A trapezoid is a quadrilateral with two parallel sides, known as bases, and two non-parallel sides, known as legs. These formulas are useful for solving math and geometry problems involving trapezoids, such as finding their area, perimeter, height, mid-segment, diagonals, and angle bisectors.
Trapezoids Formulas
Area of a Trapezoid:
- Formula: A = (1/2) * (a + b) * h
- Description: This formula calculates the area (A) of a trapezoid, where 'a' and 'b' are the lengths of the two bases, and 'h' is the height (the perpendicular distance between the bases).
Perimeter of a Trapezoid:
- Formula: P = a + b + c + d
- Description: The perimeter (P) of a trapezoid is the sum of the lengths of all its four sides, where 'a' and 'b' are the lengths of the bases, and 'c' and 'd' are the lengths of the non-parallel sides (legs).
Height of a Trapezoid (Alternative Formula):
- Formula: h = (2 * A) / (a + b)
- Description: This formula calculates the height (h) of a trapezoid using its area (A), and the lengths of the two bases ('a' and 'b').
Midsegment of a Trapezoid:
- Formula: m = (a + b) / 2
- Description: The midsegment (m) of a trapezoid is the average of the lengths of its two bases. It is a line segment that connects the midpoints of the two non-parallel sides.
- Formula: m = (a + b) / 2
Diagonal of a Trapezoid: using the Pythagorean theorem:
Formula d = √(a² + h²)
Steps to find the length of the diagonal (d) of a trapezoid using the Pythagorean theorem:
- Identify the longer base (a), the shorter base (b), and the height (h) of the trapezoid.
- Draw a perpendicular line from one of the trapezoid's vertices to the longer base (a). This creates a right triangle with d as the hypotenuse, a as one side, and h as the other side.
- Use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (d²) is equal to the sum of the squares of the other two sides: d² = a² + h²
- Take the square root of both sides to solve for the diagonal (d): d = √(a² + h²).
Trapezoid Calculation Examples
Area of a Trapezoid:
- Example: Find the area of a trapezoid with bases of length 2.36 feet and 3.15 feet and a height of 1.64 feet.
- Calculation: A = (1/2) * (2.36 feet + 3.15 feet) * 1.64 feet
- Result: A ≈ 4.724 square feet
Perimeter of a Trapezoid:
- Example: Calculate the perimeter of a trapezoid with side lengths a = 1.18 feet, b = 1.97 feet, c = 1.31 feet, and d = 2.31 feet.
- Calculation: P = 1.18 feet + 1.97 feet + 1.31 feet + 2.31 feet
- Result: P ≈ 6.77 feet
Height of a Trapezoid (Alternative Formula):
- Example: Find the height of a trapezoid with an area of 6.56 square feet, a base length of 3.93 feet, and another base length of 5.24 feet.
- Calculation: h = (2 * 6.56 square feet) / (3.93 feet + 5.24 feet)
- Result: h ≈ 1.64 feet
Midsegment of a Trapezoid:
- Example: Determine the midsegment length of a trapezoid with bases of length 1.98 feet and 2.62 feet.
- Calculation: m = (1.98 feet + 2.62 feet) / 2
- Result: m ≈ 2.30 feet
5. Diagonal of a Trapezoid:
Example: Suppose we have a trapezoid with the following measurements in inches:
- Longer base (a): 8 inches
- Shorter base (b): 4 inches
- Height (h): 6 inches
Plug in the values in inches: d = √(8² + 6²)
Square the lengths: d = √(64 + 36)
Add the squared values: d = √(100)
Take the square root of the sum: d = 10 inches