Areas¶

CalcpadCE ships a worksheet for every common 2D shape, each one giving the area and the perimeter (or arc length) from a few labelled inputs.

Pick the shape you need — square, circle, rectangle, regular polygon, triangle, parabola, ellipse, or one of the sector / segment variants — and override the inputs in the Desktop app; the result updates instantly.

Square¶

Area \(A = a^2\) and perimeter \(P = 4a\) of a square from a single side length.

Code:

'<div style="max-width:80mm;"><img style="height:60pt;" class="side" alt="square.png" src="../../Images/math/area/square.png"></div>
'Side length
a = ?
'Area
A = a^2
'Perimeter
P = 4*a
'1

Circle¶

Area \(A = \pi d^2/4\) and circumference \(P = \pi d\) of a circle, parameterised by diameter.

Code:

'<div style="max-width:100mm;"><img style="height:70pt;" class="side" alt="circle.png" src="../../Images/math/area/circle.png"></div>
'Diameter
'2&middot;<i>r</i> ='d = ?
'Area
A = π*d^2/4
'Circumference
P = π*d
'1

Rectangle¶

Area \(A = a\, b\) and perimeter \(P = 2(a + b)\) of a rectangle from its two side lengths.

Code:

'<div style="max-width:100mm;"><img style="height:55pt;" class="side" alt="rectangle.png" src="../../Images/math/area/rectangle.png"></div>
'Dimensions
a = ?', 'b = ?
'Area
A = a*b
'Perimeter
P = 2*(a+b)
'2    1

Trapezoid¶

Area \(A = \tfrac{1}{2}(a + b)\, h\) of a trapezoid from its two parallel sides and height.

Code:

"Area and Perimeter of Trapezoid
'<div style="max-width:90mm;"><img style="height:60pt;" class="side" alt="trapezoid.png" src="../../Images/math/area/trapezoid.png"></div>
'Base lengths
a = ?','b = ?
'Height
h = ?
'Area
A = (a+b)*h/2
'Perimeter (symmetrical figure only)
c = sqr(h^2 + (a - b)^2/4)
P = a+b + 2*c
'2    1   1

Polygon¶

Area and perimeter of a regular \(n\)-gon from side length \(a\), using \(A = \tfrac{n a^2}{4}\cot(\pi/n)\).

Code:

'<div style="max-width:110mm;"><img style="height:70pt;" class="side" alt="polygon.png" src="../../Images/math/area/polygon.png"></div>
'Number of sides -'n = ?
'Side length -'a = ?
'Area
α = 180/n'&deg;
A = a^2*n/4*cot(α)
'Perimeter
P = n*a
'8    1

Equilateral Triangle¶

Area \(A = \tfrac{\sqrt{3}}{4} a^2\) and perimeter \(P = 3a\) of an equilateral triangle.

Code:

'<div style="max-width:110mm;"><img style="height:60pt;" class="side" alt="equilateral-triangle.png" src="../../Images/math/area/equilateral-triangle.png"></div>
'Side length
a = ?
'Area
A = a^2*sqr(3)/4
'Perimeter
P = 3*a
P = 3*a
P = 3*a
P = 3*a
'1

Right Triangle¶

Area \(A = \tfrac{1}{2} a\, b\) and perimeter of a right triangle from its two legs (hypotenuse via Pythagoras).

Code:

'<div style="max-width:110mm;"><img style="height:60pt;" class="side" alt="right-triangle.png" src="../../Images/math/area/right-triangle.png"></div>
'Dimensions
a = ?','b = ?
c = sqr(a^2 + b^2)
'Area
A = a*b/2
'Perimeter
P = a + b + c
'4    3

Triangle¶

Area and perimeter of a general triangle from three side lengths via Heron formula, with a triangle-inequality guard.

Code:

'<div style="max-width:110mm;"><img style="height:55pt;" class="side" alt="triangle.png" src="../../Images/math/area/triangle.png"></div>
'Side lengths
a = ?','b = ?','c = ?
'<!--
#if (a + b > c)*(b + c > a)*(c + a > b)
    '<-->
    'Perimeter
    P = a + b + c
    'Area (Heron formula)
    'Semiperimeter -'p = 0.5*P
    A = sqr(p*(p - a)*(p - b)*(p - c))
    '<!--
#else
    '<-->
    'The sum of the lengths of each two sides must be greater than the third side.
    '<!--
#end if
'<-->5  4   3

Triangle (Height)¶

Area \(A = \tfrac{1}{2} a\, h_a\) of a triangle from base and corresponding height.

Code:

'<div style="max-width:100mm;"><img style="height:55pt;" class="side" alt="triangle.png" src="../../Images/math/area/triangle.png"></div>
'Base -'a = ?
'Height -'h_a = ?
'Area -'A = a*h_a/2
'2    1

Parabola¶

Area of a parabolic segment \(A = \tfrac{2}{3} a\, h\) from chord length and height.

Code:

"Area and Arc Length of Parabolic Segment
'<div style="max-width:120mm;"><img style="height:60pt;" class="side" alt="parabola.png" src="../../Images/math/area/parabola.png"></div>
'Dimensions
'Length -'a = ?
'Height -'b = ?
'Area
A = 2/3*a*b
'Arc length
'Assume'h = 4*b/a
'and'q = sqr(1 + h^2)
S = 0.5*a*(q + ln(h + q)/h)
'4    2

Ellipse¶

Area \(A = \pi\, a\, b\) and approximate circumference of an ellipse from its two semi-axes.

Code:

'<div style="max-width:110mm;"><img style="height:65pt;" class="side" alt="ellipse.png" src="../../Images/math/area/ellipse.png"></div>
'Dimensions
a = ?','b = ?
'Area
A = π*a*b/4
'Circumference
'Assume -'h = (a-b)^2/(a+b)^2
'Using the Ramanujan equation
P = π*(a+b)/2*(1 + 3*h/(10 + sqr(4 - 3*h)))
'Using series (up to the 4-th member)
P = π*(a+b)/2*(1 + 1/4*h + 1/64*h^2 + 1/256*h^3)
'2    1

Sector (Angle)¶

Area and arc length of a circular sector from radius and central angle, valid for \(\alpha \le 360°\).

Code:

"Area and Arch Length of Circular Sector
'<div style="max-width:110mm;"><img style="height:60pt;" class="side" alt="sector-angle.png" src="../../Images/math/area/sector-angle.png"></div>
'Radius -'r = ?
'Angle -'α = ?'&deg;
'<!--
#if α ≤ 180
    '<-->
    'Area
    A = π*r^2*α/360
    'Arch length
    S = π*r*α/180
    '<!--
#else
    '<-->
    'Angle must be less or equal to 180 &deg;
    '<!--
#end if
'<-->1  60

Sector (Chord)¶

Area and arc length of a circular sector specified by its chord and height instead of the central angle.

Code:

"Area and Arch Length of Circular Sector
'<div style="max-width:110mm;"><img style="height:60pt;" class="side" alt="sector-chord.png" src="../../Images/math/area/sector-chord.png"></div>
'Dimensions
'Radius -'r = ?
'Chord -'a = ?
'<!--
#if a ≤ 2*r
    '<-->
    'Angle
    α = 2*asin(a/(2*r))'&deg;
    'Area
    A = π*r^2*α/360
    'Arch length
    S = π*r*α/180
    '<!--
#else
    '<-->
    'Chord <i>a</i> must be less or equal to 2&middot;<i>r</i>
    '<!--
#end if
'<-->1  1

Segment (Angle)¶

Area and arc length of a circular segment from radius and central angle.

Code:

"Area and Arch Length of Circular Segment
'<div style="max-width:120mm;"><img style="height:70pt;" class="side" alt="segment-angle.png" src="../../Images/math/area/segment-angle.png"></div>
'Radius -'r = ?
'Angle -'α = ?'&deg;
'<!--
#if α ≤ 180
    '<-->
    'Area
    A = 0.5*(π*α/180 - sin(α))*r^2
    'Arch length
    S = π*r*α/180
    '<!--
#else
    '<-->
    'Angle must be less or equal to 180 &deg;
    '<!--
#end if
'<-->1  60

Segment (Chord)¶

Area and arc length of a circular segment specified by chord and height, with a chord-vs-height feasibility guard.

Code:

'<div style="max-width:120mm;"><img style="height:60pt;" class="side" alt="segment-chord.png" src="../../Images/math/area/segment-chord.png"></div>
'Dimensions
'Chord -'a = ?
'Height -'b = ?
'<!--
#if a ≥ 2*b
    '<-->
    'Radius
    r = b/2 + a^2/(8*b)
    'Angle
    α = 2*asin(a/(2*r))'&deg;
    'Area
    A = 0.5*(π*α/180 - sin(α))*r^2
    'Arc length
    S = π*r*α/180
    '<!--
#else
    '<-->
    'Chord <i>a</i> must be greater or equal to 2&middot;<i>b</i>
    '<!--
#end if
'<-->2  1

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