Question
The mayor is trying to decide if she wants a triangular sitting area or a parallelogram sitting area. The formulas for area of a triangle and area of a parallelogram are similar. Describe how to calculate the area of each shape and how the area of a triangle is related to that of a parallelogram.
Answers
GPT-4o mini
To calculate the area of both a triangle and a parallelogram, you can use the following formulas:
### Area of a Parallelogram
The formula for the area \( A \) of a parallelogram is given by:
\[
A = b \times h
\]
where \( b \) is the length of the base, and \( h \) is the height (the perpendicular distance from the base to the opposite side).
### Area of a Triangle
The formula for the area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times b \times h
\]
where \( b \) is the length of the base, and \( h \) is the height (the perpendicular distance from the base to the apex of the triangle).
### Relationship Between the Areas
The area of a triangle is directly related to the area of a parallelogram. Specifically, when you draw a triangle with the same base and height as a parallelogram, the area of the triangle is exactly half of the area of the parallelogram.
To visualize this relationship, you can think of a parallelogram as being composed of two identical triangles. If you take one of those triangles and separate it from the parallelogram, you'll see that the area of that triangle is half the area of the entire parallelogram:
\[
\text{Area of Triangle} = \frac{1}{2} \times \text{Area of Parallelogram}
\]
Therefore, if the mayor is deciding between a triangular sitting area and a parallelogram sitting area, each with the same base and height, the area of the triangular sitting area will be half the area of the parallelogram sitting area. This is an important consideration when planning the space and determining how many people can comfortably sit in each area.
### Area of a Parallelogram
The formula for the area \( A \) of a parallelogram is given by:
\[
A = b \times h
\]
where \( b \) is the length of the base, and \( h \) is the height (the perpendicular distance from the base to the opposite side).
### Area of a Triangle
The formula for the area \( A \) of a triangle is given by:
\[
A = \frac{1}{2} \times b \times h
\]
where \( b \) is the length of the base, and \( h \) is the height (the perpendicular distance from the base to the apex of the triangle).
### Relationship Between the Areas
The area of a triangle is directly related to the area of a parallelogram. Specifically, when you draw a triangle with the same base and height as a parallelogram, the area of the triangle is exactly half of the area of the parallelogram.
To visualize this relationship, you can think of a parallelogram as being composed of two identical triangles. If you take one of those triangles and separate it from the parallelogram, you'll see that the area of that triangle is half the area of the entire parallelogram:
\[
\text{Area of Triangle} = \frac{1}{2} \times \text{Area of Parallelogram}
\]
Therefore, if the mayor is deciding between a triangular sitting area and a parallelogram sitting area, each with the same base and height, the area of the triangular sitting area will be half the area of the parallelogram sitting area. This is an important consideration when planning the space and determining how many people can comfortably sit in each area.