Right triangle area calculator

What is the area of a right triangle?

The area of a right triangle is the space occupied by the triangle on a plane. A right triangle has one angle measuring 90 degrees, and the two sides, known as legs, are adjacent to this angle. Calculating the area is significant in geometry, science, engineering, and many other fields.

How to calculate the area given the lengths of the legs

The formula for finding the area of a right triangle when the lengths of the legs $a$ and $b$ are known is as follows:

$$A = \frac{1}{2} \times a \times b$$

This formula implies that the area is half the product of the lengths of the two legs. If you imagine a square with a side length equal to one leg, such a square would be twice the size of the triangle.

How to calculate the area given one leg and an angle

If only one leg and an angle are known, trigonometric functions are required:

• If leg $a$ and angle $\beta$ are known, the area can be found by the formula:

$$A = \frac{1}{2} \times a \times b = \frac{1}{2} \times a \times (a \times \tan(\beta)) = \frac{1}{2} \times a^2 \times \tan(\beta)$$

• If leg $b$ and angle $\alpha$ are known, the area can be calculated as follows:

$$A = \frac{1}{2} \times b^2 \times \tan(\alpha)$$

The tangent of an angle is the ratio of the length of the opposite leg to the length of the adjacent leg:

$$\tan(\theta) = \frac{\text{opposite leg}}{\text{adjacent leg}}$$

In this case, the angle $\alpha$ is opposite to leg $a$, and angle $\beta$ is opposite to leg $b$.

Formulas

• When the legs are known:

  $$A = \frac{1}{2} \times a \times b$$

• With known leg $a$ and angle $\beta$:

  $$A = \frac{1}{2} \times a^2 \times \tan(\beta)$$

• With known leg $b$ and angle $\alpha$:

  $$A = \frac{1}{2} \times b^2 \times \tan(\alpha)$$

Examples

Example 1: Known two legs

Suppose the legs of a triangle are $3$ and $4$. Then, using the formula, the area can be found as follows:

$$A = \frac{1}{2} \times 3 \times 4 = 6$$

Example 2: Known leg $a$ and angle $\beta$

Let $a = 5$, $\beta = 45^\circ$. The area can then be calculated as:

$$A = \frac{1}{2} \times 5^2 \times \tan(45^\circ) = \frac{1}{2} \times 25 \times 1 = 12.5$$

Example 3: Known leg $b$ and angle $\alpha$

Let $b = 7$, $\alpha = 30^\circ$. The area is calculated as follows:

$$A = \frac{1}{2} \times 7^2 \times \tan(30^\circ) = \frac{1}{2} \times 49 \times 0.577 \approx 14.14$$

Example 4: Historical structure base area calculation

Imagine needing to calculate the base area of a pyramid, assuming it had the shape of a right triangle. For example, if one leg of the base, $a$, is 150 meters, and the other leg, $b$, is 200 meters, the base area would be:

$$A = \frac{1}{2} \times 150 \times 200 = 15,000 \,\text{square meters}$$

Notes

• The angle $\alpha$ or $\beta$ should be given in degrees when using tangent.
• Trigonometric calculations can be challenging without a calculator.
• If you need to find the perimeter of a right triangle, you can use our right triangle calculator.

Frequently asked questions

How to find the area of a right triangle if only the hypotenuse is known?

To calculate the area, you must know the length of at least one leg or the angle adjacent to the hypotenuse.

Can I use the same formula for non-right triangles?

The formulas above are specific to right triangles. Other types of triangles use different approaches, such as Heron’s formula. To calculate the area of other triangles, use our triangle area calculator.

Why are triangle area calculations important?

Area is important in architecture, construction, cartography, and physics. Knowing the area of an object helps in planning the right use of materials and resources.

What role do angles and sides play in determining the area?

The length of the sides and the magnitude of the angles determine the possible scale and shape of the triangle, which directly influences its area.

How does rounding affect the accuracy of results when using trigonometric values?

Rounding can introduce small errors in calculations, so for accuracy, it’s essential to consider all decimal places in intermediate calculations.

What is the area of a right triangle with legs 3 and 4?

The area of a right triangle with legs 3 and 4 units long is:

$$A = \frac{1}{2} \times 3 \times 4 = 6 \,\text{square units}$$