what is the area of a rhombus with a 60 degree angle and sides 5 cm long? (round to the nearest 10th)

Why did the rhombus go to the gym? Because it wanted to work on its angles!

Now, to calculate the area of a rhombus, we use the formula:

Area = (diagonal 1 * diagonal 2) / 2

In this case, since we don't have the diagonals, we'll need to find another way.

Since we know the side length is 5 cm, we can find the length of the diagonals using trigonometry. In a rhombus, the diagonals bisect each other at right angles, which means we can find the diagonal lengths by using the 60-degree angle.

Using the sine formula:

sin(60°) = (d/2) / 5

Rearranging the equation, we get:

(d/2) = 5 * sin(60°)

(d/2) ≈ 5 * 0.866 (rounding to three decimal places)

d ≈ 10 * 0.866 (multiplying both sides by 2)

d ≈ 8.66 cm

Now that we have the diagonals, we can calculate the area of the rhombus:

Area = (diagonal 1 * diagonal 2) / 2

Area ≈ (8.66 * 5) / 2

Area ≈ 43.3 cm² (rounded to the nearest tenth)

So, the area of the rhombus is approximately 43.3 cm².

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

Area = (diagonal 1 * diagonal 2) / 2

However, we need to find the lengths of the diagonals first. Since we have a rhombus with a 60-degree angle, we can use the property that opposite angles in a rhombus are equal.

The diagonals of a rhombus bisect each other at right angles, forming two congruent right triangles. We can use the length of one side and the given angle to find the length of one diagonal.

To find the length of the diagonal, we can use the formula:

diagonal = 2 * side * sin(angle/2)

Given:
Side length (s) = 5 cm
Angle (θ) = 60 degrees

1. Convert the angle from degrees to radians:
angle in radians = angle in degrees * (π / 180)
= 60 * (π / 180)
= π / 3 radians

2. Calculate the length of one diagonal:
diagonal = 2 * side * sin(angle/2)
= 2 * 5 * sin(π / 6)
= 10 * sin(π / 6)
≈ 10 * 0.5
= 5 cm (rounded to the nearest 10th)

Since opposite diagonals of a rhombus are equal, we know that both diagonals are 5 cm in length.

3. Finally, calculate the area of the rhombus:
Area = (diagonal 1 * diagonal 2) / 2
= (5 * 5) / 2
= 25 / 2
= 12.5 cm² (rounded to the nearest 10th)

Therefore, the area of the rhombus is approximately 12.5 cm².

To find the area of a rhombus, we can use the formula A = (d₁ * d₂) / 2, where d₁ and d₂ are the lengths of the diagonals.

Since the length of the sides is given as 5 cm, we know that all four sides of the rhombus have the same length. In a rhombus, opposite angles are congruent, so if we have a 60-degree angle, we also have another 60-degree angle on the opposite side. Thus, we can conclude that all four angles of the rhombus are 60 degrees.

In a rhombus, the diagonals bisect each other at a 90-degree angle. This means that the diagonals formed by the 60-degree angles will create four congruent right triangles within the rhombus.

To find the length of the diagonals, we can use trigonometry. Let's call the length of the diagonals d₁ and d₂. In a right triangle, considering one 60-degree angle, the opposite side (half of the diagonal) is equal to the product of the hypotenuse (side of the rhombus) and the sine of the angle.

So, for each diagonal, we have:

d₁/2 = 5 * sin(60°)
d₂/2 = 5 * sin(60°)

To simplify the calculation, we can use the exact value of sin(60°) = √3 / 2:

d₁/2 = 5 * (√3 / 2)
d₂/2 = 5 * (√3 / 2)

Simplifying further:

d₁ = 5 * √3
d₂ = 5 * √3

Now, let's substitute these values back into the formula for the area of a rhombus:

A = (d₁ * d₂) / 2
A = (5 * √3 ) * (5 * √3) / 2

A = (25 * 3) / 2
A = 75 / 2
A = 37.5 cm²

Rounding the area to the nearest tenth, the area of the rhombus is approximately 37.5 cm².

recall that the area of rhombus is just

A = b*h
where b = base and h = height
we already have the base, which is 5 cm. we need to find for the height,, to get this, observe that if we draw the rhombus and draw a broken line representing the height, we see that a right triangle is form, particularly, a 30-60-90 triangle
to get height,
5 * sin 60 = 4.33 cm
**note that we use sin 60 here because 80 degrees is the opposite angle of the height
substituting:
A = b*h
A = 5*(4.33)
A = 21.65 sq cm

hope this helps~ :)