Two parallel sides of a trapezium are 29 cm and 21 cm. Non-parallel sides are equal and each is of length 8.5 cm. What is the area of the trapezium?
Finding the Area of an Isosceles Trapezium
The question asks us to find the area of a trapezium where we know the lengths of the two parallel sides and the lengths of the two equal non-parallel sides. This specific type of trapezium, where the non-parallel sides are equal, is called an isosceles trapezium.
The given information is:
- Length of one parallel side (\(a\)) = 29 cm
- Length of the other parallel side (\(b\)) = 21 cm
- Length of each non-parallel side (\(c\)) = 8.5 cm (since they are equal)
To find the area of any trapezium, we use the formula:
\(\text{Area} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}\)
Or, using the given variables:
\(\text{Area} = \frac{1}{2} \times (a + b) \times h\)
where \(h\) is the perpendicular distance between the parallel sides (the height of the trapezium).
We know \(a\) and \(b\), but we need to find the height \(h\). In an isosceles trapezium, we can find the height by drawing perpendicular lines from the endpoints of the shorter parallel side to the longer parallel side. This creates a rectangle in the middle and two congruent right-angled triangles on the sides.
Let the longer parallel side be \(a\) (29 cm) and the shorter parallel side be \(b\) (21 cm). The difference in length between the parallel sides is \(a - b = 29 - 21 = 8\) cm.
Since the two right-angled triangles formed are congruent (due to the trapezium being isosceles), the base of each triangle will be half of this difference:
\(\text{Base of triangle} (x) = \frac{a - b}{2} = \frac{29 - 21}{2} = \frac{8}{2} = 4 \text{ cm}\)
Now consider one of the right-angled triangles. The sides are:
- Base = 4 cm
- Height = \(h\) (the height of the trapezium)
- Hypotenuse = 8.5 cm (one of the non-parallel sides)
We can use the Pythagorean theorem (\( \text{hypotenuse}^2 = \text{base}^2 + \text{height}^2 \)) to find the height \(h\):
\(8.5^2 = 4^2 + h^2\)
Calculate the squares:
\(72.25 = 16 + h^2\)
Subtract 16 from both sides to find \(h^2\):
\(h^2 = 72.25 - 16\)
\(h^2 = 56.25\)
Take the square root to find \(h\):
\(h = \sqrt{56.25}\)
\(h = 7.5 \text{ cm}\)
Now that we have the height (\(h = 7.5\) cm) and the lengths of the parallel sides (\(a = 29\) cm, \(b = 21\) cm), we can calculate the area of the trapezium:
\(\text{Area} = \frac{1}{2} \times (a + b) \times h\)
\(\text{Area} = \frac{1}{2} \times (29 + 21) \times 7.5\)
\(\text{Area} = \frac{1}{2} \times (50) \times 7.5\)
\(\text{Area} = 25 \times 7.5\)
\(\text{Area} = 187.5 \text{ square cm}\)
So, the area of the isosceles trapezium is 187.5 square cm.
Revision Table: Trapezium Area Calculation
| Concept | Formula/Method | Value Used |
|---|---|---|
| Sum of parallel sides | \(a + b\) | \(29 + 21 = 50\) cm |
| Base of right triangle | \(\frac{a - b}{2}\) | \(\frac{29 - 21}{2} = 4\) cm |
| Pythagorean Theorem | \(c^2 = x^2 + h^2\) | \(8.5^2 = 4^2 + h^2\) |
| Height (\(h\)) | \(\sqrt{c^2 - x^2}\) | \(\sqrt{8.5^2 - 4^2} = \sqrt{72.25 - 16} = \sqrt{56.25} = 7.5\) cm |
| Area of Trapezium | \(\frac{1}{2} \times (a + b) \times h\) | \(\frac{1}{2} \times 50 \times 7.5 = 187.5\) cm\(^2\) |
Additional Information: Properties of Isosceles Trapezium
- An isosceles trapezium is a quadrilateral with one pair of parallel sides (bases) and the non-parallel sides being equal in length.
- The base angles of an isosceles trapezium are equal. That is, the two angles along one parallel base are equal, and the two angles along the other parallel base are also equal.
- The diagonals of an isosceles trapezium are equal in length.
- An isosceles trapezium has a line of symmetry that passes through the midpoints of the parallel sides.
Understanding these properties helps in solving geometry problems involving trapeziums, especially when calculating lengths, angles, or areas.
Similar Questions
If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:
ABCD is a cyclic quadrilateral. The bisectors of the angles A, B, C and D cut the circle at P, Q, R and S respectively. What is ∠PQR + ∠RSP equal to?
What is the magnitude (in radian) of the interior angle of a regular pentagon?
If the sum of all interior angles of a regular polygon is twice the sum of all its exterior angles then the polygon is
OABC is a rhombus whose three vertices lie on a circle with centre at O. If the area of the rhombus is 32√3 square cm, then what is the radius of the circle ?
ABCD is a square field with AB = x. A vertical pole OP of height 2x stands at the centre O of the square field. If ∠APO = θ, then what is cot θ equal to ?
A square is inscribed in a right angled triangle with legs p and q and has a common right angle with triangle. The diagonal of the square is given by
ABCDA is a con-cyclic quadrilateral of a circle ABCD with radius r and centre at O. If AB is the diameter and CD is parallel and half of AB and if the circle completes one rotation about the centre O, then the locus of the middle point of CD is a circle of radius:
The diagonals of a rhombus are of length 20 cm and 48 cm. What is the length of a side of the rhombus?
The area of a regular hexagon of side ‘a’ is equal to
Important Questions from Quadrilaterals
The ratio between the length and breadth of a rectangular park is 3 : 2. If a man cycling along the boundary at the speed of 12 km per hour completes one round in 8 minutes, then the area of the park in square meter will be
The side of a rhombus is 26 cm. The length of one of its diagonals is 20 cm. The sum of the lengths of the diagonals of this rhombus is equal to the perimeter of a rectangle. If the difference between the length and breadth of the rectangle is 6 cm, then what is the area of the rectangle?
PQRS is a cyclic quadrilateral. If ∠P is 4 times ∠R, and ∠S is 3 times ∠Q, then the average of ∠Q and ∠R is:
ABCD is a trapezium in which AB || DC and DC is perpendicular to BC. If ∠DAB = 110°, then ∠ABC - ∠ADC =_____.
If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:
