How To Find Area Of Trapezoid Without Height

Area of Trapezoid

The area of a trapezoid is the number of unit squares that can exist fit into information technology and it is measured in square units (similar cm², m², in², etc). For instance, if 15 unit squares each of length 1 cm tin be fit inside a trapezoid, then its area is 15 cm². A trapezoid is a type of quadrilateral with 1 pair of parallel sides (which are known as bases). It means the other pair of sides can be non-parallel (which are known equally legs). It is not e'er possible to depict unit squares and measure the surface area of a trapezoid. And so, permit us learn about the formula to discover the area of a trapezoid on this page.

1.	What is the Surface area of Trapezoid?
2.	Expanse of Trapezoid Formula
3.	Area of Trapezoid without Height
4.	How to Derive Surface area of Trapezoid Formula?
5.	Area of Trapezoid Reckoner
6.	FAQs on Surface area of Trapezoid

What is the Area of Trapezoid?

The expanse of a trapezoid is the total space covered by its sides. An interesting point to exist noted here is that if nosotros know the length of all the sides we can simply split up the trapezoid into smaller polygons similar triangles and rectangles, detect their area, and add them up to get the expanse of the trapezoid. Nevertheless, there is a direct formula that is used to find the expanse of a trapezoid if we know certain dimensions.

Area of Trapezoid Formula

The surface area of a trapezoid can exist calculated if the length of its parallel sides and the distance (height) between them is given. The formula for the area of a trapezoid is expressed as,

A = ½ (a + b) h

where (A) is the area of a trapezoid, 'a' and 'b' are the bases (parallel sides), and 'h' is the height (the perpendicular altitude between a and b)

Example:

Detect the area of a trapezoid whose parallel sides are 32 cm and 12 cm, respectively, and whose superlative is 5 cm.

Solution:

The bases are given every bit, a = 32 cm; b = 12 cm; the height is h = v cm.

The expanse of the trapezoid = A = ½ (a + b) h

A = ½ (32 + 12) × (5) = ½ (44) × (5) = 110 cm².

Area of Trapezoid without Top

When all the sides of the trapezoid are known, and we practice not know the height we can find the area of the trapezoid. In this instance, nosotros first need to summate the meridian of the trapezoid. Permit us sympathise this with the help of an example.

Example: Find the expanse of a trapezoid in which the bases (parallel sides) are given equally 6 and xiv units respectively, and the non-parallel sides (legs) are given equally, five units each.

Solution: Allow the states calculate the area of the trapezoid using the following steps.

Step ane: We know that the area of a trapezoid = ½ (a + b) h; where h = height of the trapezoid which is non given in this case; a = 6 units, b = xiv units, non parallel sides (legs) = 5 units each.
Pace two: And then, if we find the summit of the trapezoid, nosotros can summate the area. If we depict the summit of the trapezoid on both sides nosotros tin meet that the trapezoid is split into a rectangle ABQP and 2 right-angled triangles, ADP and BQC.
Step three: Since a rectangle has equal opposite sides, this ways AP = BQ and it is given that the sides Advertizement = BC = v units. So, the height AP and BQ tin can be calculated using the Pythagoras theorem.
Footstep 4: Now, permit us find the length of DP and QC. Since ABQP is a rectangle, AB = PQ and DC = 14 units. This means PQ = 6 units, and the remaining combined length of DP + QC can exist calculated as follows. DC - PQ = 14 - half dozen = 8. So, eight ÷ two = 4 units. Therefore, DP = QC = four units.
Footstep 5: Now, the top of the trapezoid can be calculated using the Pythagoras theorem. Taking the right-angled triangle ADP, we know that Advertisement = 5 units, DP = 4 units, and so AP = √(AD² - DPⁱⁱ) = √(v² - 4ⁱⁱ) = √(25 - 16) = √nine = 3 units. Since ABQP is a rectangle, in which the reverse sides are equal, AP = BQ = iii units.
Pace half dozen: Now, that we know all the dimensions of the trapezoid including the peak, we can calculate its surface area using the formula, area of a trapezoid = ½ (a + b) h; where h = 3 units, a = vi units, b = xiv units. After substituting the values in the formula, we get, surface area of a trapezoid = ½ (a + b) h = ½ (6 + 14) × three = ½ × 20 × three = 30 unitⁱⁱ.

How to Derive Area of Trapezoid Formula?

We can prove the area of a trapezoid formula by using a triangle here. Taking a trapezoid of bases 'a' and 'b' and height 'h', allow united states of america prove the formula.

Step 1: Split one of the legs into two equal parts and cutting a triangular portion of the trapezoid every bit shown.
Footstep 3: Attach it at the lesser as shown, such that it forms a big triangle.

Proof of area of trapezoid formula

Step 4: This way, the trapezoid is rearranged as a triangle. Even later on we attach it this way, we know that the area of the trapezoid and the new big triangle remains the same. Nosotros can besides encounter that the base of operations of the new big triangle is (a + b) and the height of the triangle is h.
Stride 5: So, it tin be said that the expanse of the trapezoid = the area of the triangle
Step vi: This can be written as, expanse of the trapezoid = ½ × base × height = ½ (a + b) h

Thus, nosotros have proved the formula for finding the area of a trapezoid.

Area of Trapezoid Calculator

The area of a trapezoid is the number of unit of measurement squares that can fit into it. Area of trapezoid calculator is an online tool that helps to find the area of a trapezoid. If certain parameters such as the value of base or height is available we tin directly give the inputs and summate the expanse. Try Cuemath's Area of a Trapezoid Reckoner and calculate the area of a trapezoid inside a few seconds. For more practice check out the expanse of trapezoid worksheets and solve the problems with the help of the calculator.

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Area of Trapezoid Examples

Instance 1: If ane of the bases of a trapezoid is equal to 8 units, its acme is 12 units and its area is 108 square units, observe the length of the other base.

Solution:

One of the bases is 'a' = viii units.

Let the other base exist 'b'.

The area of the trapezoid is, A = 108 foursquare units.

Its pinnacle is 'h' = 12 units.

Substitute all these values in the area of trapezoid formula,

A = ½ (a + b) h

108 = ½ (viii + b) × (12)

108 = six (8 + b)

Dividing both sides by 6,

18 = eight + b

b = 10

Answer: The length of the other base of operations of the given trapezoid = 10 units.
Case 2: Find the area of an isosceles trapezoid in which the length of each leg is eight units and the bases are equal to xiii units and 17 units respectively.

Solution:

The bases are a = 13 units and b = 17 units. Let usa presume that its top is h.

We can divide the given trapezoid into two congruent right triangles and a rectangle as follows:

From the above figure,

x + ten + 13 = 17

2x + 13 = 17

2x = iv

x = 2

Using Pythagoras theorem,

10² + h² = 8ⁱⁱ

2² + hⁱⁱ = 64

4 + hⁱⁱ = 64

h² = sixty

h = √60 = √four × √15 = 2√xv

The area of the given trapezoid is,

A = ½ (a + b) h

A = ½ (thirteen + 17) × (2√xv) = thirty√15 = 116.18 foursquare units

Reply: The area of the given trapezoid = 116.eighteen square units.
Case 3: Find the surface area of a trapezoid in which the bases are given every bit 7 units and nine units and the summit is given as 5 units.

Solution: The area of a trapezoid = ½ (a + b) h; where a = seven, b = nine, h = five.

Substituting these values in the formula, nosotros get:

A = ½ (a + b) h

A = ½ (7 + nine) × 5

A = ½ × 16 × 5 = 40 unit of measurementⁱⁱ

Therefore, the expanse of the trapezoid is 40 square units.

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Practise Questions on Area of Trapezoid

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FAQs on Area of Trapezoid

What is Surface area of Trapezoid in Math?

The area of a trapezoid is the number of unit of measurement squares that tin fit into it. Nosotros know that a trapezoid is a four-sided quadrilateral in which one pair of contrary sides are parallel. The expanse of a trapezoid is calculated with the assist of the formula, Area of trapezoid = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the perpendicular height. It is represented in terms of square units.

How to Find the Area of a Trapezoid?

The expanse of a trapezoid is found using the formula, A = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the height (the perpendicular distance between the bases) of the trapezoid.

Why is the Area of a Trapezoid ½ (a + b) h?

The formula for the area of a trapezoid tin be proved easily. Consider a trapezoid of bases 'a' and 'b', and height 'h'. We can cut a triangular-shaped portion from the trapezoid and attach it at the lesser then that the unabridged trapezoid is rearranged as a triangle. Then the triangle obtained has the base (a + b) and height h. By applying the area of a triangle formula, the area of the trapezoid (or triangle) = ½ (a + b) h. For more than data, you can refer to How to Derive Surface area of Trapezoid Formula? department of this page.

How to Find the Missing Base of a Trapezoid if yous Know the Area?

Nosotros know that the area of a trapezoid whose bases are 'a' and 'b' and whose height is 'h' is A = ½ (a + b) h. If i of the bases (say 'a'), height, and area are given, and so nosotros will just substitute these values in the to a higher place formula and solve it for the missing base (a) as follows:

A = ½ (a + b) h

Multiplying both sides by 2,

2A = (a + b) h

Dividing both sides past h,

2A/h = a + b

Subtracting b from both sides,

a = (2A/h) - b

How to Find the Peak of a Trapezoid With the Expanse and Bases?

If the area and the bases of a trapezoid is known, then we tin calculate its top using the formula, Area of trapezoid = ½ (a + b) h; where 'a' and 'b' are the bases and 'h' is the height. In other words, we can detect the height of the trapezoid by substituting the given values of the area and the ii bases.

How to Observe the Area of an Isosceles Trapezoid Without the Acme?

If the height of the trapezoid is non given and all its sides are given, then nosotros can divide the trapezoid into two congruent correct triangles and a rectangle. Using the Pythagoras theorem in the right-angled triangles, we tin calculate the height. After we get the acme, we can utilize the formula, A = ½ (a + b) h, to get the area of the trapezoid.

What is the Formula for Area of Trapezoid?

The formula that is used to find the surface area of a trapezoid is expressed as, Area of trapezoid = ½ (a + b) h; where a' and 'b' are the bases (parallel sides) and 'h' is the height of the trapezoid.

Source: https://www.cuemath.com/measurement/area-of-trapezoid/

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