Introduction
An isosceles trapezoid, or isosceles trapezium in British English, is a convex quadrilateral in Euclidean geometry that has a line of symmetry cutting through one pair of opposite sides. It is an uncommon instance of a trapezoid.
Alternatively, it can be described as a trapezoid with equal amounts of both the base angles and the legs. Note that the second requirement, or the fact that it lacks a line of symmetry, prevents a non-rectangular parallelogram from being an isosceles trapezoid.
A form of trapezoid known as an isosceles trapezoid has non-parallel sides, called legs, that are of equal length. We can distinguish an isosceles trapezoid from other quadrilaterals owing to a few characteristics.
A trapezoid that has non-parallel sides and base angles of the same measure is said to be an isosceles trapezium. In other words, a trapezoid is an isosceles trapezoid if its two opposite sides (or bases) are parallel and its two non-parallel sides are of equal length.
Properties of Isosceles Trapezoid
The bases (opposite sides) of any isosceles trapezoid are parallel, while the other two sides (legs) are of equal length (properties shared with the parallelogram). Also, the diagonals are of the same length. An isosceles trapezoid’s base angles are equal in size (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base).
Formulas related to the Isosceles Trapezoid
Area of an isosceles trapezoid
The base sides or parallel sides must be added, divided by \( 2 \), and the result multiplied by the height to determine the area of an isosceles trapezoid.
Hence, the area of an isosceles trapezoid \( = (\frac{\text{Sum of parallel sides}}{2}) \times h \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= (\frac{a+b}{2}) \times h \)
Perimeter of an Isosceles Trapezoid
We must find the sum of the length of each side of an isosceles trapezoid in order to determine its perimeter.
The perimeter of an isosceles trapezoid \( = \) sum of all the sides
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= a + b + 2c \)
Where, a, b, c are the sides of the trapezoid.
Solved Examples
Example 1.If an isosceles trapezoid has bases that are \( 36 \) and \( 41 \) centimeters wide and non-parallel sides that are \( 87 \) centimeters wide, find its perimeter.
Solution:
Perimeter of an isosceles trapezoid \( = \) sum of all sides of isosceles trapezoid
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= a + b + 2c \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 36 + 41 + 2 \times 87 \) [Substitute the value]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 36 + 41 + 174 \) [Apply PEMDAS rule]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 251 \) centimeters
Therefore, the perimeter of an isosceles trapezoid is \( 251 \) centimeters.
Example 2. An isosceles trapezoid has bases that are \( 7 \) mm and \( 11 \) mm and a height of \( 22 \) mm, calculate its area.
Solution:
As stated in the question, base lengths are \( 7 \) mm and \( 11 \) mm, height is \( 22 \) mm
Area of an isosceles trapezoid \( = (\frac{a+b}{2}) \times h \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= (\frac{7+11}{2}) \times 22 \) [Substitute the value]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= (\frac{18}{2}) \times 22 \) [Apply PEMDAS rule]
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= (9) \times 22 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 198 \) sq.mm
Therefore, the area of an isosceles trapezoid is \( 198 \) square millimeters.
Example 3. If the area of the isosceles trapezoid is \( 165 \) square inches and the bases lengths are \( 17 \) inches and \( 29 \) inches. Determine its height.
Solution:
As stated in the question, area \( =165 \) in\( ^2 \)
Bases \( = 17 \) in and \( 29 \) in.
We know that the area of an isosceles trapezoid \( = (\frac{a+b}{2}) \times h \)
\( 165 = (\frac{17+29}{2}) \times h \) [Substitute the value]
\( 165 \times 2 = (17+29) \times h \) [Multiply both sides by \( 2 \)]
\( 330 = (46) \times h \) [Apply PEMDAS rule]
\( \frac{330}{46} = h \) [Divide both sides by
\( 46 \)]
\( h = 7.17 \)
Hence, the height of the isosceles trapezoid is \( 7.17 \) inches.
A four-sided isosceles trapezoid has four sides. The other two sides are equal in length but not parallel to one another, while the two opposite sides (bases) are parallel to one another.
A form of trapezoid known as an isosceles trapezoid has non-parallel sides that are equal to each other. An isosceles trapezoid is a particular kind of quadrilateral in which one set of the opposite sides is divided by the axis of symmetry. An isosceles trapezoid has equal-sized legs and bases that are parallel.
A trapezoid has sides of different lengths and incongruent diagonals. An isosceles trapezoid on the other hand has non-parallel sides that are equal, base angles that are equal, congruent diagonals, and opposite angles that are supplementary.
The base angles of an isosceles trapezoid are equal, so if one of the base angles is 30°, the other base angle will also be 30°.