Right Trapezoid Calculator

What Is a Right Trapezoid?

A right trapezoid (also known as a right-angled trapezoid or right trapezium) is a four-sided convex polygon with at least one pair of parallel sides (called bases) and two adjacent right angles ($90^\circ$). Because the two bases are parallel, any perpendicular line segment that intersects both bases meets them at right angles. Therefore, if one interior angle is $90^\circ$, its adjacent consecutive angle along the parallel leg must also be $90^\circ$. A trapezoid can have either two right angles or none at all; a trapezoid with exactly one right angle cannot exist in Euclidean geometry.

In a right trapezoid, the perpendicular leg serves double duty as the geometric height ($c = h$) of the shape. The remaining fourth side is an inclined slant leg ($d$) that forms an acute angle ($\alpha$) with the longer base and an obtuse supplementary angle ($\beta = 180^\circ - \alpha$) with the shorter base. This unique configuration makes right trapezoids a staple of geometry, carpentry, engineering trusses, and architectural framing.

Essential Right Trapezoid Formulas

Let $a$ denote the length of the longer base, $b$ the length of the shorter base, $c$ the perpendicular leg (height), and $d$ the non-parallel slant leg:

• Area ($A$): The area equals the product of the average of the bases and the perpendicular height: $$A = \frac{a + b}{2} \cdot c$$
• Perimeter ($P$): The total boundary length is the sum of all four sides: $$P = a + b + c + d$$
• Slant Leg ($d$): Dropping an altitude from the end of the shorter base forms an internal right-angled triangle with base $(a - b)$ and vertical leg $c$. By the Pythagorean theorem: $$d = \sqrt{(a - b)^2 + c^2}$$
• Height ($c$): If the slant leg $d$ and both bases are known, rearrange the Pythagorean theorem: $$c = \sqrt{d^2 - (a - b)^2}$$
• Midsegment ($m$): The line connecting the midpoints of the non-parallel legs is parallel to both bases: $$m = \frac{a + b}{2}$$
• Diagonals ($p$ and $q$): By constructing right triangles inside the trapezoid: $$p = \sqrt{a^2 + c^2} \quad \text{(diagonal to the far corner of long base)}$$ $$q = \sqrt{b^2 + c^2} \quad \text{(diagonal to the far corner of short base)}$$
• Angles ($\alpha$ and $\beta$): The two non-right interior angles satisfy: $$\tan(\alpha) = \frac{c}{a - b} \implies \alpha = \arctan\left(\frac{c}{a - b}\right)$$ $$\beta = 180^\circ - \alpha$$

Step-by-Step Example Calculation

Consider a right trapezoid where the longer base is $a = 10\text{ cm}$, the shorter base is $b = 6\text{ cm}$, and the perpendicular height is $c = 5\text{ cm}$:

1. Base difference: Compute the horizontal projection of the slant leg: $a - b = 10 - 6 = 4\text{ cm}$.
2. Slant leg ($d$): Apply the Pythagorean theorem: $$d = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \approx 6.4031\text{ cm}$$
3. Area ($A$): $$A = \frac{10 + 6}{2} \times 5 = 8 \times 5 = 40\text{ cm}^2$$
4. Perimeter ($P$): $$P = 10 + 6 + 5 + 6.4031 = 27.4031\text{ cm}$$
5. Acute angle ($\alpha$): $$\alpha = \arctan(5 / 4) = \arctan(1.25) \approx 51.34^\circ$$
6. Obtuse angle ($\beta$): $$\beta = 180^\circ - 51.34^\circ = 128.66^\circ$$

Related Geometric Calculators

Explore other polygon and triangle calculation tools on OnlineMiniTools:

• Area of a Trapezoid Calculator - Compute trapezoid areas with general side and angle inputs.
• Right Triangle Calculator - Solve side lengths, hypotenuse, and trigonometric ratios for right triangles.
• Isosceles Trapezoid Calculator - Analyze symmetric trapezoids with equal non-parallel legs.