I have 1000m of fencing to make a paddock, there is a 1km wall in which I can build the paddock onto.

I have to find the length or area (I'm not sure what) of a square shaped paddock, a triangle shaped paddock and half a circle. The notes I was given with them are -
Square - L=1000/3 to the power of 2 = 333 1/3
A= (333 1/3) to the power of 2

Triangle - Area = 1/2 ABsinC
C = 90 degrees
AB = 500
A = 490
B = 510

Half the circle - Pie R squared /2
R = (1000/pie)

To find the length or area of different shaped paddocks, let's break down the calculations for each shape separately.

Square-shaped paddock:
1. The total length of the fencing is given as 1000m.
2. To create a square-shaped paddock, you need to divide the total length of the fencing by 4 (since a square has four equal sides).
So, 1000m / 4 = 250m. This is the length of each side of the square.

The length of a single side of the square paddock is 250m.
To calculate the area of the square paddock:
1. Square the length of a single side (250m^2).
250m * 250m = 62,500m^2.
So, the area of the square paddock is 62,500 square meters.

Triangle-shaped paddock:
1. The total length of the fencing is given as 1000m.
2. Let's assume one side of the triangle (AB) is 500m.
3. Another side of the triangle (BC) is 500m (since it connects to the 1km wall).
4. The angle between the sides AB and BC is given as 90 degrees (C = 90 degrees).

To calculate the missing side length of the triangle:
1. Use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
BC^2 = AB^2 + AC^2 (using the labels from the triangle sides)
1000m^2 = 500m^2 + AC^2
AC^2 = 1000m^2 - 500m^2
AC^2 = 500,000m^2 - 250,000m^2
AC^2 = 250,000m^2

2. Take the square root of both sides to find the length of AC:
AC = sqrt(250,000m^2)
AC ≈ 500m

Now that we have all the side lengths of the triangle, we can calculate the area:
1. Use the formula for the area of a triangle: Area = 1/2 * base * height.
In this case, the base is AB (500m) and the height is AC (500m).
Area = 1/2 * 500m * 500m
Area = 125,000 square meters.

So, the area of the triangle-shaped paddock is 125,000 square meters.

Half circle-shaped paddock:
1. The total length of the fencing is given as 1000m.
2. To create a half circle-shaped paddock, you need to find the radius of the circle.

To calculate the radius:
1. Divide the total length of fencing by pi (approximately 3.14159) to get the circumference of the circle.
Radius = (1000m / pi) / 2
Radius ≈ 159.1549m

To find the area of a half circle:
1. Use the formula for the area of a circle: Area = pi * radius^2.
Half the circle has an area of only half the full circle, so divide the above result by 2.
Area = (pi * 159.1549m^2) / 2
Area ≈ 12605.9641 square meters.

So, the area of the half circle-shaped paddock is approximately 12605.9641 square meters.