The length of the two diagonals are 2.5 cm and 3 cm, find the area of the quadrilateral.

Solution:

We divide the quadrilateral into two triangles, and by using Heron’s formula, we can calculate the area of triangles.

Heron's formula for the area of a triangle is: Area = √[s(s - a)(s - b)(s - c)]

Where a, b, and c are the sides of the triangle, and s = Semi-perimeter = Half the Perimeter of the triangle

Now, ABCD is quadrilateral shown in the figure

For ∆ABC, consider

AB2 + BC2 = 32 + 42 = 25 = 52

⇒ 52 = AC2

Since ∆ABC obeys the Pythagoras theorem, we can say ∆ABC is right-angled at B.

Therefore, the area of ΔABC = 1/2 × base × height

= 1/2 × 3 cm × 4 cm = 6 cm2

Area of ΔABC = 6 cm2

Now, In ∆ADC

we have a = 5 cm, b = 4 cm and c = 5 cm

Semi Perimeter: s = Perimeter/2

s = (a + b + c)/2

s = (5 + 4 + 5)/2

s = 14/2

s = 7 cm

By using Heron’s formula,

Area of ΔADC = √[s(s - a)(s - b)(s - c)]

= √[7(7 - 5)(7 - 4)(7 - 5)]

= √[7 × 2 × 3 × 2]

= 2√21 cm2

Area of ΔADC = 9.2 cm2 (approx.)

Area of the quadrilateral ABCD = Area of ΔADC + Area of ΔABC

= 9.2 cm2 + 6 cm2

= 15.2 cm2

Thus, the area of the quadrilateral ABCD is 15.2 cm2.

☛ Check: NCERT Solutions Class 9 Maths Chapter 12

Video Solution:

Class 9 Maths NCERT Solutions Chapter 12 Exercise 12.2 Question 2

Summary:

We have found that area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm is 15.2 cm2.

☛ Related Questions:

Get a Widget for this Calculator

π = pi = 3.1415926535898

Calculator Use

Calculate certain variables of a rhombus depending on the inputs provided. Calculations include side lengths, corner angles, diagonals, height, perimeter and area of a rhombus.

A rhombus is a quadrilateral with opposite sides parallel and all sides equal length. A rhombus whose angles are all right angles is called a square. A rhombus (or diamond) is a parallelogram with all 4 sides equal length.

Units: Note that units of length are shown for convenience. They do not affect the calculations. The units are in place to give an indication of the order of the calculated results such as ft, ft2 or ft3. Any other base unit can be substituted.

Rhombus Formulas & Constraints

Corner Angles: A, B, C, D

• A = C
• B = D
• A + B = 180° = π radians
• for a rhombus that is not a square,
  • 0 < A< 90° (0 < A < π/2)
  • 90° < B < 180° (π/2 < B < π)

Area: K

with A and B in radians,

K = ah = a2 sin(A) = a2 sin(B) = pq/2

Height: h

• h = ha = hb
• h = a sin(A) = a sin(B)

Diagonals: p, q

• p = a √( 2 + 2 cos(A) ) = a √( 2 - 2 cos(B) )
• q = a √( 2 - 2 cos(A) ) = a √( 2 + 2 cos(B) )
• p2 + q2 = 4a2

Perimeter: P

P = 4a

Rhombus Calculations:

The following formulas, based on those above, are used within this calculator for the selected calculation choices.

• Calculate B, C, D | Given A
  Given angle A calculate angles B, C and D
• Calculate A, C, D | Given B
  Given angle B calculate angles A, C and D
• Calculate a | Given P
  Given the perimeter calculate side a
• Calculate P | Given a
  Given side length a calculate the perimeter
• Calculate B, p, q, h, P, K | Given a, A
  Given side length a and angle A calculate the diagonals, perimeter, height, area and angles B, C and D
  • p = √( 2a2 + 2a2 cos(A) )
  • q = √( 2a2 - 2a2 cos(A) )
  • P = 4a
  • h = a sin(A)
  • K = ah
  • B = 180° - A
  • C = A
  • D = B
• Calculate A, B, q, h, P, K | Given a, p
  Given side length a and diagonal p calculate diagonal q, perimeter, height, area and angles A, B, C and D
  • A = arccos( 1 - (p2 / 2a2) )
  • q = √( 2a2 - 2a2 cos(A) )
  • h = a sin(A)
  • P = 4a
  • K = a2 sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate A, B, p, h, P, K | Given a, q
  Given side length and diagonal q calculate diagonal p, perimeter, height, area and angles A, B, C and D
  • A = arccos( 1 + (q2 / 2a2) )
  • p = √( 2a2 + 2a2 cos(A) )
  • h = a sin(A)
  • P = 4a
  • K = a2 sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate A, B, p, q, P, K | Given a, h
  Given side length and height calculate the diagonals, perimeter, area and angles A, B, C and D
  • A = arcsin(h/a)
  • p = √( 2a2 + 2a2 cos(A) )
  • q = √( 2a2 - 2a2 cos(A) )
  • P = 4a
  • K = a2 sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate A, B, p, q, h, P | Given a, K
  Given side length and area calculate the diagonals, perimeter, height and angles A, B, C and D
  • A = arcsin(K/a2)
  • p = √( 2a2 + 2a2 cos(A) )
  • q = √( 2a2 - 2a2 cos(A) )
  • h = a sin(A)
  • P = 4a
  • B = 180° - A
  • C = A
  • D = B
• Calculate a, A, B, p, q, P | Given K, h
  Given area and height calculate side length, diagonals, perimeter and angles A, B, C and D
  • a = K / h
  • P = 4a
  • A = arcsin(K/a2)
  • p = √( 2a2 + 2a2 cos(A) )
  • q = √( 2a2 - 2a2 cos(A) )
  • B = 180° - A
  • C = A
  • D = B
• Calculate a, A, B, q, h, P | Given K, p
  Given diagonal p and area calculate the perimeter, height, side length, diagonal q and angles A, B, C and D
  • q = 2K / p
  • a = √(p2 + q2) / 2
  • P = 4a
  • A = arccos( 1 - (p2 / 2a2) )
  • h = a sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate a, A, B, p, h, P | Given K, q
  Given diagonal q and area calculate the perimeter, height, side length, diagonal p and angles A, B, C and D
  • p = 2K / q
  • a = √(p2 + q2) / 2
  • P = 4a
  • A = arccos( 1 + (q2 / 2a2) )
  • h = a sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate a, B, p, q, P, K | Given A, h
  Given angle A and height calculate side a, angles B, C and D, diagonals, perimeter and area
  • a = h / sin(A)
  • P = 4a
  • p = √( 2a2 + 2a2 cos(A) )
  • q = √( 2a2 - 2a2 cos(A) )
  • K = a2 sin(A)
  • B = 180° - A
  • C = A
  • D = B
• Calculate a, A, B, h, P, K | Given p, q
  Given diagonal p and diagonal q calculate the side length, angles A, B, C and D, height, perimeter, and area
  • a = √(p2 + q2) / 2
  • P = 4a
  • K = (p * q) / 2
  • A = arcsin( K / a2)
  • B = 180° - A
  • C = A
  • D = B
  • h = a sin(A)

References

Zwillinger, Daniel (Editor-in-Chief). CRC Standard Mathematical Tables and Formulae, 31st Edition New York, NY: CRC Press, p. 323, 2003.

Weisstein, Eric W. "Rhombus." From MathWorld--A Wolfram Web Resource. Rhombus.

Find out the length of square diagonal with our diagonal of a square calculator. Check out below how to find the diagonal of a square formula or simply give our tool a try - you won't be disappointed.

Diagonal is a line segment that joins two non-neighboring vertices. Each quadrilateral has two diagonals, a square too. Its diagonals are:

• equal in length
• perpendicular bisectors of each other
• bisectors of square angles

Each diagonal divides the square into two congruent isosceles right triangles - 45 45 90 special right triangles. Such triangle has a half of the area of a square, its legs are square sides and hypotenuse equals to the length of the diagonal of a square.

To calculate the diagonal of a square, multiply the length of the side by the square root of 2:

d = a√2

So, for example, if the square side is equal to 5 in, then the diagonal is 5√2 in ≈ 7.071 in. Type that value into the diagonal of a square calculator to check it yourself!

Where does this equation come from? You can derive this diagonal of square formula e.g. from Pythagorean theorem.

1. As we noticed before, the diagonal of a square divides the square into two congruent right triangles.

2. Legs are square sides and the length of diagonal of the square is the hypotenuse. Given the side length of the square:

   a² + a² = diagonal²

   diagonal = √(a² + a²) = √(2 * a²) which simplifies to

   diagonal = a√2

If you don't have the side of a square given, use other formulas:

• d = √(2*area) if area is given
• d = (perimeter/4)*√2 knowing square perimeter.

With our calculator it's a piece of cake!

1. Type the given value into a proper box. Let's say we know that the square side is equal to 8 in.
2. Diagonal of a square calculator displays the result! In our case, the diagonal is 15.556 in.
3. If you want to check the result in a different unit, click on the unit name and select the one that meets your needs. For instance, 15.556 in is 39.51 cm and around 1 ft 3 in.
4. The diagonal of a square calculator works the other way round as well - knowing the diagonal, you can calculate the square side.

Now you're the expert and you know exactly how to find the diagonal of a square given square sides. However, if you don't have it provided, use this general square calculator where you can type the area or perimeter, and the tool will find the diagonal as well.