
|---|---|
| Mixing centimetres and metres | Convert all lengths into the same unit before substituting. |
| Using a sloping side as height | Find the perpendicular distance to the base. |
| Using diameter in πr² | Halve the diameter first. |
| Putting degrees into ½r²θ | Use radians, or use (θ/360)πr² for degrees. |
| Using full ellipse axes | Halve both axis lengths first. |
| Converting area like length | Square the length-conversion factor. |
| Rounding each intermediate step | Keep full precision until the final answer. |
A 2 m by 50 cm rectangle is 2 m by 0.5 m, so its area is 1 m². In centimetres, it is 200 cm by 50 cm, giving 10,000 cm². The two answers describe the same area: 1 m² = 100 cm × 100 cm = 10,000 cm², not 100 cm².
Perimeter uses length units; area uses square units; volume uses cubic units. For a solid such as a cube, surface area adds up the areas of its outside faces. It is a separate problem from finding the area of one flat face.
Try the method before checking the answer
Write a formula and a substitution for each question. Then open the answer to compare your working. A correct number with the wrong units still needs fixing.
A triangle has base 12 cm and perpendicular height 7 cm. What is its area?
A = ½ × 12 × 7 = 42 cm². The height must meet the base at 90°.
A circle has diameter 14 mm. What is its area?
r = 14 / 2 = 7 mm. A = π × 7² = 49π mm² ≈ 153.94 mm².
A trapezium has parallel sides 5 m and 9 m, separated by 3 m. What is its area?
A = ½ × (5 + 9) × 3 = 21 m². Only the parallel sides go inside the brackets.
For more revision practice, see our guide to study methods, or explore Stepler’s mathematics support for worked exam questions.
Frequently asked questions
How do you find the area of any shape?
Identify a known shape and use its formula. For compound shapes, split the region into non-overlapping pieces and add their areas, or subtract holes from a larger shape. Arbitrary curved boundaries may need grid estimates or integration; there is no single elementary formula for every outline.
What is the difference between area and perimeter?
Area measures the region inside a boundary in square units. Perimeter measures the length of the boundary in ordinary length units. A rectangle measuring 8 cm by 5 cm has area 40 cm² and perimeter 26 cm.
How do you find the area of a triangle without its height?
If you know all three sides, use Heron’s formula: p = (a + b + c) / 2, then A = √[p(p − a)(p − b)(p − c)]. If you know two sides and their included angle C, use A = ½ab sin C. Two side lengths alone do not generally determine the area.
How do you find the area of a circle using its diameter?
Divide the diameter d by 2 to get the radius. Then use A = π(d / 2)². For a diameter of 10 cm, the radius is 5 cm and the area is 25π cm², approximately 78.54 cm².
Are trapezoid and trapezium area formulas different?
In the usual US and UK school terminology, a US trapezoid is a UK trapezium. For a quadrilateral with parallel sides a and b separated by perpendicular height h, the area is ½(a + b)h. Check the diagram and the parallel sides, since naming conventions vary.
Can you find area from perimeter alone?
Generally, no. A 6 cm by 4 cm rectangle and an 8 cm by 2 cm rectangle both have a perimeter of 20 cm, but their areas are 24 cm² and 16 cm². You need more information about the shape.
What about the area of 3D shapes?
For a solid, surface area is the total area of its outside faces or curved surfaces. A cube with edge s has six square faces and total surface area 6s². Volume measures space inside the solid in cubic units. The calculator here covers flat 2D regions.
Sources and calculation notes
These are mathematical references, not studies of how well a learning technique works. OpenStax provides textbook explanations; MathWorld supplies formulas and derivations; NIST defines the SI unit. Our examples and diagrams were made for this guide.
- [1]OpenStax, Prealgebra 2e, §9.4
Rectangles, triangles, trapezoids, and square units.
- [2]OpenStax, Prealgebra 2e, §9.5
Circles and splitting irregular figures into familiar shapes.
- [3]NIST, SI Units: Area
The square metre as the SI unit of area.
- [4]Eric W. Weisstein, Heron’s Formula, MathWorld
Area from three sides, with an algebraic derivation.
- [5]Eric W. Weisstein, Circular Sector, MathWorld
Sector area in radians.
- [6]Eric W. Weisstein, Ellipse, MathWorld
Semi-axes and the derivation of A = πab.
- [7]Eric W. Weisstein, Regular Polygon, MathWorld
Area from side count and side length.
- [8]Eric W. Weisstein, Kite, MathWorld
Perpendicular diagonals and the half-product formula.
Worked examples round to two decimal places where needed. The interactive calculator keeps JavaScript’s numerical precision until display and shows up to four decimal places, using scientific notation for very small or large results. All illustrations are schematic and not to scale.