Area & Perimeter in the Coordinate Plane | Geometry | Eat Pi

Summary

This educational video script offers a clear guide on calculating the area and perimeter of shapes and polygons in the coordinate plane. The presenter explains the formula for a triangle's area as half the base times the height and demonstrates using a grid. For the perimeter, the Pythagorean theorem is applied to find the hypotenuse. The script then shifts to rectangles, introducing the distance formula to find the length and width when the grid isn't aligned, and uses it to calculate the area and perimeter. The explanation is practical, with step-by-step instructions and a friendly tone, making it accessible for viewers to grasp geometrical concepts.

Takeaways

• 📐 The area of a triangle can be found using the formula: \( \frac{1}{2} \times \text{base} \times \text{height} \).
• 📏 When shapes are aligned with the grid, it's easier to determine the base and height for calculating area.
• 📍 The base and height of a triangle in the coordinate plane can be measured directly from the grid.
• 🔢 For the given triangle example, the base is 7 units and the height is 5 units, resulting in an area of 17.5 square units.
• 🧩 To find the perimeter of a triangle, sum all its sides, including the hypotenuse found using the Pythagorean theorem.
• 📐 The Pythagorean theorem states that \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
• 🔑 The hypotenuse in the triangle example is approximately 8.6 units, calculated using the Pythagorean theorem with sides 7 and 5.
• 📏 The perimeter of the triangle is the sum of its sides, which is approximately 20.6 units.
• 📐 For shapes not aligned with the grid, like a rectangle, use the distance formula to find the length and width.
• 📏 The distance formula is \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) for finding the distance between two points.
• 🔢 The area of a rectangle is calculated by multiplying its length by its width, which are not necessarily aligned with the grid.
• 📐 The perimeter of a rectangle can be found by adding twice the length and twice the width.
• 📝 The video provides a step-by-step guide on using the distance formula and the Pythagorean theorem for calculating areas and perimeters in the coordinate plane.

Q & A

What is the formula to calculate the area of a triangle?

The formula to calculate the area of a triangle is one half the base times the height (A = 1/2 * base * height).

How can you determine the base and height of a triangle on a coordinate grid?

On a coordinate grid, the base can be determined by counting the horizontal spaces from one vertex to another, and the height is the vertical distance from the base to the opposite vertex.

What is the area of the triangle in the video if the base is 7 units and the height is 5 units?

The area of the triangle would be 17.5 square units (A = 1/2 * 7 * 5 = 17.5 units²).

How do you find the perimeter of a polygon when you know the lengths of its sides?

To find the perimeter, you add up the lengths of all the sides of the polygon.

What is the Pythagorean theorem used for in the context of the video?

The Pythagorean theorem (a² + b² = c²) is used to find the length of the hypotenuse of a right-angled triangle when the lengths of the other two sides are known.

How is the distance formula applied to find the length of a side of a rectangle on a coordinate grid?

The distance formula (√((x2 - x1)² + (y2 - y1)²)) is used to calculate the distance between two points, which can be the length or width of a rectangle if the points are its vertices.

What is the formula for the area of a rectangle?

The formula for the area of a rectangle is the length times the width (A = length * width).

How do you calculate the perimeter of a rectangle?

The perimeter of a rectangle is calculated by adding twice the length and twice the width (P = 2 * length + 2 * width).

What is the approximate perimeter of the rectangle in the video with a length of 12.7 units and a width of 4.2 units?

The approximate perimeter of the rectangle is 33.8 units (P = 2 * 12.7 + 2 * 4.2 ≈ 33.8 units).

What is the distance formula and how is it used to find the area of a shape on a coordinate grid?

The distance formula is used to calculate the distance between two points in a coordinate plane. It is used to find the lengths of sides of shapes, which can then be used to calculate the area of polygons by applying the appropriate area formula for the shape.

How can you find the area of a shape that is not aligned with the grid on a coordinate plane?

For shapes not aligned with the grid, you can use the distance formula to find the lengths of the sides of the shape, and then apply the area formula specific to that shape (e.g., for a rectangle, use length * width).