Question

a parallelogram is shown with some of its dimensions.

if the area of the parallelogram is 30 cm², what is the length of the height, h?
h = \square cm

Explanation:

Step1: Recall area formula of parallelogram

The area \( A \) of a parallelogram is given by \( A = base \times height \). Here, the base corresponding to the height \( h \) should be the side that is perpendicular to \( h \). Wait, actually, looking at the diagram, the side of length 5 cm – no, wait, the height here: Wait, the area is \( 30 \, \text{cm}^2 \), and we need to find \( h \). Wait, maybe the base is 5 cm? Wait, no, let's check. Wait, the formula is \( \text{Area} = base \times height \). Let's assume that the base is 5 cm? Wait, no, maybe the height is related to the side. Wait, no, the diagram shows a parallelogram with one side 5 cm, and the height \( h \) corresponding to that side? Wait, no, the area is \( 30 = 5 \times h \)? Wait, no, wait, maybe I got the base wrong. Wait, no, let's re-express. Wait, the area of a parallelogram is \( \text{Area} = base \times height \). Let's denote the base as \( b \) and height as \( h \), so \( A = b \times h \). Here, if the area is 30, and let's see the given side: the side labeled 5 cm – maybe that's the base? Wait, no, maybe the height is \( h \), and the base is 5 cm? Wait, no, let's solve for \( h \). Wait, if \( \text{Area} = base \times height \), then \( h = \frac{\text{Area}}{base} \). Wait, but what's the base? Wait, the diagram: the parallelogram has a side of 5 cm, and the height \( h \) is the distance between the two sides of length 5 cm? Wait, no, maybe the base is 5 cm, so \( 30 = 5 \times h \), then \( h = 30 / 5 = 6 \)? Wait, no, that can't be. Wait, maybe the base is different. Wait, no, maybe I misread. Wait, the problem says "the length of the height, \( h \)". Wait, let's check again. The area of a parallelogram is \( \text{Area} = base \times height \). Let's suppose that the base is the side to which the height \( h \) is perpendicular. Wait, in the diagram, the side with length 5 cm – maybe the height is \( h \), and the base is 5 cm? Then \( 30 = 5 \times h \), so \( h = 30 / 5 = 6 \)? Wait, no, that seems off. Wait, maybe the base is another side. Wait, no, maybe the height is \( h \), and the base is 5 cm. Wait, let's do the calculation. If \( \text{Area} = base \times height \), then \( h = \frac{\text{Area}}{base} \). So if the area is 30, and the base is 5 cm, then \( h = 30 / 5 = 6 \)? Wait, but that seems too big. Wait, no, maybe the base is 5 cm, and the height is \( h \), so \( 30 = 5 \times h \), so \( h = 6 \). Wait, but let's confirm. Wait, the formula for the area of a parallelogram is indeed \( \text{Area} = base \times height \), where height is the perpendicular distance between the two bases (the parallel sides). So if one of the parallel sides is 5 cm (the base), then the height \( h \) is the distance between them, so \( 30 = 5 \times h \), so \( h = 30 / 5 = 6 \). Wait, but that would mean \( h = 6 \) cm. Wait, but let's check again. Wait, maybe the base is not 5 cm. Wait, the diagram shows a parallelogram with a side of 5 cm, and the height \( h \) is the other dimension. Wait, no, maybe I made a mistake. Wait, let's re-express: \( \text{Area} = base \times height \). So \( h = \text{Area} / base \). If the area is 30, and the base is 5, then \( h = 30 / 5 = 6 \). So the height \( h \) is 6 cm.

Step1: Identify the formula for the area of a parallelogram

The area \( A \) of a parallelogram is given by the formula \( A = \text{base} \times \text{height} \).

Step2: Substitute the known values into the formula

We know the area \( A = 30 \, \text{cm}^2 \) and the base (assuming the base is \( 5 \, \text{cm} \), as per the diagram) is \( 5 \, \text{cm} \). Substituting into the formula:
\( 30 = 5 \times h \)

Step3: Solve for \( h \)

To find \( h \), divide both sides of the equation by \( 5 \):
\( h = \frac{30}{5} = 6 \)

Answer:

\( \boxed{6} \)