Geometric shapes and their area and perimeter formulas.

Problem Analysis:

The task involves understanding and summarizing the properties of various geometric shapes, including their area and perimeter formulas. The table provided lists eight different shapes along with their names, area formulas, and perimeter formulas. The goal is to ensure that the information in the table is accurate and complete.

Step-by-Step Solution:

#### 1. Triangle
- Shape: Triangle
- Area: $\frac{1}{2} \times b \times h$
- Explanation: The area of a triangle is given by half the product of its base ($b$) and height ($h$).
- Perimeter: Sum of all three sides
- Explanation: The perimeter of a triangle is the sum of the lengths of its three sides.

#### 2. Equilateral Triangle
- Shape: Equilateral Triangle
- Area: $\frac{\sqrt{3}}{4} a^2 \left(h = \frac{\sqrt{3}}{2} a\right)$
- Explanation: For an equilateral triangle with side length $a$, the height $h$ can be derived using the Pythagorean theorem in one of the 30-60-90 right triangles formed by dropping a perpendicular from a vertex to the opposite side. The area formula simplifies to $\frac{\sqrt{3}}{4} a^2$.
- Perimeter: $3a$
- Explanation: Since all three sides of an equilateral triangle are equal, the perimeter is simply three times the side length.

#### 3. Quadrilateral
- Shape: Quadrilateral
- Area: $\frac{1}{2} \times d \times (h_1 + h_2)$
- Explanation: This formula applies to a general quadrilateral where $d$ is the length of a diagonal, and $h_1$ and $h_2$ are the perpendicular distances from the diagonal to the opposite vertices. The area is half the product of the diagonal and the sum of these perpendicular heights.
- Perimeter: Sum of all the four sides
- Explanation: The perimeter of any quadrilateral is the sum of the lengths of its four sides.

#### 4. Parallelogram
- Shape: Parallelogram
- Area: $b \times h$
- Explanation: The area of a parallelogram is the product of its base ($b$) and the corresponding height ($h$).
- Perimeter: $2(a + b)$
- Explanation: A parallelogram has opposite sides that are equal in length. If the lengths of the two adjacent sides are $a$ and $b$, the perimeter is twice the sum of these two sides.

#### 5. Rectangle
- Shape: Rectangle
- Area: $l \times b$
- Explanation: The area of a rectangle is the product of its length ($l$) and breadth ($b$).
- Perimeter: $2(l + b)$
- Explanation: A rectangle has opposite sides that are equal. The perimeter is twice the sum of its length and breadth.

#### 6. Trapezium
- Shape: Trapezium
- Area: $\frac{1}{2} \times h \times (a + b)$
- Explanation: The area of a trapezium is half the product of its height ($h$) and the sum of its two parallel sides ($a$ and $b$).
- Perimeter: Sum of all the four sides
- Explanation: The perimeter of a trapezium is the sum of the lengths of its four sides.

#### 7. Rhombus
- Shape: Rhombus
- Area: $\frac{1}{2} \times d_1 \times d_2$
- Explanation: The area of a rhombus is half the product of its diagonals ($d_1$ and $d_2$).
- Perimeter: $4a$
- Explanation: All four sides of a rhombus are equal. If the side length is $a$, the perimeter is four times $a$.

#### 8. Square
- Shape: Square
- Area: $a^2$
- Explanation: The area of a square is the side length ($a$) squared.
- Perimeter: $4a$
- Explanation: All four sides of a square are equal. If the side length is $a$, the perimeter is four times $a$.

Final Answer:

The table accurately summarizes the area and perimeter formulas for the given geometric shapes. Each formula is consistent with standard mathematical definitions and derivations.

$$
\boxed{\text{The table is correct and complete.}}
$$