Free Right Trapezoid Calculator

What Is a Right Trapezoid?

A right trapezoid is a quadrilateral with one pair of parallel sides (the bases) and one leg that is perpendicular to both bases. This perpendicular side introduces two right angles into the shape, distinguishing it from a general trapezoid. Because the height coincides with the length of that vertical leg, calculations become simpler than for other trapezoid types.

This free right angle trapezoid calculator online is designed to compute all the major properties of such a shape with minimal input. It serves as a right trapezoid area calculator, a perimeter calculator, and a tool that returns slant side length, angles, and the median in one go. Whether you’re solving geometry homework or designing a physical object, this tool saves time by applying the underlying right trapezoid formulas automatically.

Key Formulas and Properties

The following relationships are used by the calculator to derive unknown values from known ones.

Area

The area of any trapezoid is the average of the bases multiplied by the height. For a right trapezoid, the height equals the perpendicular leg (usually denoted as $c$):

$$\text{Area} = \frac{a + b}{2} \times c$$

where $a$ is the longer base, $b$ the shorter base, and $c$ the height (short side).

Slant Side (Long Leg)

The non‑vertical leg, $d$, forms the hypotenuse of a right triangle together with the height $c$ and the difference of the bases $(a - b)$. By the Pythagorean theorem:

$$d = \sqrt{(a - b)^{2} + c^{2}}$$

Short Base (When Slant Side Is Known)

If the longer base, the height, and the slant side are given, the shorter base can be found by rearranging the same triangle:

$$b = a - \sqrt{d^{2} - c^{2}}$$

Perimeter

The perimeter is the sum of all four sides:

$$P = a + b + c + d$$

Angles

Besides the two right angles, the remaining angles can be determined using basic trigonometry. The acute angle $\alpha$ adjacent to the longer base satisfies:

$$\tan\alpha = \frac{c}{a - b}$$

The obtuse angle adjacent to the shorter base is simply $180^{\circ} - \alpha$.

Median (Midsegment)

The segment joining the midpoints of the legs is parallel to the bases and equals their arithmetic mean:

$$M = \frac{a + b}{2}$$

A rectangle is a special case of a right trapezoid where both legs are perpendicular to the bases, making it a fully symmetric example of the same family.

Practical Usage Example

Suppose you have a right trapezoid with longer base $a = 12\ \text{cm}$, shorter base $b = 7\ \text{cm}$, and height $c = 5\ \text{cm}$. Using the formulas above:

• Area: $\displaystyle \frac{12 + 7}{2} \times 5 = 47.5\ \text{cm}^{2}$
• Slant side: $\displaystyle d = \sqrt{(12-7)^{2} + 5^{2}} = \sqrt{25 + 25} \approx 7.07\ \text{cm}$
• Perimeter: $12 + 7 + 5 + 7.07 = 31.07\ \text{cm}$
• Acute angle: $\displaystyle \tan\alpha = \frac{5}{12-7} = 1$, so $\alpha = 45^{\circ}$.

Entering any three of these values into the right trapezoid calculator yields the remaining properties instantly, making it a versatile tool for both students and professionals.