What is the area of a rhombus with:

Side=8 cm
Acute angle between one pair of sides=30 degrees

Please don't use trigonometry as I'm in Class 7.

Thanks in advance.

Very nonsense

To find the area of a rhombus, you can use the formula:

Area = (diagonal1 * diagonal2) / 2

However, since we don't have the diagonals, we can use a different approach. In a rhombus, the diagonals bisect each other into four congruent right triangles. So, if we find the length of one of these right triangles' height (altitude), we can calculate the area of the rhombus.

To find the altitude, follow these steps:

Step 1: Divide the rhombus into two congruent right triangles by drawing a line perpendicular to one side.

Step 2: Since the acute angle between one pair of sides is given as 30 degrees, we can use this information to divide the 8 cm side into two equal lengths.

Step 3: Now, take the divided side length (4 cm) as the base of the right triangle and the height (altitude) as the unknown value we need to find.

Step 4: We have a right triangle with the base (4 cm) and the acute angle (30 degrees). To find the height, we can use the trigonometric function tangent (tan).

Step 5: The tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the height of the right triangle is the opposite side, and the base (4 cm) is the adjacent side.

Step 6: Now, use the tangent function to find the height (altitude):
tan(30 degrees) = height / 4 cm

Rearranging the equation:
height = 4 cm * tan(30 degrees)

Step 7: Calculate the height:
height = 4 cm * tan(30 degrees)
height ≈ 4 cm * 0.577
height ≈ 2.308 cm

Step 8: Now that we have the height of the right triangle, we can double it to find the height of the rhombus. Since the diagonals of a rhombus bisect each other, the height of the right triangle is half of the rhombus's height.

Height of the rhombus = 2 * 2.308 cm
Height of the rhombus ≈ 4.616 cm

Step 9: Finally, calculate the area of the rhombus using the formula:
Area = (side length * height)
Area = 8 cm * 4.616 cm
Area ≈ 36.928 cm²

Therefore, the approximate area of the rhombus is 36.928 square cm.

To find the area of a rhombus, you can use the formula: Area = (diagonal1 * diagonal2) / 2.

However, since you don't know the diagonals of the rhombus, we'll need to find them first.

In a rhombus, the diagonals bisect each other at right angles, dividing it into four congruent right-angled triangles. The given angle of 30 degrees represents one of these right-angled triangles.

Here's how we can find the diagonals without using trigonometry:

Step 1: Draw the rhombus. Label one side as 8 cm.

Step 2: Since the diagonals bisect each other at right angles, draw a perpendicular line segment from one side of the rhombus to the other side, passing through the midpoint of the side. This perpendicular line segment represents the height of the right-angled triangle.

Step 3: Since the rhombus is symmetrical, the perpendicular line segment is also a bisector of the 30-degree angle. This means that the triangle is isosceles, with the base angles each measuring 75 degrees (180 - 30 = 150 degrees divided by 2).

Step 4: Using a protractor or a compass, construct the base angles of 75 degrees on either side of the perpendicular line segment.

Step 5: Draw the diagonal from the vertex of the 30-degree angle to the opposite vertex of the rhombus. This diagonal represents the hypotenuse of the right-angled triangle.

Step 6: Since the triangle is isosceles, the two sides adjacent to the 75-degree angles are congruent. We know that one side is 8 cm. Bisect it and label the midpoint as point A.

Step 7: Using a ruler, measure the length from point A to the point where the diagonal intersects the rhombus's side (label this point as B). This length represents half of the diagonal.

Step 8: Double the length measured in step 7 to find the full length of the diagonal.

Step 9: Repeat steps 6-8 to find the length of the other diagonal.

Step 10: Now that you have the diagonals, you can find the area of the rhombus by applying the formula: Area = (diagonal1 * diagonal2) / 2.

I hope this explanation helps you find the area of the rhombus without using trigonometry.

In your sketch that you made, you will notice that the diagonals right-bisect each other, creating 4 congruent right-angled triangles.

Each of them has angles 30:60:90 degrees
the ratio of sides for such a triangle is 1:√3:2
where 2 is the hypotenuse
Our hypotenuse is 8
so the other sides are 4 and 4√3
the area of one of them = (1/2)base x height
= (1/2)(4)(4√3)
= 8√3
= appr ....