Given:
Perimeter of the rhombus = 40 cm
Measure of an angle = 60°

To find:
Area of the rhombus

Solution:
Step 1: Finding the length of one side
The perimeter of a rhombus is the sum of the lengths of all four sides. Since a rhombus has all four sides equal, we can divide the perimeter by 4 to find the length of one side.
Perimeter of the rhombus = 4s, where s is the length of one side.
Given perimeter = 40 cm
40 = 4s
Dividing both sides by 4, we get:
s = 10 cm

Step 2: Finding the area of the rhombus
The area of a rhombus can be found using the formula: Area = (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals.
In a rhombus, the diagonals bisect each other at right angles. This means that each diagonal divides the rhombus into two congruent right-angled triangles.
Since the measure of one angle in the rhombus is 60°, each right-angled triangle has an angle of 30°.
Let's label the diagonals as d1 and d2. The diagonals d1 and d2 form two congruent right-angled triangles.

Step 2.1: Finding the length of the diagonals
In a rhombus, the diagonals are perpendicular bisectors of each other. This means that they form right angles with each other and divide each other into two equal parts.
Since the diagonals bisect each other, they divide the rhombus into four congruent right-angled triangles.
Let's call the point of intersection of the diagonals O.

Step 2.2: Finding the length of one-half of the diagonals
In each right-angled triangle, the side opposite to the 30° angle is half the length of the diagonal.
Using trigonometry, we can find the length of one-half of the diagonals.
sin(30°) = opposite/hypotenuse
sin(30°) = (s/2) / d1/2
s/2 = d1/2 * sin(30°)
s/2 = (10/2) * (1/2)
s/2 = 5/2
s/2 = 2.5 cm

Step 2.3: Finding the length of the diagonals
Since the diagonals bisect each other at right angles, the length of the diagonals can be found using the Pythagorean theorem.
In each right-angled triangle, the hypotenuse is the length of the diagonal.
Using Pythagoras, we can find the length of the diagonal.
d1^2 = (s/2)^2 + (s/2)^2
d1^2 = (2.5)^2 + (2.5)^2
d1^2 = 6.25 + 6.25
d1^2 = 12.5
d1 = √12.5
d1 ≈ 3.54 cm