Trapezoids: Examples, Formulas & Characteristics - StudySmarter

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Fact Checked Content
Last Updated: 16.06.2022
Published at: 16.06.2022
9 min reading time

Applied Mathematics
Calculus
Decision Maths
Discrete Mathematics
Geometry

2 Dimensional Figures
3 Dimensional Vectors
3-Dimensional Figures
Affine geometry
Altitude
Analytic geometry
Angle Sum Property
Angles in Circles
Arc Measures
Area and Volume
Area of Circles
Area of Circular Sector
Area of Parallelograms
Area of Plane Figures
Area of Rectangles
Area of Regular Polygons
Area of Rhombus
Area of Trapezoid
Area of a Kite
Birational geometry
Central Angle
Complex geometry
Composition
Computational geometry
Cones
Conformal geometry
Congruence Transformations
Congruent Triangles
Convex geometry
Convexity in Polygons
Coordinate Systems
Cube Properties
Cylinders
Differential geometry
Differential topology
Dilations
Discrete geometry
Distance and Midpoints
Equation Of Circles
Equilateral Triangles
Ergodic theory
Euclidean geometry
Exterior Angle Theorem
Figures
Finsler geometry
Fractal geometry
Fundamentals of Geometry
Galois geometry
Geodesy
Geometric Inequalities
Geometric Mean
Geometric Probability
Geometric Transformations
Geometric analysis
Geometric function theory
Geometric topology
Glide Reflections
HL ASA and AAS
Harmonic analysis
Hodge theory
Hyperbolic geometry
Identity Map
Inscribed Angles
Integral geometry
Interior Angles
Inversive geometry
Isometry
Isosceles Triangles
Kähler manifolds
Law of Cosines
Law of Sines
Linear Measure and Precision
Manifolds
Mathematical Physics
Median
Metric geometry
Minimal surfaces
Mirror symmetry
Moduli spaces
Morse theory
Möbius transformations
Non-Euclidean geometry
Parallel Lines Theorem
Parallelograms
Perpendicular Bisector
Plane Geometry
Poisson geometry
Polygon Properties
Polygons
Projections
Projective geometry
Properties of Chords
Proportionality Theorems
Pyramids
Pythagoras Theorem
Quantum geometry
Quaternionic analysis
Rectangle
Rectangular Prism
Reflection in Geometry
Regular Polygon
Rhombus Properties
Rhombuses
Ricci flow
Right Triangles
Rotations
SSS and SAS
Sasaki geometry
Scalene Triangles
Segment Length
Similarity
Similarity Transformations
Simplicial complexes
Special Right Triangles
Special quadrilaterals
Spherical geometry
Square Properties
Squares
Sub-Riemannian geometry
Surface Area of Cone
Surface Area of Cylinder
Surface Area of Prism
Surface Area of Sphere
Surface Area of a Solid
Surface of Pyramids
Symmetry
Symplectic geometry
Toric geometry
Transformation groups
Translations
Transversal Lines
Trapezoid Properties
Trapezoids
Triangle Inequalities
Triangles
Twistor theory
Using Similar Polygons
Vector Addition
Vector Product
Vector bundles
Volume of Cone
Volume of Cylinder
Volume of Pyramid
Volume of Solid
Volume of Sphere
Volume of prisms
Vortex dynamics
Weyl geometry
What is Point Slope Form

Logic and Functions
Mechanics Maths
Probability and Statistics
Pure Maths
Statistics
Theoretical and Mathematical Physics

Contents

Applied Mathematics
Calculus
Decision Maths
Discrete Mathematics
Geometry

2 Dimensional Figures
3 Dimensional Vectors
3-Dimensional Figures
Affine geometry
Altitude
Analytic geometry
Angle Sum Property
Angles in Circles
Arc Measures
Area and Volume
Area of Circles
Area of Circular Sector
Area of Parallelograms
Area of Plane Figures
Area of Rectangles
Area of Regular Polygons
Area of Rhombus
Area of Trapezoid
Area of a Kite
Birational geometry
Central Angle
Complex geometry
Composition
Computational geometry
Cones
Conformal geometry
Congruence Transformations
Congruent Triangles
Convex geometry
Convexity in Polygons
Coordinate Systems
Cube Properties
Cylinders
Differential geometry
Differential topology
Dilations
Discrete geometry
Distance and Midpoints
Equation Of Circles
Equilateral Triangles
Ergodic theory
Euclidean geometry
Exterior Angle Theorem
Figures
Finsler geometry
Fractal geometry
Fundamentals of Geometry
Galois geometry
Geodesy
Geometric Inequalities
Geometric Mean
Geometric Probability
Geometric Transformations
Geometric analysis
Geometric function theory
Geometric topology
Glide Reflections
HL ASA and AAS
Harmonic analysis
Hodge theory
Hyperbolic geometry
Identity Map
Inscribed Angles
Integral geometry
Interior Angles
Inversive geometry
Isometry
Isosceles Triangles
Kähler manifolds
Law of Cosines
Law of Sines
Linear Measure and Precision
Manifolds
Mathematical Physics
Median
Metric geometry
Minimal surfaces
Mirror symmetry
Moduli spaces
Morse theory
Möbius transformations
Non-Euclidean geometry
Parallel Lines Theorem
Parallelograms
Perpendicular Bisector
Plane Geometry
Poisson geometry
Polygon Properties
Polygons
Projections
Projective geometry
Properties of Chords
Proportionality Theorems
Pyramids
Pythagoras Theorem
Quantum geometry
Quaternionic analysis
Rectangle
Rectangular Prism
Reflection in Geometry
Regular Polygon
Rhombus Properties
Rhombuses
Ricci flow
Right Triangles
Rotations
SSS and SAS
Sasaki geometry
Scalene Triangles
Segment Length
Similarity
Similarity Transformations
Simplicial complexes
Special Right Triangles
Special quadrilaterals
Spherical geometry
Square Properties
Squares
Sub-Riemannian geometry
Surface Area of Cone
Surface Area of Cylinder
Surface Area of Prism
Surface Area of Sphere
Surface Area of a Solid
Surface of Pyramids
Symmetry
Symplectic geometry
Toric geometry
Transformation groups
Translations
Transversal Lines
Trapezoid Properties
Trapezoids
Triangle Inequalities
Triangles
Twistor theory
Using Similar Polygons
Vector Addition
Vector Product
Vector bundles
Volume of Cone
Volume of Cylinder
Volume of Pyramid
Volume of Solid
Volume of Sphere
Volume of prisms
Vortex dynamics
Weyl geometry
What is Point Slope Form

Logic and Functions
Mechanics Maths
Probability and Statistics
Pure Maths
Statistics
Theoretical and Mathematical Physics

Contents

Fact Checked Content
Last Updated: 16.06.2022
9 min reading time

Content creation process designed by Lily Hulatt
Content sources cross-checked by Gabriel Freitas
Content quality checked by Gabriel Freitas

Jump to a key chapter

Parallelogram shape of a handbag and takeaway box, StudySmarter Originals

Now, notice how both the bases of the handbag and takeaway box are parallel to their tops. Since this shape has four sides, it is classified as a type of quadrilateral. However, it is neither a square, a rectangle nor a parallelogram. These shapes have two pairs of parallel sides while the shape described by this handbag and takeaway box has only one pair. Have you got any guesses as to what this shape might be? Let me give you a hint: it's called a trapezoid.

This article will explore the definition of a trapezoid along with its characteristics and types. We shall also look into the formulas used to find the perimeter and area of a trapezoid.

What is a Trapezoid?

As mentioned before, a trapezoid falls under the category of a quadrilateral as it contains four sides. This special type of quadrilateral actually has two names: a trapezoid and a trapezium. The name varies from where you are in the world. Here in the United States, it is typically called a trapezium. However, in the United Kingdom, it is usually called a trapezium. How interesting is that? With that in mind, let us begin our discussion with the definition of a trapezoid.

A trapezoid is a quadrilateral with one set of parallel sides.

Below is a graphical representation of a trapezoid. We shall call this trapezoid ABCD.

Illustration of a trapezoid, StudySmarter Originals

We shall now move on to listing the properties of a trapezoid. By doing so, we can observe how different they are compared to a regular quadrilateral.

Characteristics of a Trapezoid

Let us now refer back to our trapezoid ABCD above. There are several notable characteristics of trapezoids we should familiarize ourselves with. These are listed below.

A trapezoid has a pair of parallel sides and a pair of non-parallel sides;
Usually, the bases (the top and bottom) of ABCD are parallel to each other. This can be written as AD // BC;

By the definition of a trapezoid.

A pair of adjacent angles formed between one parallel side and one non-parallel side of a trapezoid add up to 180°. Here, ∠ABC + ∠BAD = 180° and ∠BCD + ∠ADC = 180°;
The sum of the interior angles of a trapezoid is 360°;
The diagonals of a trapezoid bisect each other;
The median (midline or midsegment) of a trapezoid is parallel to both bases. This is shown by the pink line below;

Median of a trapezoid, StudySmarter Originals

The median (or mid-section) of a trapezoid is the line segment connecting the midpoints of the two non-parallel sides of a trapezoid.

The length of the median is the average of both bases. Say a = AD and b = BC, then m=a+b2 , where m is the median.

Forming Other Quadrilaterals from Trapezoids

There are three types of quadrilaterals that can stem from a trapezoid, namely a parallelogram, a square and a rectangle. These instances are described in the table below.

Type of Quadrilateral	Description
Parallelogram Parallelogram, StudySmarter Originals	A trapezoid where both pairs of opposite sides are parallel to each other
Square Square, StudySmarter Originals	A trapezoid where both pairs of opposite sides are parallel to each other All four sides are of equal length and at right angles to each other
Rectangle Rectangle, StudySmarter Originals	A trapezoid where both pairs of opposite sides are parallel to each other The opposite sides are of equal length and at right angles to each other

Types of Trapezoids

There are five types of trapezoids we should consider, namely

Scalene trapezoid
Isosceles trapezoid
Right trapezoid
Obtuse trapezoid
Acute trapezoid

The table below describes each of these trapezoids in turn along with their pictorial representation and distinct traits.

Type of Trapezoid	Pictorial Representation	Description
Scalene Trapezoid	Scalene trapezoid, StudySmarter Originals	A trapezoid with no sides or angles of equal measure.
Isosceles Trapezoid	Isosceles trapezoid, StudySmarter Originals	A trapezoid with opposite sides of the same length. Usually, represented by the non-parallel sides (or legs) of a trapezoid. The angles of the parallel sides (or bases) are equal to each other.
Right Trapezoid	Right trapezoid, StudySmarter Originals	A trapezoid with two adjacent right angles (equal to 90o).
Obtuse Trapezoid	Obtuse trapezoid, StudySmarter Originals	A trapezoid with two opposite obtuse angles (more than 90o).
Acute Trapezoid	Acute trapezoid, StudySmarter Originals	A trapezoid with two adjacent acute angles (less than 90o).

The Perimeter of a Trapezoid

A trapezoid is a two-dimensional polygon that lies on a two-dimensional plane. The perimeter of a trapezoid is described as the total length of its boundary. In other words, it is the sum of all its sides. Say we have a trapezoid ABCD with sides a, b, c, and d.

The perimeter of a trapezoid, StudySmarter Originals

Then the perimeter of a trapezoid formula is

P = a + b + c + d,

where P is the perimeter, a = AB, b = BC, c = CD and d = AD. This can also be written as

P = AB + BC + CD + AD.

Examples Using the Perimeter of a Trapezoid Formula

Let us now look at some worked examples involving the formula for finding the perimeter of a trapezoid.

Given the trapezoid below, find its perimeter.

Example 1, StudySmarter Originals

Solution

To find the perimeter of this trapezoid, we shall simply add the measures of all four sides together.

P=13+21+19+34⇒P=87 units

Thus, the perimeter of this trapezoid is 87 units.

An isosceles trapezoid has a perimeter of 35 units. What is the length of each (equal) opposite side given that the bases are 5 units and 8 units, respectively?

Solution

Here, we are given the perimeter of a trapezoid and the lengths of the bases. We are also told that this trapezoid is an isosceles trapezoid, meaning that there is a pair of equal opposite sides. We shall name these two identical sides by x.

Example 2, StudySmarter Originals

Since the perimeter is the sum of all four sides of this trapezoid, we can write this as the equation below.

P=5+8+x+x⇒35=13+2x

Rearranging this equation, we obtain

2x=35-13⇒2x=22

Simplifying this, we obtain

x=222⇒x=11 units

Thus, the value of each opposite side is 11 units.

The Area of a Trapezoid

The area of a trapezoid is defined by the space enclosed within its boundary. It is found by calculating the average length between two given parallel sides and multiplying this product with the height of the trapezoid. Observe the illustration of trapezoid ABCD below.

Area of a trapezoid, StudySmarter Originals

Here, the bases are a = BC and b = AD. The height is denoted by the letter h.

The height, h of a trapezoid is at a perpendicular distance between bases, a and b. It is also referred to as the altitude of a trapezoid.

Thus, the area of a trapezoid is

A=12(a+b)×h ,

where A = area, a = length of the shorter base, b = length of the longer base and h = height. Similarly, we can express this formula as

A=BC+AD2×h.

Examples Using the Area of a Trapezoid Formula

Let us now look at some worked examples applying the area of a trapezoid formula.

Identify the area of the following trapezoid.

Example 3, StudySmarter Originals

Solution

Here,

a = 6 units;

b = 8 units;

h = 5 units.

Don't get yourselves confused by the other two sides given! They are not parallel to each other so we cannot use their measures in our formula.

Now, using the area of a trapezoid formula, we obtain

A=12(a+b)×h⇒A=12(6+8)×5

Simplifying this, we get a final answer of

A=12(14)×5⇒A=7×5⇒A=35 units2

Thus, the area of this trapezoid is 35 units2.

Find the length of the shorter base of a trapezoid given that the area is 232 units2, the height is 16 units and the length of the longer base is 17 units.

Solution

In this case,

A = 232 units2

b = 17 units;

h = 16 units.

Substituting these values into our formula, we obtain

A=12(a+b)×h⇒232=12(a+17)×16

Solving this, we have

232=16(a+17)2⇒232=8(a+17)

Expanding this, we get

232=8a+136⇒8a+136=232

Rearranging this equation and solving for a, we obtain the following final answer.

8a=232-136⇒8a=96⇒a=968⇒a=12 units

Hence, the length of the shorter base of this trapezoid is 12 units.

Example Involving Trapezoids

We shall end this topic with an example that encompasses everything we have learnt throughout this discussion.

Given the trapezoid ABCD below, determine its type, perimeter and area.

Example 4, StudySmarter Originals

Solution

Type

Let us first deduce what type of trapezoid this is. Looking at the diagram above, observe that ∠BAD = 103o and ∠BCD = 118o. Both these angles are greater than 90o and are located opposite each other. Thus, we have an obtuse trapezoid.

Perimeter

Next, we shall find the perimeter of this trapezoid. Adding all four sides of this trapezoid, we obtain

P=AB+BC+CD+AD⇒P=14+16+18+22⇒P=70 units

Thus, the perimeter of this trapezoid is 70 units.

Area

Here, BC (shorter base) is parallel to AD (longer base). The height is perpendicular to both these bases. Thus,

a = 16 units;

b = 22 units;

h = 11 units.

Applying the formula of the area of a trapezoid, we obtain

A=12(16+22)×11⇒A=12(38)×11⇒A=19×11⇒A=209 units2

Thus, the area of this trapezoid is 209 units2.

Bonus Question

What is the value of angle ∠ADC given that ∠ABC = 88o?

By the property of trapezoids, the sum of its interior angles adds up to 360°. Since we have the measures of three angles, we can find the value of the missing angle below.

∠ABC+∠BCD+∠ADC+∠BAD=360°⇒88°+118°+∠ADC+103°==360°

Rearranging this and solving for the unknown angle, we obtain

∠ADC=360°-88°-118°-103°⇒∠ADC=51°

Thus, angle ∠ADC is 51o.

Trapezoids - Key takeaways

A trapezoid is a quadrilateral with one set of parallel sides.
There are 5 types of trapezoids: scalene, isosceles, right, obtuse and acute.
The perimeter of a trapezoid is given by P = a + b + c + d.
The area of a trapezoid is given by A=12(a+b)×h.

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Frequently Asked Questions about Trapezoids

What is a trapezoid?

A quadrilateral with one set of parallel sides.

What are the characteristics of trapezoids?

The main characteristics of a trapezoid are:

it has a pair of parallel sides;
it has a pair of adjacent angles formed between one parallel side and one non-parallel side;
its diagonals bisect each other;
its median is parallel to the parallel sides.

Are all trapezoids parallelograms?

No, not all trapezoids are parallelograms.

What is an example of a trapezoid?

A right trapezoid is an example of a trapezoid.

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Tag » What Two Shapes Make A Trapezoid

Trapezoids: Examples, Formulas & Characteristics - StudySmarter

What is a Trapezoid?

Characteristics of a Trapezoid

Forming Other Quadrilaterals from Trapezoids

Types of Trapezoids

The Perimeter of a Trapezoid

Examples Using the Perimeter of a Trapezoid Formula

The Area of a Trapezoid

Examples Using the Area of a Trapezoid Formula

Example Involving Trapezoids

Trapezoids - Key takeaways

Similar topics in Math

Related topics to Geometry

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Frequently Asked Questions about Trapezoids

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