BIZOMAT MAKEOS · Geometry
Plane Figure Area and Perimeter
What area and perimeter does this flat shape have?
Area, perimeter and equivalent circle and square of common plane figures: circle, ring, sector, ring sector, rectangle, parallelogram, trapezoid, fillet (spandrel), parabolic segment and cycloid.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shape | choice | The figure; the option text says which dimension each input means. |
| Dimension A | mm | Main dimension as described for the shape. |
| Dimension B | mm | Second dimension where the shape needs it. |
| Dimension C (height) | mm | Height where the shape needs it. |
| Angle (sectors) | ° | Included angle of a sector or ring sector. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Shape
- What it is
- The flat figure whose area you want. Each option names what Dimensions A, B and C and the angle mean for that figure, so read the option text before filling them in.
- Units and format
- Choose one of 10 options from the list: Circle (A = diameter); Ring (A = outer dia, B = inner dia); Circle sector (A = radius, angle); Ring sector (A = outer radius, B = inner radius, angle); Rectangle (A × B); Parallelogram (A = base, C = height, B = side); Trapezoid (A, B = parallel sides, C = height); Fillet or spandrel (A = radius); Parabolic segment (A = chord, C = height); Cycloid (A = rolling circle radius).
- Example
- The form opens with “Trapezoid (A, B = parallel sides, C = height)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Some shapes use A as a diameter (circle, ring) and others as a radius (sectors, fillet, cycloid); check the option text.
What this means: Dimension A
- What it is
- The main size of the chosen figure: the diameter of a circle or the outer diameter of a ring, the radius of a sector, fillet or cycloid’s rolling circle, a side or base of a rectangle, parallelogram or trapezoid, or the chord of a parabolic segment.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Radius and diameter are easy to swap: the circle and ring take a diameter, the sectors, fillet and cycloid take a radius.
What this means: Dimension B
- What it is
- The second size where the figure has one: the inner diameter of a ring, the inner radius of a ring sector, the second side of a rectangle, the sloping side of a parallelogram or the second parallel side of a trapezoid. Other figures ignore it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For a ring or ring sector it must be smaller than A. For a parallelogram it only affects the perimeter, not the area.
What this means: Dimension C (height)
- What it is
- The perpendicular height of a parallelogram, trapezoid or parabolic segment, measured square to the base or chord, not along a sloping side. Other figures ignore it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 40 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- For the parallelogram, trapezoid and parabolic segment the area is directly proportional to this height.
- Get it right
- Use the square (perpendicular) height; the length of a slanted side gives too large an area.
What this means: Angle (sectors)
- What it is
- The included angle between the two straight edges of a circle sector or ring sector, measured at the centre. Only the two sector shapes use it.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sector and ring sector area rise in direct proportion to the angle.
- Get it right
- Give the full included angle in degrees; 360 gives a complete circle or ring.
Method and formulas
- Circle πd²/4; ring π(D² − d²)/4; sector πr²θ/360; ring sector π(R² − r²)θ/360; rectangle AB; parallelogram base × height; trapezoid (a + b)h/2.
- Fillet (spandrel) between a quarter circle and its tangents: r²(1 − π/4) = 0.2146 r².
- Parabolic segment 2/3 × chord × height; cycloid 3πr² with arc length 8r.
- Equal-area circle diameter = √(4A/π); equal-area square side = √A.
- Example: a trapezoid with parallel sides 100 and 60 mm and height 40 mm has an area of 3,200 mm².
Limitations
- Ideal geometric figures; fillets and corner radii on real parts change the area.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find plate, gasket and section areas
- Estimate material for cut shapes
- Check areas on drawings
Who it is for
- Design engineers
- Toolmakers
- Estimators
What you get
- Area
- Perimeter (where defined)
- Diameter of a circle and side of a square with the same area
Interpreting the result
- Use the polygon area tool for irregular shapes defined by corner coordinates.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 66–81. Topic reference; no source tables or recommended settings are reproduced.
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