The diagonal of a parallelogram are 6cm and 8cm long and they intersect at an angle of 55°.calculate the area if the parallelogram
The diagonals of a parallelogram bisect each other and form 4 triangles of equal area
So looking at one these:
area of triangle = (1/2)(3)(4)sin55° = 4.9149...
So the area of the parallelogram is 4(4.9149..) = appr 19.66 cm^2
Well, well, well, we have ourselves a parallelogram in the house! Let's calculate its area and have some fun with math, shall we?
First things first, we need to find the length of one side of the parallelogram. Lucky for us, the diagonals of a parallelogram bisect each other, so we can split the larger diagonal into two equal parts.
Using some trigonometric magic, we can find that each half of the larger diagonal is equal to:
(1/2) * 8cm = 4cm
Now that we have one side of the parallelogram, we can use the formula for finding the area of a parallelogram: Area = base * height.
The base of our parallelogram is one side, which we just calculated to be 4cm.
To find the height, we need a little more math humor... Pythagorean humor, to be precise! Using the Pythagorean theorem, we can find the height as the square root of:
(6cm)^2 - (4cm)^2
Solving this equation will give us the height of the parallelogram. Once we have that, we can plug it into our area formula.
So, grab your math cap and calculator, and let's solve this riddle together!
To calculate the area of a parallelogram, you can use the formula: Area = base × height.
In this case, the diagonals of the parallelogram act as the base and height.
Step 1: Determine the lengths of the diagonals.
Diagonal 1 = 6 cm
Diagonal 2 = 8 cm
Step 2: Determine the angle between the diagonals.
Angle = 55°
Step 3: Find the length of the base and the height using the diagonals and angle.
Let the length of the base be "b" and the length of the height be "h".
Using the Law of Cosines, we can find the length of the base:
b² = 6² + 8² - (2 × 6 × 8 × cos(55°))
b² = 36 + 64 - (96 × cos(55°))
b² = 100 - (96 × cos(55°))
b ≈ √(100 - 96 × cos(55°))
b ≈ √(100 - 96 × 0.5736)
b ≈ √(100 - 55.1376)
b ≈ √(44.8624)
b ≈ 6.7 cm
Using the sine of the angle, we can find the length of the height:
h = 8 × sin(55°)
h = 8 × 0.819
h ≈ 6.55 cm
Step 4: Calculate the area of the parallelogram:
Area = base × height
Area = 6.7 cm × 6.55 cm
Area ≈ 43.885 cm²
Therefore, the area of the parallelogram is approximately 43.885 square centimeters.
To calculate the area of a parallelogram, you need the length of one of its sides and the length of the perpendicular distance (height) between that side and its opposite side. However, in this case, we are given the lengths of the diagonals instead of the sides. So, we'll have to calculate the lengths of the sides using the diagonals and the angle between them.
To find the lengths of the sides, we can use the law of cosines. The law of cosines states that in any triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle.
Let's label the diagonals of the parallelogram as d1 = 6 cm and d2 = 8 cm, and the angle between them as θ = 55°.
Using the law of cosines, we can find the lengths of the sides:
a² = d₁² + d₂² - 2 * d₁ * d₂ * cos(θ)
a² = 6² + 8² - 2 * 6 * 8 * cos(55°)
a² = 36 + 64 - 96 * cos(55°)
a² = 100 - 96 * cos(55°)
Now, let's calculate a:
a = √(100 - 96 * cos(55°))
a ≈ √(100 - 96 * 0.5736)
a ≈ √(100 - 55.0656)
a ≈ √44.9344
a ≈ 6.709 cm
We have found the length of one side of the parallelogram to be approximately 6.709 cm.
Next, we need to calculate the area of the parallelogram using the formula:
Area = base * height
The base of the parallelogram is one of its sides, which we already found to be approximately 6.709 cm.
To find the height, we can use the formula:
height = d₂ * sin(θ)
Substituting the given values:
height = 8 * sin(55°)
height ≈ 8 * 0.8192
height ≈ 6.5536 cm
Finally, we can calculate the area:
Area = 6.709 cm * 6.5536 cm
Area ≈ 43.9375 cm²
Therefore, the approximate area of the parallelogram is 43.9375 cm².