**What Is The Area Of A Rectangle With Vertices**

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**What are the vertices of a rectangle?**

While a vertex is the highest point or apex, in geometry it is each angular point of a polygon, polyhedron, or other figure.

Given a rectangle ABCD, each of the four vertices is formed at its corners; ABC, BCD, CDA, DAB.

**How To Find The Area Of A Rectangle With Vertices**

According to Gary Ward (MaEd in Education & Mathematics, Austin Peay State University, 1997), pick any vertex of a rectangle and call it P1(x1, y1)

Pick the two adjacent vertices of the rectangle and call them P2(x2, y2) and P3(x3, y3).

The area of the rectangle becomes the distance between P1 and P2 times the distance between P1 and P3.

Thus, Area Of The Rectangle With Vertices = sqrt((x2-x1)^2 + (y2-y1)^2) * sqrt((x3-x1)^2 + (y3-y1)^2)

**Example:** How do you find the area of the rectangle with vertices A(-5,1), B(-3,-1), C(3,5), D(1,7)?

**Solution By Alan P:**

The Area of the □ ABCD = 24 square units

This can be explained in bits given the graphical representation below:

The area of □ ABCD could be calculated in several ways. One would be to divide the rectangle into two triangles, then use the Pythagorean Theorem to find the distances between the points, and finally use Heron’s formula to find the areas of the two triangles.

However, I think the method below is easier.

Enclose the rectangle ABCD in another rectangle □PQRS with sides parallel to the x and y axis as seen below:

Area of □ABCD = Area of □PQRS – (Area of △DPA +

Area of △AQB + Area of △BRC + Area of △CSD)

Area of □PQRS = 8 × 8 = 64

Area Of △DPA = Area Of △BRC = 6 × 6/2 = 18

Area Of △AQB = Area Of △CSD = 2 × 2/2 = 2

Therefore

Area Of □PQRS = 64 −(18 + 2 + 18 + 2) = 24 Square Units.

**Practice Questions**

- What is the area of a rectangle with vertices at (7 3)
- Explain how to use a coordinate plane to find the area of a rectangle with vertices
- What is the area of a rectangle with vertices at (4 3)
- What is the area of a rectangle with vertices at (2 3)
- What is the area of a rectangle with vertices at (2 3) (7 3)
- What is the area of a rectangle with vertices at (-6 3)
- What is the area of a rectangle with vertices 8 2
- Find the area of a rectangle with vertices calculator
- What is the area of a rectangle with vertices at (-4 0)
- How to find the area and perimeter of a rectangle with vertices.

Also, see **How To Find The Surface Area Of A Rectangular Prism** and also check out **Powerful Prayer Points For Miracles** here. Cheers!