Area and Perimeter Calculator Guide: Formulas for Rectangles, Triangles, Circles, Polygons, and More
Area and perimeter describe different geometric properties. Perimeter measures the total distance around the boundary of a two-dimensional figure, while area measures the amount of two-dimensional region enclosed by that boundary.
Because perimeter measures length, it uses linear units such as meters, feet, centimeters, or inches. Area measures two-dimensional extent, so its units are squared: square meters, square feet, square centimeters, square inches, and so on.
This distinction matters in practical problems. Fencing around a rectangular yard depends on perimeter, while the amount of sod needed to cover the yard depends on area. A calculator that reports only one quantity can therefore answer the wrong real-world question even when the arithmetic is correct.
Different shapes require different measurements. A rectangle needs length and width. A circle needs radius or diameter. A triangle can use base and perpendicular height for area, but its perimeter requires all three side lengths.
Some shapes support multiple valid area formulas depending on the known information. A triangle can use one-half base times height, Heron’s formula when all three sides are known, or trigonometric formulas when two sides and an included angle are known.
Circles use π because circumference and area depend on the geometric relationship between radius and circular distance. Circumference is 2πr or πd, while area is πr².
Regular polygons can be calculated efficiently from side length, perimeter, apothem, or number of sides because all sides and central angles follow a common structure.
Composite or irregular figures often require decomposition. A complex floor plan, for example, can be split into rectangles and triangles, whose areas are calculated separately and then added or subtracted.
Perimeter of a composite shape requires additional care. Shared internal edges do not belong to the exterior perimeter, so adding the perimeters of component shapes directly can double-count interior boundaries.
Units also must remain consistent. Multiplying a length in meters by a width in centimeters without conversion produces a numerically meaningless area unless the units are first converted to a common system.
This calculator therefore identifies the selected shape, requests only the dimensions required by the relevant formula, calculates area and perimeter or circumference where sufficient information is available, and displays the units explicitly.
How to Calculate Area, Perimeter, and Circumference for Common 2D Shapes
- Choose the shape: Select rectangle, square, triangle, circle, parallelogram, trapezoid, rhombus, kite, sector, semicircle, or regular polygon. For a composite shape, calculate its nonoverlapping components separately and then combine their areas.
- Choose the known measurements: Enter the complete measurement set shown for the selected shape. The guide also explains alternate formulas you can use when different measurements are known.
- Use one consistent unit system: Convert all input lengths into the same unit before applying a geometry formula.
- Enter perpendicular height where required: For area formulas such as 1/2 bh or bh, height means perpendicular distance rather than a slanted side unless the formula specifically says otherwise.
- Calculate area: The calculator applies the selected shape formula and returns the result in squared units.
- Calculate perimeter or circumference: When all required boundary lengths are known, the calculator adds the exterior edges or uses the appropriate circular formula.
- Review exact and approximate results: For circle-based formulas, the displayed formula preserves π while the numerical result provides a decimal approximation.
- Convert units separately if needed: Use the Area Converter when the mathematical area is already known and only the unit needs to change.
Formula and variables
Area formulas measure two-dimensional extent and therefore produce squared units. Perimeter and circumference formulas add or integrate boundary lengths and produce linear units. Each shape uses dimensions appropriate to its geometry.
Rectangle: A = lw, P = 2(l + w); Triangle: A = 1/2 bh, P = a + b + c; Circle: A = πr², C = 2πr- A — Area
- The two-dimensional region enclosed by the shape, expressed in square units.
- P — Perimeter
- The total length of the outer boundary, expressed in linear units.
- C — Circumference
- The perimeter of a circle.
- l — Length
- One dimension of a rectangle or related shape.
- w — Width
- The perpendicular second dimension of a rectangle.
- b — Base
- A selected side used with its corresponding perpendicular height.
- h — Perpendicular height
- The shortest perpendicular distance from a base to the opposite vertex or parallel side.
- r — Radius
- Distance from the center of a circle to its boundary.
Scenario 1: Rectangle With Length 12 m and Width 7 m
A rectangular garden measures 12 meters long and 7 meters wide. The user wants both surface area and boundary length.
- Length
- 12 m
- Width
- 7 m
- Area = length × width.
- A = 12 × 7 = 84 m².
- Perimeter = 2(length + width).
- P = 2(12 + 7) = 2 × 19 = 38 m.
Result: Area = 84 m²; perimeter = 38 m.
The area represents the surface inside the garden, while 38 meters represents one complete trip around its outer boundary.
Understanding your results
Area
Area measures the amount of two-dimensional surface enclosed by the figure.
The result uses squared units because two independent length dimensions are involved.
Perimeter
Perimeter measures the total exterior boundary length.
It remains in the original linear unit.
Circumference
Circumference is the perimeter of a circle.
It can be calculated from radius or diameter.
Exact π form
Circle and sector results can often be written exactly as a multiple of π.
The decimal result is an approximation of that exact expression.
Composite-shape result
Composite area is built by adding and subtracting component areas.
Composite perimeter includes only exterior boundary segments.
Assumptions
- All lengths used together in one formula have been converted to the same unit.
- Shapes are ideal Euclidean plane figures.
- Heights used in base-height formulas are perpendicular to the corresponding base.
- Circle radius is nonnegative.
- Triangle side lengths satisfy the triangle inequality before Heron’s formula is used.
- Regular polygons have equal side lengths and equal interior-angle structure.
- Composite-shape components do not overlap unless overlap is explicitly subtracted.
- Displayed decimal results can be rounded while exact symbolic forms are preserved when practical.
Limitations
- A two-dimensional calculator does not account for material thickness, slope, curvature outside the selected formula, or three-dimensional surface area.
- Real construction surfaces may not form perfect rectangles, circles, or polygons.
- Area cannot be calculated from perimeter alone for most shape classes without additional geometric information.
- Triangle area from base and height requires a perpendicular height, not simply any side length.
- Heron’s formula becomes invalid when the three entered sides cannot form a triangle.
- Circle calculations assume a perfect circle.
- Composite-shape calculations require correct decomposition and must account for overlaps or cutouts explicitly.
- Adding the perimeters of component shapes can double-count internal edges.
- Area-unit conversion is not the same as length-unit conversion because the conversion factor must be squared.
- Measured dimensions can carry uncertainty, and area uncertainty can become larger when multiple measurements are multiplied.
Common mistakes
- Confusing perimeter with area.
- Reporting area in linear units.
- Reporting perimeter in square units.
- Adding length and width to calculate rectangle area.
- Multiplying length and width to calculate rectangle perimeter.
- Using diameter as radius without dividing by two.
- Using radius instead of radius squared in circle area.
- Using a slanted side as triangle height when it is not perpendicular to the base.
- Forgetting one side when calculating polygon perimeter.
- Using inconsistent measurement units in the same formula.
- Adding component perimeters in a composite figure and double-counting interior edges.
- Rounding π too early.
Practical use cases
Scenario 2: Square
A square has side length 9 cm.
Area = 9² = 81 cm² and perimeter = 4 × 9 = 36 cm.
Scenario 3: Triangle
A triangle has base 10 m and perpendicular height 6 m.
Area = 1/2 × 10 × 6 = 30 m².
Scenario 4: Circle
A circle has radius 5 cm.
Area = 25π cm² and circumference = 10π cm.
Scenario 5: Trapezoid
Parallel sides are 8 m and 14 m with height 5 m.
Area = 1/2(8 + 14) × 5 = 55 m².
Scenario 6: Composite room
An L-shaped floor can be split into two nonoverlapping rectangles.
Calculate each rectangle area and add them to obtain total floor area.
Planning and decision guide
Rectangle area is length times width
A rectangle can be viewed as rows and columns of equal square units.
Therefore A = lw.
Rectangle perimeter adds all four sides
Opposite sides are equal.
Therefore P = l + w + l + w = 2(l + w).
Scenario 7: Rectangle 15 by 4
Area = 15 × 4 = 60 square units.
Perimeter = 2(15 + 4) = 38 units.
Square formulas are rectangle formulas with equal sides
If every side has length s, area is s².
Perimeter is 4s.
Scenario 8: Find square side from area
If A = 144, then s = √144 = 12.
Perimeter = 48.
Triangle area is one-half base times perpendicular height
A triangle with base b and perpendicular height h has area A = bh/2.
This can be understood by pairing two congruent triangles to form a parallelogram.
Triangle perimeter requires all three side lengths
P = a + b + c.
Base and height alone generally do not determine the perimeter.
Scenario 9: 3-4-5 triangle
Perimeter = 3 + 4 + 5 = 12.
Using base 4 and height 3 gives area = 6.
Heron’s formula uses all three triangle sides
When a, b, and c are known, first compute semiperimeter s = (a + b + c)/2.
Then A = √[s(s − a)(s − b)(s − c)].
Scenario 10: Heron for 5-5-6 triangle
s = (5 + 5 + 6)/2 = 8.
A = √[8 × 3 × 3 × 2] = √144 = 12.
Triangle sides must satisfy the triangle inequality
The sum of any two side lengths must be greater than the third for a nondegenerate triangle.
Inputs 2, 3, and 6 cannot form a triangle.
Scenario 11: Invalid triangle
2 + 3 = 5, which is less than 6.
The calculator should reject the side set rather than feed it into Heron’s formula.
Two sides and included angle can determine triangle area
A = 1/2 ab sin(C).
The angle must be the included angle between sides a and b.
Angle unit matters in trigonometric triangle formulas
If the included angle is entered in degrees, the sine function must interpret it as degrees.
Use the Angle Conversion Calculator when unit conversion is needed.
Scenario 12: Two sides of 8 and 10 with included angle 30°
A = 1/2 × 8 × 10 × sin(30°).
Since sin(30°) = 0.5, area = 20 square units.
Parallelogram area uses base times perpendicular height
A = bh.
The slanted side does not replace perpendicular height unless it happens to be perpendicular.
Parallelogram perimeter uses adjacent side lengths
If adjacent sides are a and b, P = 2(a + b).
Height does not generally contribute directly to perimeter.
Scenario 13: Parallelogram
Base = 12, side = 7, height = 5.
Area = 60 and perimeter = 38.
Trapezoid area averages the parallel sides
For parallel bases a and b and perpendicular height h, A = (a + b)h/2.
The formula is the average base length multiplied by height.
Trapezoid perimeter requires all four sides
P = a + b + c + d.
The two bases and height alone generally do not determine the nonparallel side lengths.
Rhombus area has multiple useful formulas
A = bh when base and perpendicular height are known.
If diagonals d1 and d2 are known, A = d1d2/2.
Rhombus perimeter depends only on side length
All four sides are equal.
Therefore P = 4s.
Scenario 14: Rhombus from diagonals
d1 = 10 and d2 = 8.
Area = 10 × 8 / 2 = 40.
Kite area also uses perpendicular diagonals
For a standard kite, A = d1d2/2.
Perimeter requires the two pairs of adjacent equal side lengths.
Circle area uses radius squared
A = πr².
Squaring radius is essential because area is two-dimensional.
Circle circumference is linear
C = 2πr.
Using diameter d = 2r gives the equivalent formula C = πd.
Scenario 15: Radius 4
Area = 16π ≈ 50.265.
Circumference = 8π ≈ 25.133.
Radius and diameter should not be confused
Diameter spans the full circle through the center.
Radius is half the diameter.
Scenario 16: Diameter 18
Radius = 9.
Area = 81π and circumference = 18π.
Semicircle area is half the circle area
A = πr²/2.
Perimeter is not half the circle circumference alone because the diameter is also part of the boundary.
Semicircle perimeter includes the diameter
P = πr + 2r.
πr is the half-circle arc and 2r is the straight diameter.
Scenario 17: Semicircle radius 3
Area = 9π/2.
Perimeter = 3π + 6.
Sector area depends on the fraction of the circle
In degrees, A = (θ/360)πr².
In radians, A = 1/2 r²θ.
Sector arc length has parallel formulas
In degrees, arc length = (θ/360)2πr.
In radians, arc length = rθ.
Sector perimeter includes two radii
Sector perimeter = arc length + 2r.
The arc alone is not the complete sector boundary.
Scenario 18: 90° sector radius 8
Area = 90/360 × π × 64 = 16π.
Arc length = 90/360 × 16π = 4π.
Perimeter = 16 + 4π.
Regular polygons can be calculated from perimeter and apothem
For a regular polygon, A = Pa/2 where P is perimeter and a is apothem.
The apothem is the perpendicular distance from the center to a side.
Regular polygon perimeter is number of sides times side length
For n sides each of length s, P = ns.
This works because every side is equal.
Scenario 19: Regular hexagon side 5
Perimeter = 6 × 5 = 30.
If the apothem is known, area follows from Pa/2.
Regular polygon area can be calculated directly from side length
A = ns² / [4 tan(π/n)].
This follows by dividing the polygon into n congruent isosceles triangles.
Scenario 20: Square through regular-polygon formula
Set n = 4.
A = 4s²/[4 tan(π/4)] = s².
The general formula reduces correctly to the familiar square area.
Compound area uses addition and subtraction
Break the figure into nonoverlapping simple regions.
Add included regions and subtract cutouts or holes.
Scenario 21: Rectangle with circular hole
Calculate rectangle area.
Calculate circle area.
Subtract circle area from rectangle area.
Composite perimeter is not component-perimeter sum
Only edges exposed on the outside of the final figure belong to its perimeter.
Shared internal edges must be removed from the calculation.
Scenario 22: Two rectangles joined along one side
Adding both rectangle perimeters counts the shared edge twice.
Subtract twice the shared-edge length to obtain the exterior perimeter.
Area scaling is quadratic
If every length dimension is multiplied by scale factor k, area is multiplied by k².
Perimeter is multiplied only by k.
Scenario 23: Double every side
Perimeter doubles.
Area becomes four times as large.
This explains why area-unit conversions square the factor
One meter equals 100 centimeters.
Therefore one square meter equals 100² = 10,000 square centimeters.
Use the Area Converter for unit changes
Area and Perimeter determines geometric quantities from dimensions.
The Area Converter converts an already known area among square units.
Scenario 24: Convert after solving
A floor area is calculated as 50 m².
The geometry calculation is complete; any conversion to ft² belongs to the Area Converter.
Length units must be normalized before multiplication
A rectangle 2 m by 50 cm should not be calculated as 100 “m·cm.”
Convert 50 cm to 0.5 m first, giving area 1 m².
Scenario 25: Mixed-unit rectangle
Length = 3 ft and width = 24 in.
Convert 24 in to 2 ft.
Area = 6 ft².
Perimeter conversion uses linear factors
If a perimeter is 2 meters, that equals 200 centimeters.
Do not square the conversion factor for linear boundary measurements.
Area conversion uses squared factors
If one foot equals 12 inches, then one square foot equals 12² = 144 square inches.
This is why area conversion requires its own logic.
Area from coordinate points requires a different method
A polygon specified by vertices can be calculated using coordinate geometry, such as the shoelace formula.
This can be added as an advanced polygon mode without mixing it into simple side-length input.
Shoelace formula supports simple polygons from vertices
For ordered polygon coordinates, area can be calculated from alternating cross-products of adjacent vertices.
Vertex order matters and self-intersecting polygons require special interpretation.
Perimeter from coordinates uses distance
Calculate the distance between each pair of consecutive vertices and add the closing segment.
The distance formula follows from the Pythagorean theorem.
Scenario 26: Rectangle coordinates
Vertices (0,0), (4,0), (4,3), (0,3) form a 4-by-3 rectangle.
Area = 12 and perimeter = 14.
Measurement precision limits result precision
An input measured only to the nearest centimeter does not justify reporting an area to many meaningless decimal places.
The calculator can compute precisely while the user should interpret output in light of measurement accuracy.
Rounding π early reduces accuracy
Use the calculator’s π constant internally.
Round only the final displayed circle or sector result.
The Scientific Calculator handles underlying advanced functions
Polygon and triangle formulas can involve square roots, tangent, or sine.
Area and Perimeter should package those functions into geometry-specific workflows rather than requiring users to assemble formulas manually.
The Angle Conversion Calculator handles angle units
A geometry mode that accepts degrees should state that clearly.
Use explicit conversion before applying radian-specific formulas.
The strongest geometry result shows both quantities only when supported
Do not invent perimeter from insufficient dimensions.
If the user provides enough information for area but not perimeter, calculate area and explain what additional boundary measurements are needed.
Frequently asked questions
What is area?
Area measures the amount of two-dimensional region inside a shape and is expressed in square units.
What is perimeter?
Perimeter is the total length around the outside boundary of a two-dimensional shape.
What is the difference between area and perimeter?
Area measures enclosed surface, while perimeter measures boundary length.
Why is area measured in square units?
Area combines two length dimensions, so the unit is multiplied by itself.
What units are used for perimeter?
Linear units such as meters, feet, centimeters, or inches.
How do I calculate rectangle area?
Multiply length by width: A = lw.
How do I calculate rectangle perimeter?
P = 2(l + w).
How do I calculate square area?
Square the side length: A = s².
How do I calculate square perimeter?
Multiply the side length by four.
How do I calculate triangle area?
With base and perpendicular height, A = 1/2 bh.
How do I calculate triangle perimeter?
Add the three side lengths.
Can I calculate triangle area from three sides?
Yes. Use Heron’s formula after confirming the sides can form a triangle.
What is Heron’s formula?
With semiperimeter s = (a+b+c)/2, area is √[s(s−a)(s−b)(s−c)].
What is the triangle inequality?
For a nondegenerate triangle, the sum of any two side lengths must be greater than the third.
How do I calculate circle area?
A = πr².
How do I calculate circumference?
C = 2πr or πd.
What is the difference between radius and diameter?
Diameter is twice the radius.
How do I calculate a semicircle area?
A = πr²/2.
How do I calculate a semicircle perimeter?
P = πr + 2r because the boundary contains the half-circle arc and the diameter.
How do I calculate parallelogram area?
Multiply base by perpendicular height.
How do I calculate trapezoid area?
A = 1/2(a+b)h where a and b are the parallel sides.
How do I calculate rhombus area?
Use base × perpendicular height or one-half the product of the diagonals when appropriate.
How do I calculate kite area?
For the standard perpendicular-diagonal geometry, A = d1d2/2.
How do I calculate regular polygon perimeter?
Multiply the number of sides by the side length.
How do I calculate regular polygon area?
One common formula is A = Pa/2 where P is perimeter and a is apothem.
What is an apothem?
For a regular polygon, it is the perpendicular distance from the center to one side.
How do I calculate sector area?
In degrees, A = (θ/360)πr². In radians, A = 1/2 r²θ.
How do I calculate arc length?
In degrees, s = (θ/360)2πr. In radians, s = rθ.
Does sector perimeter equal arc length?
No. Sector perimeter includes the arc plus two radii.
Can I calculate area from perimeter alone?
Not generally. Different shapes can have the same perimeter and different areas.
Can I calculate perimeter from area alone?
Not generally unless the shape and sufficient geometric constraints are known.
How do I calculate the area of an irregular shape?
Break it into simpler nonoverlapping shapes, add included areas, and subtract holes or cutouts.
How do I calculate perimeter of a composite shape?
Add only the exposed outer boundary segments. Do not include shared interior edges.
Why can’t I mix feet and inches directly?
Geometry formulas require consistent units. Convert dimensions to one unit before calculating.
Is 1 square foot equal to 12 square inches?
No. Since 1 foot = 12 inches, 1 ft² = 12² = 144 in².
How do I convert area units?
Use the Area Converter, which applies squared conversion factors correctly.
Can this calculator use degrees and radians?
Angle-based geometry modes can support either when clearly specified. Use the Angle Conversion Calculator to convert between them.
How accurate is an area and perimeter calculator?
The formulas are exact for ideal geometric shapes. Real-world accuracy depends on the precision of the measurements and how closely the object matches the selected shape.
Sources and review
- Perimeter and Area — OpenStax. Accessed 2026-09-01.
- Geometry Concepts — OpenStax. Accessed 2026-09-01.
- Trigonometric Functions and Geometry — OpenStax. Accessed 2026-09-01.
- Elementary Geometry — Encyclopedia of Mathematics. Accessed 2026-09-01.
Reviewed 2026-09-01 by Dr Akawak Ejigu, DBA.