Find the area of the polygon with vertices at (1,3) , (7,3) ,(7,7) and (4,7)?
Two ways to do it.
If you plot the points, it shows a trapezium with parallel sides 6 (=7-1) and 3(=7-4) units long, and the height is 4 (=7-3).
So the area is (6+3)*4/2=18.
If you would like to calculate the area of a polygon given in corner coordinates, you can list the points in sequence like
a,b
c,d
e,f
g,h
a,b
and repeat a,b at the bottom.
Calculate the crossed sum for each successive pair of coordinates, as in:
ad-bc + cf-de + eh-fg + gb-ha
and divide the sum by 2.
The sum may be negative depending the direction of the points. So just take the absolute value.
For the given points, list:
1,3
7,3
7,7
4,7
1,3 (repeat of first point)
Products:
1*3-3*7 + 7*7-3*7 + 7*7-7*4 + 4*3-7*1
=-18 + 28 + 21 + 5
= 36
Divide by 2 to get 18.
It may look complicated, but when the figure does not happen to be a known shape, the second method is more systematic, and easier.
18units
To find the area of a polygon with given vertices, you can use the Shoelace Formula. The Shoelace Formula calculates the area of a polygon by multiplying the sums of the products of the coordinates of consecutive vertices, with the first vertex appearing twice.
The formula is as follows:
Area = 1/2 * | (x1*y2 + x2*y3 + ... + xn-1*yn + xn*y1) - (y1*x2 + y2*x3 + ... + yn-1*xn + yn*x1) |
In this case, the given vertices are:
A = (1, 3)
B = (7, 3)
C = (7, 7)
D = (4, 7)
Let's calculate the area step by step using the Shoelace Formula:
1. Calculate the sum of the products of the x-coordinates of consecutive vertices, starting with the first vertex appearing twice:
x1 * y2 + x2 * y3 + x3 * y4 + x4 * y1 = 1*3 + 7*7 + 7*7 + 4*3 = 3 + 49 + 49 + 12 = 113
2. Calculate the sum of the products of the y-coordinates of consecutive vertices:
y1 * x2 + y2 * x3 + y3 * x4 + y4 * x1 = 3*7 + 7*7 + 7*4 + 3*1 = 21 + 49 + 28 + 3 = 101
3. Subtract the result from step 2 from the result from step 1:
113 - 101 = 12
4. Take the absolute value of the result:
|12| = 12
5. Multiply the absolute value by 1/2:
12 * 1/2 = 6
Therefore, the area of the polygon with vertices (1,3), (7,3), (7,7), and (4,7) is 6 square units.
To find the area of a polygon, you can use the shoelace formula, also known as the Gauss area formula. Here's how you can apply it to find the area of the given polygon with vertices at (1,3), (7,3), (7,7), and (4,7):
1. Write down the coordinates of the vertices in a clockwise or counterclockwise order. Let's use a counterclockwise order: (1, 3), (7, 3), (7, 7), and (4, 7).
Vertex A = (1, 3)
Vertex B = (7, 3)
Vertex C = (7, 7)
Vertex D = (4, 7)
2. Multiply the x-coordinate of each vertex by the y-coordinate of the next vertex, and subtract the product of the y-coordinate of the current vertex and the x-coordinate of the next vertex. Repeat this for all vertices in order.
A = (1 × 3) - (7 × 3) = -18
B = (7 × 3) - (7 × 7) = -14
C = (7 × 7) - (4 × 7) = 7
D = (4 × 7) - (1 × 3) = 25
3. Add up all the results from step 2 and take the absolute value:
|A + B + C + D| = |-18 + (-14) + 7 + 25| = 40
4. Divide the absolute value obtained in step 3 by 2 to get the area of the polygon:
Area = 40 / 2 = 20 square units
So, the area of the polygon with vertices at (1, 3), (7, 3), (7, 7), and (4, 7) is 20 square units.