Trapezoids can seem tricky with their slanted sides and all. But don’t worry we’re here to walk you through everything you need to know about finding the perimeter of these funky quadrilaterals. In this article, we’ll start with a quick definition and properties refresher. Then we’ll break down the formula step-by-step, you’ll learn how to calculate the perimeter of a trapezium. We’ve even included some solved examples so you can see the formula in action.
What Is a Trapezium?
A trapezium is a quadrilateral with one pair of parallel opposite sides. The parallel sides are called the bases of the trapezium. The non-parallel sides are called the legs. A trapezium has four vertices and four sides enclosing an area. The angles on the same side of the bases are equal in measure.
A trapezium is a two-dimensional closed figure with some special properties that make it distinct from other quadrilaterals. The most prominent characteristic is that it has at least two sides that are parallel to each other. Trapeziums are found in many areas of science and mathematics. For example, trapeziums are used to calculate the area under curves and are found in shapes like the trapezoidal rule. The various properties of a trapezium also make it an important concept in geometry.
Types of Trapezium
A trapezium is a quadrilateral with one pair of parallel opposite sides. The two parallel sides are called bases while the non-parallel sides are called legs. There are three main types of trapezium:
Isosceles Trapezium
An isosceles trapezium has two non-parallel sides of equal length. The angles on each side of the parallel sides are also equal. An isosceles trapezium has line symmetry and rotational symmetry of order 2.
Right Trapezium
A right trapezium has two adjacent right angles. The right angles are formed between one leg and one base. The legs in a right trapezium are perpendicular to the bases. A right trapezium does not have any line of symmetry but has rotational symmetry of order 2. The adjacent interior angles sum up to 180°.
Scalene Trapezium
A scalene trapezium has no equal sides and no equal angles. None of the interior angles measure 90 degrees. A scalene trapezium lacks both line symmetry and rotational symmetry. The properties of a scalene trapezium depend on its specific dimensions.
In summary, the three types of trapeziums are:
- Isosceles Trapezium: Two non-parallel sides and angles are equal. Has line and rotational symmetry.
- Right Trapezium: Has two right angles between the legs and bases. Legs are perpendicular to bases. Has rotational symmetry only.
- Scalene Trapezium: No equal sides or angles. Lacks both line and rotational symmetry. Properties depend on dimensions.
Knowing the different types of trapeziums and their properties is useful when solving problems involving perimeter, area, angles and symmetry of trapeziums. With practice, identifying and applying the properties of each trapezium type will become second nature.
Properties of Trapezium
Some important properties of a trapezium include the following:
Only one pair of parallel sides
A trapezium has only one pair of parallel sides. The parallel sides are called bases while the non-parallel sides are called legs. The lengths of the bases can be different or equal.
Sum of interior angles is 360°
The sum of interior angles of a trapezium is 360°. This is a property of all quadrilateral figures. Each angle of a trapezium is less than 180°.
Diagonals intersect each other
The diagonals of a trapezium intersect each other at a point. The point of intersection divides each diagonal into two segments. The lengths of diagonal segments depend on the lengths of the parallel bases.
Mid-segment theorem
The mid-segment of a trapezium is parallel to its bases and its length is half the sum of the lengths of the bases. This is an important property used to calculate the area of a trapezium.
If a pair of opposite angles are equal, then it is an isosceles trapezium.
In an isosceles trapezium, the non-parallel sides have equal length. The angles on each side of the bases are also equal. An isosceles trapezium has one line of symmetry.
The perimeter of a trapezium
The perimeter of a trapezium can be calculated by adding the length of its 4 sides. If ‘a’ and ‘b’ are the lengths of the parallel bases and ‘c’ and ‘d’ are the lengths of the non-parallel sides, then the perimeter = a + b + c + d.
Height of a Trapezium
The perpendicular distance between the two bases is called the height of the trapezium.
Knowing these properties of trapezium helps in calculating angles, lengths, area and perimeter of trapeziums in various mathematical calculations and real-world applications.
How to Calculate the Perimeter of Trapezium
The perimeter of a trapezium refers to the total distance around its outer edge. To calculate the perimeter, you need to know the length of all four sides of the trapezium.
The Formula to calculate the perimeter of a Trapezium
The formula for calculating the perimeter of a trapezium is:
Perimeter = a + b + c + d
Where:
- a and b are the lengths of the parallel sides
- c and d are the lengths of the non-parallel sides
For example, if a trapezium has:
- Parallel sides a = 8 inches and b = 12 inches
- Non-parallel sides c = 10 inches and d = 14 inches
The perimeter would be:
Perimeter = 8 + 12 + 10 + 14 = 44 inches
Double check that you’ve used the correct measurements for the bases and legs. It also helps to visualize the trapezium shape in your mind to make sure the values you’re inputting match what you see. If the perimeter seems very large or small for the trapezium size, recheck your calculations. It’s an easy step but can help avoid mistakes.
Finding the perimeter of a trapezium is a useful skill in geometry and measurements. With some practice, calculating the perimeter will become second nature!
Why the Formula Works
This formula simply adds the four side lengths together because the perimeter refers to the total distance around the outer edge of the shape. By measuring and summing all four sides, you can calculate the complete perimeter of any trapezium.
The properties of a trapezium state that it must have at least two sides that are parallel. The non-parallel sides join the parallel sides at an angle, forming the four-sided shape. So the perimeter formula accounts for all components that make up the trapezium – the parallel sides, non-parallel sides, and angles between the sides.
So in short, to find the perimeter of any trapezium, you simply need to measure all four sides and plug those values into the formula: Perimeter = a + b + c + d. Add up the sides and you’ll have the complete perimeter in no time!
Solved Example Perimeter of Trapezium Formula Using Sides
A trapezium is a quadrilateral with one pair of parallel sides. To find the perimeter of a trapezium, you need to know the length of all four sides. Let’s look at the formula and a couple of examples to clarify how it works.
The perimeter of a trapezium is calculated as:
Perimeter = a + b + c + d
Where a and b are the lengths of the parallel sides, and c and d are the lengths of the non-parallel sides.
For example: A trapezium’s parallel sides measure 15 inches and 9 inches. The non-parallel sides are 8 inches and 12 inches long. What is the perimeter of the trapezium?
- Parallel sides: 15 inches and 9 inches
- Non-parallel sides: 8 inches and 12 inches
- Perimeter = 15 inches + 9 inches + 8 inches + 12 inches = 44 inches
So the perimeter of the trapezium is 44 inches.
A trapezium has one pair of parallel sides. Let’s say the length of the parallel sides are 15 cm and 9 cm. And the non-parallel sides are 12 cm and 10 cm.
- Find the perimeter. Perimeter = 15 cm + 9 cm + 12 cm + 10 cm = 46 cm
Another trapezium has bases of length 8 cm and 18 cm. The non-parallel sides are of length 12 cm and 15 cm.
- What is the perimeter? Perimeter = 8 cm + 18 cm + 12 cm + 15 cm = 53 cm
One more example – A trapezium has bases of length 20 cm and 30 cm. The height of the trapezium is 15 cm. If one of the non-parallel sides is 25 cm, find the length of the other non-parallel side and hence find the perimeter.
- Let the length of the other non-parallel side be ‘x’ cm.
- By the properties of trapezium: (Sum of parallel sides) / (Height) = (Non-parallel side) / (Other non-parallel side)
- 20 cm + 30 cm = 50 cm (Sum of parallel sides)
- x = 25 * 15/50 = 15 cm (Other non-parallel side)
- Perimeter = 20 cm + 30 cm + 25 cm + 15 cm = 90 cm
By going through these examples, you’ll understand how to calculate the perimeter of any trapezium. The formula is simple – just add the lengths of all the four sides. But you need to first find the unknown sides using the properties of trapezium. Practice a few more examples and the perimeter of trapezium will become easy for you!
Frequently Asked Questions about the Perimeter of Trapezium
Perimeter can be a confusing concept, especially when it comes to trapeziums. Here are some common questions people have about finding the perimeter of a trapezium:
Is trapezoid and trapezium the same?
A trapezium and trapezoid refer to the same quadrilateral shape. The names are used interchangeably, so you can call it either one. The key is that it has at least two parallel sides of different lengths.
Does the perimeter Formula Works Regardless of the Shape or the Trapezium?
The answer is yes, it works for all trapeziums, whether isosceles (with two equal non-parallel sides), right-angled (with one 90° angle), obtuse-angled (with one angle over 90°), or acute-angled (with all angles under 90°).
The only time the formula changes is if you have a trapezium where the non-parallel sides have different lengths, say c and e. Then the formula becomes:
Perimeter = a + b + c + e
You simply use the actual lengths of all four sides, whether or not they are equal. As long as you have a shape with at least two parallel sides of different lengths, this formula can be used to calculate its perimeter.
How do I determine which sides are the bases and which are the legs?
The bases of a trapezium are its two parallel sides. The legs are the two non-parallel sides that join the bases. The legs are often not the same length.
Can I calculate the perimeter if I only have the length of one base?
Unfortunately, with only the length of one base and one leg, you cannot calculate the full perimeter. You need the measurements of both bases and both legs to determine the complete perimeter using the formula.
What if the trapezium is irregular?
An irregular trapezium has bases and legs of different lengths. The perimeter formula still applies. You simply measure all four sides, the two bases and two legs and plug those values into the formula. The irregularity of the shape does not change how you calculate perimeter.
Perimeters can seem complicated, but by understanding what measurements you need and how the formula works, you’ll be calculating the perimeter of trapeziums in no time. Let me know if you have any other questions!
Conclusion
So there you have it! Now you’re a pro at finding the perimeter of a trapezium. With the simple formula and step-by-step examples, you can calculate the perimeter of any trapezoid that comes your way. Whether you need to find the perimeter for a math test or real-world project, you’ve got this. Trapezoids aren’t so tricky after all. The next time you see a trapezoid shape, put your new skills to work. Find the lengths of the sides, plug them into the formula, and determine that perimeter.