Calculate Isosceles Trapezoid

Calculator and formulas for the isosceles (symmetrical) trapezoid

Isosceles Trapezoid Calculator

Isosceles trapezoid

An isosceles (symmetrical) trapezoid has equal legs (b = d) and is symmetric about the mid-perpendicular. Calculation possible using sides and height or sides and angle.

Select calculation method
Choose between height- or angle-based calculation
Enter parameters
Distance between the parallel sides
First parallel side
Second parallel side
Calculation results
Side a:
Leg b/d:
Side c:
Diagonal e:
Height h:
Middle width m:
Area A:
Perimeter P:
Angle α/β:
Angle γ/δ:
Excess x:

Isosceles Trapezoid

The diagram shows an isosceles trapezoid with equal legs (b = d).
The axis of symmetry is perpendicular through the middle of the parallel sides.

Isosceles Trapezoid

Properties of the isosceles trapezoid

An isosceles (symmetrical) trapezoid has special geometric properties:

  • Equal legs: b = d (slanted sides have equal length)
  • Axis of symmetry: Perpendicular through the midpoint of the parallel sides
  • Equal base angles: α = β and γ = δ
  • Equal diagonals: Both diagonals have the same length
  • Axis symmetry: Reflection along the mid-perpendicular
  • Special formulas: Simplified calculations are possible

Formulas for the isosceles trapezoid

Side lengths
Side a: \[a = \frac{2A}{h} - c = 2m - c\] Leg b = d: \[b = d = \frac{h}{\sin(\alpha)}\] Side c: \[c = \frac{2A}{h} - a = 2m - a\]

Basic side calculations

Area and perimeter
Area: \[A = \frac{(a + c) \cdot h}{2} = m \cdot h\] Perimeter: \[P = a + 2b + c\] Middle width: \[m = \frac{a + c}{2}\]

Since b = d the perimeter simplifies

Angles
Base angles: \[\alpha = \beta = \arcsin\left(\frac{h}{b}\right)\] Top angles: \[\gamma = \delta = 180° - \alpha\]

Symmetry: α = β and γ = δ

Diagonals and height
Diagonals: \[e = f = \sqrt{a^2 + b^2 - 2ab\cos(\beta)}\] Height: \[h = \frac{2A}{a + c} = b \sin(\alpha)\] Excess: \[x = \sqrt{b^2 - h^2}\]

Both diagonals have equal length

Calculation example

Given
Side a = 7 Side c = 5 Height h = 4

Find: All parameters of the isosceles trapezoid

1. Middle width and area
\[m = \frac{a + c}{2} = \frac{7 + 5}{2} = 6\] \[A = m \cdot h = 6 \cdot 4 = 24\]

Basic calculations

2. Legs via excess
\[x = \frac{|a - c|}{2} = \frac{|7 - 5|}{2} = 1\] \[b = d = \sqrt{h^2 + x^2} = \sqrt{16 + 1} = \sqrt{17} ≈ 4.12\]

Calculation of the equal legs

3. Compute angles
\[\alpha = \beta = \arcsin\left(\frac{h}{b}\right) = \arcsin\left(\frac{4}{4.12}\right) ≈ 76.0°\] \[\gamma = \delta = 180° - 76.0° = 104.0°\]

Symmetric angle pairing

4. Perimeter and diagonals
\[P = a + 2b + c = 7 + 2(4.12) + 5 = 20.24\] \[e = f ≈ 7.35\] (equal diagonals)
Isosceles trapezoid fully computed

All symmetry properties satisfied

Applications of the isosceles trapezoid

Isosceles trapezoids are frequently used due to their special properties:

Architecture & Design
  • Roof constructions with even inclination
  • Symmetric window and door openings
  • Bridge arches and vaults
  • Stair railings and balustrades
Engineering & Production
  • Gear wheels with symmetric profiles
  • V-belt pulleys and belt drives
  • Flow channels with optimal distribution
  • Tool cutting edges and milling profiles

The isosceles trapezoid in geometry

To compute the isosceles trapezoid (also called symmetric trapezoid) provide either sides a and c together with the height, or sides a and d together with the angle Alpha. The isosceles trapezoid is a special case of the general trapezoid with distinct symmetry properties.

The equality of the legs (b = d) leads to simplified calculation formulas and special geometric properties such as equal diagonals and symmetric angles. These properties make the isosceles trapezoid an important geometric form in architecture, engineering and design.

For further trapezoid computations there are specialized calculators available optimized for specific use cases.

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