draw at least two shapes with an area of 6 square units and a perimeter of 12 units
To draw two shapes with an area of 6 square units and a perimeter of 12 units, we can consider a rectangle and a square as the two shapes.
Shape 1: Rectangle
Step 1: Let's start by assuming the width of the rectangle as 'w' units.
Step 2: Since the perimeter of a rectangle is calculated by adding the lengths of all four sides, we have 2w + 2l = 12, where 'l' is the length of the rectangle.
Step 3: We can rearrange the equation, 2w + 2l = 12, to get l in terms of w, which gives l = 6 - w.
Step 4: The area of the rectangle is given by A = l × w. Substituting l = 6 - w, we get A = w(6 - w).
Step 5: To find the dimensions that give an area of 6 square units, we need to solve the equation w(6 - w) = 6.
Step 6: Upon solving, we find that w = 2 or w = 3. We can substitute these values back into l = 6 - w to find the corresponding lengths.
Using w = 2, we have l = 6 - 2 = 4. So one rectangle with an area of 6 square units and a perimeter of 12 units can have dimensions 2 units by 4 units.
Using w = 3, we have l = 6 - 3 = 3. So another rectangle with an area of 6 square units and a perimeter of 12 units can have dimensions 3 units by 3 units.
Shape 2: Square
To draw a square with an area of 6 square units and a perimeter of 12 units, we can follow these steps:
Step 1: Let's assume the side length of the square as 's' units.
Step 2: The perimeter of a square is calculated by multiplying the side length by 4, so we have 4s = 12.
Step 3: Solving the equation, we find that s = 3.
Hence, a square with a side length of 3 units will have an area of 6 square units and a perimeter of 12 units.
To draw at least two shapes with an area of 6 square units and a perimeter of 12 units, we can consider different geometric shapes and solve for their dimensions. Let's explore two options: a rectangle and a right-angled triangle.
1. Rectangle:
The formula for the area of a rectangle is A = length × width, and the formula for the perimeter is P = 2(length + width).
Let's assume the length of the rectangle as L units and the width as W units. So, we have:
Area of the rectangle: A = L × W
Perimeter of the rectangle: P = 2(L + W)
Given that the area (A) is 6 square units and the perimeter (P) is 12 units, we can set up the following equations:
Equation 1: 6 = L × W
Equation 2: 12 = 2(L + W)
To find two possible solutions, we can substitute different values for L and solve for W.
- Let's assume L = 3 units and W = 2 units:
Equation 1: 6 = 3 × 2 (True)
Equation 2: 12 = 2(3 + 2) (True)
So, a rectangle with dimensions 3 units by 2 units satisfies the given conditions.
- Now let's assume L = 6 units and W = 1 unit:
Equation 1: 6 = 6 × 1 (True)
Equation 2: 12 = 2(6 + 1) (True)
Thus, a rectangle with dimensions 6 units by 1 unit also meets the given criteria.
You can draw these rectangles using the given dimensions.
2. Right-angled triangle:
The formula for the area of a right-angled triangle is A = 0.5 × base × height, and the formula for the perimeter is P = base + height + hypotenuse.
Considering the same principle as before, let's assume the base of the triangle as B units and the height as H units. The hypotenuse can be calculated using the Pythagorean theorem: hypotenuse = √(base² + height²).
Given that the area (A) is 6 square units and the perimeter (P) is 12 units, we can set up the following equations:
Equation 3: 6 = 0.5 × B × H
Equation 4: 12 = B + H + √(B² + H²)
To find two possible solutions, we can use trial and error or numerical methods to solve these equations. Here are two solutions:
- B = 4 units, H = 2 units:
Equation 3: 6 = 0.5 × 4 × 2 (True)
Equation 4: 12 = 4 + 2 + √(4² + 2²) (True)
- B = 3 units, H = 4 units:
Equation 3: 6 = 0.5 × 3 × 4 (True)
Equation 4: 12 = 3 + 4 + √(3² + 4²) (True)
These are two right-angled triangles that satisfy the given conditions. You can draw these triangles using the given dimensions.
Note: There may be other shapes or combinations of dimensions that fulfill the given conditions. These are just two examples to help you get started.