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Trapezium

A trapezium is a quadrilateral in Euclidean geometry, though its definition differs regionally: in British English, it is a convex quadrilateral with exactly one pair of parallel sides (known as a trapezoid in American English), while in American English, it refers to a quadrilateral with no parallel sides.^[1] This terminological variation stems from historical shifts, originating with ancient Greek mathematicians like Euclid, who broadly described trapezia as quadrilaterals excluding squares, oblongs, rhombuses, or rhomboids, and later refined by Proclus and Archimedes to emphasize parallel sides in some cases.^[1] The confusion was exacerbated in the 18th century by Charles Hutton's Mathematical Dictionary (1795), which reversed the terms in American usage, leading to ongoing debate without a universal resolution.^[1] In the predominant British and international mathematical context, where a trapezium has one pair of parallel sides (the bases) and two non-parallel sides (the legs), key properties include that the parallel sides maintain constant separation, and the figure is convex.^[2] The area of such a trapezium is calculated using the formula A = \frac{1}{2} \times (b_1 + b_2) \times h, where b_1 and b_2 are the lengths of the parallel bases and h is the perpendicular height between them.^[3] A notable subtype is the isosceles trapezium, in which the non-parallel legs are equal in length, resulting in congruent base angles, equal diagonals, and symmetry across the midline connecting the midpoints of the non-parallel sides.^[4]

Terminology and Definitions

Regional Differences in Usage

In British English, the term "trapezium" refers to a quadrilateral with exactly one pair of parallel sides, while "trapezoid" denotes an irregular quadrilateral with no parallel sides.^[1] This convention is standard in UK mathematical education and publications. In contrast, North American English reverses these terms: "trapezoid" is used for a quadrilateral with exactly one pair of parallel sides, and "trapezium" describes a quadrilateral with no parallel sides.^[1] This usage prevails in American textbooks and geometric standards, leading to frequent confusion in international contexts. For example, a shape known as a trapezium in the UK would be called a trapezoid in the US, potentially causing miscommunication in cross-regional mathematical discussions. Indian English generally aligns with the British convention, defining "trapezium" as a quadrilateral with one pair of parallel sides, as seen in curricula like the CBSE syllabus.^[5]

Inclusive and Exclusive Definitions

In geometry, the exclusive definition of a trapezium describes it as a quadrilateral possessing exactly one pair of parallel sides, thereby excluding any quadrilaterals with two pairs of parallel sides, such as parallelograms.^[1] This definition emphasizes a strict distinction, focusing on the presence of precisely one pair of parallel sides, often referred to as the bases, with the non-parallel sides known as the legs.^[5] Conversely, the inclusive definition characterizes a trapezium as a quadrilateral with at least one pair of parallel sides, encompassing parallelograms, rectangles, rhombuses, and squares as special cases. Under this broader interpretation, the defining feature is the minimum requirement of one pair of parallel sides, allowing for greater flexibility in classification and alignment with hierarchical structures in quadrilateral properties.^[6] The inclusive definition is preferred in advanced mathematics because it facilitates more efficient theorem statements and proofs; for instance, any property established for a trapezium automatically applies to parallelograms, enhancing deductive consistency and reducing redundancy in geometric arguments.^[7] This approach mirrors the classification where rectangles include squares, promoting a logical hierarchy that simplifies higher-level analysis.^[6] In contrast, the exclusive definition is often favored in introductory educational settings for its clarity and ease in distinguishing basic shapes, avoiding overlap with parallelograms during early learning. The exclusive definition tends to align more closely with traditional British and Indian usage, while the inclusive is more debated in American contexts.^[8] For example, a rectangle qualifies as a special trapezium under the inclusive definition due to its two pairs of parallel sides but is excluded under the exclusive definition.^[9] Similarly, a square, with all sides parallel in pairs, fits the inclusive category as a particular type of trapezium but not the exclusive one.^[10] The exclusive definition aligns with traditional British usage in some educational standards, where "trapezium" specifically denotes exactly one pair of parallel sides.^[11]

Etymology and History

Ancient Origins

The earliest evidence of trapezoidal figures appears in ancient Mesopotamian and Egyptian mathematics, where they were employed for practical purposes such as land measurement. In Babylonian clay tablets from the Old Babylonian period (c. 1800–1600 BCE), surveyors approximated the areas of irregular fields using trapezoidal shapes, applying a formula that averaged the lengths of parallel sides multiplied by the height, as seen in exercises like tablet YBC 7290 from the Yale Babylonian Collection.^[12] Similarly, Egyptian scribes in the Middle Kingdom (c. 2000–1800 BCE) calculated trapezoid areas for surveying and taxation, as documented in Problem 53 of the Rhind Mathematical Papyrus (c. 1650 BCE), which involves computing the areas of two isosceles trapezia within a larger triangle using dimensions like bases of varying cubits and a height of 3 cubits.^[13] In ancient Greek mathematics, the concept of the trapezium was formalized by Euclid around 300 BCE in his Elements. There, in Book I, Definition 22, Euclid classified quadrilaterals, designating "trapezia" as those figures neither equilateral nor right-angled, excluding squares, oblongs, rhombuses, and rhomboids—effectively quadrilaterals with no parallel sides, contrasting with some later interpretations.^[1] This definition laid the groundwork for subsequent geometric theory, emphasizing irregular quadrilaterals in propositions on parallelograms and areas. Proclus, in his 5th-century CE commentary on Euclid's Elements, elaborated on these four-sided figures, integrating trapezia into discussions of parallelograms while distinguishing them by properties like parallel sides; he reassigned "trapezium" to quadrilaterals with exactly two parallel sides, diverging from Euclid's broader usage and influencing later terminological shifts.^[1] Non-Western traditions also utilized similar shapes contemporaneously. The Indian Sulba Sutras (c. 800–200 BCE), ritual texts guiding Vedic altar construction, describe trapezoidal forms such as the smasana citi altar, with dimensions like east-west bases of 3 and 2 purusas and a length of 6 purusas, whose area is computed as half the sum of the bases times the height to ensure precise ritual proportions.^[14]

Evolution of Terminology

The term "trapezium" traces its origins to ancient Greek geometry, where Euclid in his Elements (c. 300 BCE) defined trapezia as quadrilaterals that are neither equilateral nor right-angled, effectively excluding those with any parallel sides from the category of more structured figures like parallelograms. A significant shift occurred in the late 18th century with Charles Hutton's A Mathematical and Philosophical Dictionary (1795), which introduced the first major transposition of terms in English mathematical literature by defining "trapezium" as a quadrilateral with no parallel sides and "trapezoid" as one with exactly one pair of parallel sides, reversing earlier continental usages and marking an influential error that persisted.^[1] This redefinition quickly influenced American mathematical texts, where the Hutton-inspired terminology took hold; for instance, in Charles Davies' 1837 adaptation and translation of Adrien-Marie Legendre's Éléments de géométrie, "trapezoid" was explicitly defined as a quadrilateral with only two sides parallel, solidifying the term's association with the one-pair-parallel figure and contributing to the growing divergence from British conventions during the 19th-century standardization of geometry education in the United States.^[15] By the 20th century, the terminological inconsistencies persisted, with ongoing debates in mathematical communities about inclusive versus exclusive definitions of trapezoid and trapezium. As of 2025, there remains no universal resolution to the regional variations.^[1]

Special Types

Isosceles Trapezium

An isosceles trapezium is a trapezium with exactly one pair of parallel sides, known as the bases, and the two non-parallel sides, called the legs, of equal length.^[4] This configuration also results in the base angles being equal, specifically the two angles adjacent to each base are congruent.^[16] Key properties of an isosceles trapezium include the equality of its diagonals, which connect opposite vertices and are congruent in length.^[17] Additionally, it possesses a line of symmetry that passes through the midpoints of the two bases, bisecting both the trapezium and its diagonals perpendicularly.^[18] If the trapezium is labeled with bases AB (longer) and CD (shorter), and legs AD and BC, then the base angles satisfy ∠DAB = ∠CBA (adjacent to the longer base) and ∠ADC = ∠BCD (adjacent to the shorter base).^[19] To construct an isosceles trapezium using a compass and straightedge, given the lengths of the bases a (longer) and d (shorter) and equal legs b = c, follow these steps: First, construct segment e = a - d. Next, build an isosceles triangle with base e and equal sides b and c by drawing arcs from the endpoints of e with radius b to find the third vertex. Extend the base to full length a by adding segments of length d/2 on each end, then draw a line parallel to the base through the triangle's apex at distance corresponding to the height implied by the sides, connecting to form the shorter base d.^[20] An isosceles trapezium is always cyclic, meaning it can be inscribed in a circle, because the sums of its pairs of opposite angles each equal 180 degrees due to the supplementary nature of consecutive angles between the parallel bases and the equal base angles.^[21] This property distinguishes it as a special case among trapeziums, where cyclicity holds if and only if the legs are equal.^[22]

Right and Obtuse Trapezia

A right trapezium is a quadrilateral with exactly one pair of parallel sides, known as the bases, and two adjacent right angles formed between one non-parallel side (the leg) and the bases, making that leg perpendicular to both bases. The other leg is slanted, resulting in the remaining two angles being supplementary: one acute and one obtuse. This configuration distinguishes the right trapezium from other variants by providing a right-angled corner suitable for alignments requiring perpendicularity.^[23]^[24] In practical applications, right trapeziums are frequently employed in drafting and technical drawing due to their ease of construction with standard tools. For example, a right trapezium can be precisely drawn using a ruler to mark the bases and a set square to ensure the perpendicular leg aligns at 90 degrees to the bases, facilitating accurate representations of structural elements or mechanical components.^[25] An obtuse trapezium has one acute and one obtuse angle adjacent to each base. This configuration arises when the non-parallel legs are slanted asymmetrically—one inward and one outward relative to the bases—creating a pronounced leaning appearance. The angles adjacent to each leg are supplementary, as required by the parallel sides. Obtuse trapeziums find use in architectural designs, particularly in roof profiles that emphasize dynamic lines and visual interest. For instance, certain modern roof structures, such as the clustered trapezoidal roofs in the Ørestad Church, incorporate slanted forms that produce obtuse angles for a leaning effect, enhancing both aesthetic appeal and functional shading.^[26]

Existence and Characterizations

Conditions for Formation

A trapezium, defined as a quadrilateral with exactly one pair of parallel sides (the bases), can be formed with base lengths a > b and leg lengths c and d if and only if the side lengths satisfy the strict inequality |d - c| < a - b < d + c.^[27] This condition guarantees the existence of a non-degenerate convex trapezium with positive height and exactly one pair of parallel sides, excluding parallelograms (which require equal bases). The inequality derives from the triangle inequality theorem applied to the projections of the legs. To see this, drop perpendiculars from the endpoints of the shorter base b to the longer base a, dividing the difference a - b into two segments of lengths p and q where p + q = a - b and p, q \geq 0. These segments form the horizontal projections of the legs, leading to a configuration equivalent to forming a triangle with sides c, d, and a - b. For such a triangle to exist without degeneracy, the strict triangle inequalities must hold, yielding |d - c| < a - b < d + c. When equality holds in the inequality, the trapezium degenerates. If a - b = d + c, the height collapses to zero, reducing the figure to a straight line segment of length a. If a - b = |d - c| (assuming without loss of generality d > c), one projection becomes negative, which is impossible, preventing formation of a convex quadrilateral.^[27] For example, consider proposed side lengths with bases a = 5 and b = 3 (so a - b = 2) and legs c = 4, d = 2. The condition requires |2 - 4| < 2 < 2 + 4, or $2 < 2 < 6, which fails the left strict inequality. Thus, no such trapezium exists, as verified by attempting to solve for projections: the equation for equal heights leads to an invalid negative projection length.^[27] In the special case of an isosceles trapezium where the legs are equal (c = d), the condition simplifies to $0 < a - b < 2c, which is readily satisfied for appropriate lengths.^[28]

Equivalent Properties

A trapezium, defined as a quadrilateral with exactly one pair of opposite parallel sides (the bases), possesses several equivalent characterizations that allow identification without directly verifying parallelism. These properties provide alternative criteria for recognition and are useful in proofs, such as establishing the existence of parallel sides from angle or diagonal relations.^[29] One key equivalent property concerns the angles: in a trapezium, the two angles adjacent to each non-parallel side (leg) are supplementary, meaning their measures sum to 180°. Specifically, if the bases are the parallel sides, then the consecutive interior angles formed by each leg and the bases satisfy α + β = 180°, where α and β are the angles at the ends of a leg. This holds because the legs act as transversals to the parallel bases, invoking the co-interior angles theorem in Euclidean geometry.^[30]^[31] Another characterization involves the diagonals: the diagonals of a trapezium intersect at a point that divides each diagonal in the ratio of the lengths of the two bases. If the bases have lengths a and b (with a > b), and the diagonals intersect at point O, then for each diagonal, the segments satisfy \frac{AO}{OC} = \frac{BO}{OD} = \frac{a}{b}, where A and B are vertices on the longer base and C and D on the shorter. This ratio arises from the similarity of the triangles formed by the diagonals and bases, as corresponding angles are equal due to the parallel lines.^[32] The midsegment property offers yet another equivalent: the line segment joining the midpoints of the two non-parallel sides is parallel to the bases and has length equal to the average of the base lengths, \frac{a + b}{2}. This midsegment's parallelism to the bases directly implies the original sides' parallelism via vector geometry, where the vector from one midpoint to the other is the average of the base vectors, ensuring collinearity with the direction of the bases. A proof sketch using congruence involves dropping perpendiculars from the ends of the shorter base to the longer, forming congruent right triangles and a rectangle, whose midline aligns parallel to the bases.^[30]^[33] These properties are mutually equivalent in the sense that any one implies the existence of exactly one pair of parallel sides, distinguishing the trapezium from other quadrilaterals like parallelograms (which have two pairs) or general quadrilaterals (with none). For instance, if a quadrilateral has supplementary adjacent angles as described, it must have parallel sides to satisfy the transversal angle condition.^[31]^[34]

Geometric Properties

Midsegment and Height

In a trapezium, the midsegment (also known as the median or midline) is the line segment that connects the midpoints of the two non-parallel sides, referred to as the legs. According to the midsegment theorem, this segment is parallel to the two bases (the parallel sides) and its length m is equal to the average of the lengths of the bases, given by the formula m = \frac{a + b}{2}, where a and b are the lengths of the bases.^[35]^[36] The height h of a trapezium is defined as the perpendicular distance between its two parallel bases. To calculate the height when the leg lengths and base lengths are known, perpendiculars are dropped from the ends of the shorter base to the longer base, forming right triangles on either side. The horizontal projection x (or the difference in base lengths divided appropriately) allows the use of the Pythagorean theorem on each leg: h = \sqrt{c^2 - x^2}, where c is the length of a leg and x is the length of the projection segment adjacent to that leg. For an isosceles trapezium, the projections are equal on both sides, simplifying the calculation.^[37] Visually, the midsegment can be illustrated in a diagram of trapezium ABCD with bases AB and CD (AB longer than CD), where points E and F are the midpoints of legs AD and BC, respectively; the segment EF forms the midsegment, parallel to AB and CD, and spans halfway between them in height.^[35] A key structural relation arises when perpendiculars are dropped from the ends of one base to the other, dividing the trapezium into a central rectangle (with height h and width equal to the shorter base) flanked by two right triangles (each with height h and base equal to the projection x). This decomposition highlights how the midsegment aligns with the average base length across the height.^[38]

Area and Diagonals

The area K of a trapezium with parallel bases of lengths a and b (a > b) and height h (the perpendicular distance between the bases) is given by the formula

K = \frac{a + b}{2} \cdot h.

This derives from decomposing the trapezium into a central rectangle of width b and height h, plus two right triangles on either side with total base a - b and height h; the area of the rectangle is b h, and the combined area of the triangles is \frac{1}{2} (a - b) h, yielding the total \frac{a + b}{2} h.^[39]^[40] An alternative formula for the area, applicable to trapeziums and more generally to any convex quadrilateral, expresses it in terms of the diagonals and their intersection angle:

K = \frac{1}{2} d_1 d_2 \sin \theta,

where d_1 and d_2 are the diagonal lengths and \theta is the angle between them; this holds because the diagonals divide the trapezium into four triangles whose areas sum using the sine of the intersection angle.^[41] The lengths of the diagonals in a general trapezium are determined by dividing the figure into triangles and applying the law of cosines. For instance, drawing one diagonal splits the trapezium into two triangles sharing that diagonal as a side; the cosine rule then relates the diagonal to the adjacent sides and the included angle, requiring knowledge of the base angles or height to solve. Equivalently, dropping perpendiculars from the ends of the shorter base to the longer base creates a rectangle and two right triangles, enabling the Pythagorean theorem to compute each diagonal as the hypotenuse of a right triangle formed by the height h and the horizontal projection of a leg.^[42]^[39] In a right trapezium, where one leg is perpendicular to both bases (forming two right angles), the diagonals simplify significantly using the Pythagorean theorem. With bases a (longer) and b (shorter), and height h, one diagonal is the hypotenuse spanning the longer base and height, \sqrt{a^2 + h^2}, while the other spans the shorter base and height, \sqrt{b^2 + h^2}.^[23]

Applications

In Mathematics and Calculus

In numerical integration, the trapezoidal rule provides an approximation for the definite integral of a function f(x) over an interval [a, b] by treating the area under the curve as that of a trapezium with bases f(a) and f(b), and height h = b - a. The formula for a single interval is \int_a^b f(x) \, dx \approx \frac{h}{2} (f(a) + f(b)). This method extends to composite rules by dividing the interval into n subintervals, yielding \int_a^b f(x) \, dx \approx \frac{h}{2} (f(a) + 2\sum_{i=1}^{n-1} f(a + i h) + f(b)), where h = (b - a)/n. The local truncation error for one subinterval is -\frac{(b-a)^3}{12} f''(\xi) for some \xi \in (a, b), which is of order O(h^3), making it suitable for smooth functions but less accurate than higher-order methods like Simpson's rule for the same computational effort.^[43] Historically, trapezia have featured in geometric proofs, notably in James A. Garfield's 1876 demonstration of the Pythagorean theorem. Garfield constructed a trapezium with bases of lengths a and b (the legs of a right triangle) and height a + b. The area of this trapezium equals \frac{1}{2}(a + b)(a + b) = \frac{a^2 + 2ab + b^2}{2}. Equivalently, the figure decomposes into three right triangles: two with legs a and b (each of area \frac{ab}{2}) and one with legs a and b along the hypotenuse c (area \frac{c^2}{2}), so the total area is ab + \frac{c^2}{2}. Setting these equal gives \frac{a^2 + 2ab + b^2}{2} = ab + \frac{c^2}{2}, simplifying to a^2 + b^2 = c^2. In coordinate geometry, a trapezium is conveniently represented by specifying the coordinates of its four vertices, ensuring one pair of sides is parallel. For instance, place the parallel bases along the x-axis with vertices at (0, 0), (p, 0), (q, h), and (r, h), where p and r - q define the lengths of the bases (assuming $0 < p < r - q) and h > 0 is the height; the non-parallel sides connect (0, 0) to (q, h) and (p, 0) to (r, h). These coordinates allow computation of properties like area via the shoelace formula or diagonals through distance metrics. To parameterize the vertices for variable configurations, one can express them in terms of parameters such as base lengths a and b (with a < b), height h, and offset d for asymmetry: vertices at (0, 0), (a, 0), (d + b, h), and (d, h), where b - a \leq d \leq 0 ensures convexity.^[44] Advanced applications include the role of trapezia in quadrilateral midpoint theorems, particularly extensions of Varignon's theorem. Varignon's theorem states that connecting the midpoints of any quadrilateral's sides forms a parallelogram whose area is half that of the original figure. For a trapezium with parallel sides a and b, the Varignon parallelogram inherits specific symmetries: its sides are parallel to the diagonals of the trapezium, and if the trapezium is isosceles (equal non-parallel sides), the Varignon figure is a rhombus. This property arises because the midsegments of the non-parallel sides contribute equal vectors in the parallelogram construction, highlighting trapezia's utility in vector geometry and complex plane representations where vertices are points in \mathbb{C}.^[45]^[46]

In Architecture and Everyday Use

In architecture, geometric shapes have been employed both historically and in modern engineering for structural efficiency. In Gothic buildings, such as cathedrals, tracery designs optimize stress distribution under vertical loads, allowing for larger openings while maintaining stability through pointed arches that direct forces downward.^[47] More contemporarily, trapezoidally corrugated steel plate shear walls (TCSPSWs) are widely used in seismic-resistant structures; these panels provide enhanced out-of-plane stiffness and buckling resistance, dissipating earthquake energy effectively in high-rise buildings.^[48] Experimental studies confirm that TCSPSWs outperform traditional stiffened shear walls by reducing lateral displacements during seismic events.^[49] In everyday applications, the isosceles trapezium's symmetrical design enhances visibility and aesthetic appeal in common objects. Road signs, particularly those indicating recreational or cultural areas, frequently adopt trapezoidal shapes to stand out against natural backgrounds, ensuring quick recognition by drivers; for instance, the U.S. Manual on Uniform Traffic Control Devices specifies rectangular or trapezoidal signs for such purposes to improve legibility from a distance.^[50] Similarly, table tops and picture frames often feature slanted trapezoidal sides for stability and visual interest; self-standing acrylic frames with trapezoidal bases, for example, are popular for desktop displays, allowing horizontal or vertical orientation without additional supports.^[51] The trapezium shape also appears in biological and engineering contexts, drawing analogies from natural forms. In human anatomy, the trapezium bone in the wrist—named for its quadrilateral, table-like structure—facilitates thumb opposition and grip strength, illustrating how the shape's stability aids precise movements.^[52] In mechanical engineering, gear teeth and timing belt profiles commonly use trapezoidal forms to minimize backlash and ensure smooth power transmission; trapezoidal tooth designs in synchronous belts, such as those with pitches like XL (0.2 inches), provide high load capacity with low slippage under moderate speeds.^[53] Trapezia feature in symbolic designs, particularly in national emblems and flags, where their geometric balance conveys authority and harmony. The flag of Brunei incorporates two yellow trapeziums alongside parallelograms.^[54] In modern technology, trapezoidal approximations optimize aerodynamics and ergonomics. LCD and LED screen bezels sometimes adopt trapezoidal contours for custom installations, such as in curved or irregular displays, enhancing fit and reducing light leakage.^[55] Drone propellers often employ trapezoidal blade shapes to improve efficiency; studies show these forms require less power for equivalent thrust compared to rectangular blades, due to better airflow distribution and reduced induced drag.^[56]

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Dec 12, 2021 · A quadrilateral is a polygon in Euclidean plane geometry with four edges (sides) and four vertices (corners).
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[PDF] A trapezoid is a quadrilateral with exactly one pair of parallel sides ...
5. The lower base angle is supplementary to any upper base angle. 6. The diagonals of an isosceles trapezoid are congruent.
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[PDF] QUADRILATERALS
QUADRILATERALS – PROPERTIES. 1. Trapezoid: (a) Consecutive angles between parallel sides of a trapezoid are supplementary. 2. Parallelogram: (a) Consecutive ...
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Trapezoids
A quadrilateral having two and only two sides parallel is called a trapezoid. However, most mathematicians would probably define the concept with the Inclusive ...Missing: trapezium | Show results with:trapezium<|control11|><|separator|>
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[PDF] 90 90 90 and 180 180 0 ≠ c - cs.wisc.edu
4.4.4 The median of a trapezoid is parallel to each base. 4.4.5 If two base angles of a trapezoid are congruent, the trapezoid is an isosceles trapezoid.<|separator|>
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[PDF] Chapter 10 Quadrilaterals - HUFSD
The sum of the measures of the angles of a trapezoid is 360°. 14. In a trapezoid, there are always two pairs of supplementary angles. Applying Skills. 15. In ...
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Midsegment or Median of a Trapezoid - Math Monks
Dec 8, 2021 · The trapezoid mid-segment theorem states that a line connecting the midpoints of the non-parallel sides (legs) is parallel to the bases. It ...
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Midsegment of a Trapezoid - Varsity Tutors
•The midsegment is parallel to the bases. •Length of midsegment: A B + C D 2; •A midsegment theorem exists for both trapezoids and triangles.
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How to Find the Height of a Trapezoid - Math Monks
The height (altitude) of a trapezoid is the distance between its two bases (long base and short base). It is always perpendicular to either of the bases.Height of a Trapezoid · Formula · How to Find the Height of a...<|control11|><|separator|>
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Trapezoid area proof by dividing it into two triangles?
Jan 13, 2013 · The formula for the area of a trapezoid is deduced by dividing the trapezoid into two triangles, one with base a and height h, and one with base b and height h.Deriving the area of a trapezoid by converting it into a rectangleHow to bisect an isosceles trapezoid into two equal area partsMore results from math.stackexchange.com
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Area of trapeziums - KS3 Maths - BBC Bitesize
The area of a trapezium is the average length of its parallel sides multiplied by the perpendicular height. This is shown by the formula
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Area of Quadrilateral – Definition, Formulas, Types & Examples
Measure both diagonals d₁ and d₂. · Find the angle θ where the diagonals meet. · Substitute the values into the formula: Area = (1/2) × d₁ × d₂ × sin(θ) · Multiply ...<|control11|><|separator|>
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How to find the length of the diagonal of a trapezoid - Varsity Tutors
The formula for the length of diagonal uses the Pythagoreon Theorem: , where is the point between and representing the base of the triangle. Plugging in our ...
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[PDF] TRAPEZOIDAL METHOD ERROR FORMULA Theorem Let f(x) have ...
There are two stages in deriving the error: (1) Obtain the error formula for the case of a single subinterval (n = 1); (2) Use this to obtain the general error ...
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Trapezoid and its properties. (Coordinate Geometry)
A trapezoid is a quadrilateral with one pair of parallel sides. (See Trapezoid definition). In coordinate geometry, each of the four vertices (corners) also ...Missing: parametric | Show results with:parametric
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Varignon's Theorem -- from Wolfram MathWorld
The area of the Varignon parallelogram of a convex quadrilateral is half that of the quadrilateral, and the perimeter is equal to the sum of the diagonals ...Missing: trapezoid | Show results with:trapezoid
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[PDF] Consequences of the Varignon Parallelogram Theorem
What figure is formed when the consecutive midpoints of the sides of a quadrilateral are joined? What if the origi- nal quadrilateral were a rectangle? A kite?
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Gothic Windows: A Comprehensive Guide
Jun 11, 2024 · Gothic arched windows help distribute weight, allowing for taller and wider window openings fit with heavy glass. The glass is supported by ...<|separator|>
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On the probabilistic seismic damage assessment of trapezoidally ...
The unique properties of trapezoidally corrugated steel plate shear walls (TCSPSWs) have made them promising alternatives to stiffened SPSWs. The aim of this ...
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Seismic vulnerability evaluation of trapezoidally Corrugated Steel ...
These plates have high buckling resistance due to their out-of-plane stiffness. This research investigated the seismic behavior and vulnerability of steel ...
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IXITON Rustic Picture frame,Solid Wood Base photo ... - Amazon.com
Clear acrylic panel self standing frame for table top decor and desktop display home (trapezoid-vertical 5"×7")
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Trapezium bone: anatomy, function and features | Kenhub
The trapezium bone is one of the eight carpal bones of the hand. It is a cube shaped bone located on the radial side of the hand, in the distal carpal row.Missing: analogy | Show results with:analogy
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https://www.bruneiresources.com/flagandcrest.html
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National Flag, Emblem and Anthem - BRUNEI RESOURCES
Four colours are incorporated in the flag, red for the crest, yellow for the trapeziums, white for the upper parallelogram and black for the lower parallelogram ...
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Custom any Size and Shape LED Screen One Stop - iDISPLAY
Custom LED Display Screen of any Size and Shape, like Triangle, Trapezoid, Column, Curved, Twisted, Square, Rings, etc. product images. Letter Shaped LED ...
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Impact of blade shape on the aerodynamic performance and ...
May 8, 2024 · The trapezoidal and elliptical blades are more aerodynamically efficient, requiring less power from the motor to produce the same thrust, as ...Missing: trapezium | Show results with:trapezium