Find the area of a pentagon ABCDE with verticles A(0,4),B(3,2),C(3,-1), D(-3,-1) and E (-3,2)
If the pentagon's boundary does not cross itself, I would use the surveyor's formula:
1. lay out the coordinates in order, repeat the first at the end.
A(0,4)
B(3,2)
C(3,-1)
D(-3,-1)
E (-3,2)
A(0,4)
2. Cross multiply between each successive pair and sum.
For A and B, we get 0*2-3*4=-12
For B and C, we get 3*-1 - 3*2 = -9
Continue until the last pair.
The sum of all 5 products is twice the area. If the vertices have been named clockwise, the sum will be negative.
3. Divide the sum by 2. If the sum is negative, write down only the positive value.
For the example:
vertex sum
A -12
B -9
C -6
D -9
E -12
Sum -48
Area = 48/2=24
Check my calculations.
To find the area of a pentagon, we can split it into triangles and calculate the area of each triangle.
Let's calculate the area of triangle ABD first.
1. Calculate the length of side AB using the distance formula:
AB = √[(x2 - x1)² + (y2 - y1)²]
AB = √[(3 - 0)² + (2 - 4)²]
AB = √[9 + 4]
AB = √13
2. Calculate the length of the altitude from point D to line AB:
From the given points, we can see that the y-coordinate of point D is the same as that of point B (both are 2).
3. Calculate the area of triangle ABD using the formula:
Area = (1/2) * base * height
Area = (1/2) * AB * altitude
Area = (1/2) * √13 * 2 = √13
Next, let's calculate the area of triangle BCD.
1. Calculate the length of side BC using the distance formula:
BC = √[(x2 - x1)² + (y2 - y1)²]
BC = √[(3 - 3)² + (-1 - 2)²]
BC = √[0 + 9]
BC = √9 = 3
2. Calculate the length of the altitude from point D to line BC:
From the given points, we can see that the x-coordinate of point D is the same as that of point C (both are -3).
3. Calculate the area of triangle BCD using the formula:
Area = (1/2) * base * height
Area = (1/2) * BC * altitude
Area = (1/2) * 3 * 6 = 9
Now, let's calculate the area of triangle CDE.
1. Calculate the length of side CD using the distance formula:
CD = √[(x2 - x1)² + (y2 - y1)²]
CD = √[(-3 - 3)² + (-1 - (-1))²]
CD = √[36 + 0]
CD = √36 = 6
2. Calculate the length of the altitude from point E to line CD:
From the given points, we can see that the y-coordinate of point E is the same as that of point C (both are -1).
3. Calculate the area of triangle CDE using the formula:
Area = (1/2) * base * height
Area = (1/2) * CD * altitude
Area = (1/2) * 6 * 3 = 9
Finally, adding up the areas of all three triangles, we get the area of the pentagon ABCDE:
Area of pentagon ABCDE = Area of triangle ABD + Area of triangle BCD + Area of triangle CDE
= √13 + 9 + 9
= √13 + 18 square units