Area of a Triangle Using Trigonometry

This calculator finds the area of a triangle when you know two side lengths and the included angle (the angle between them). Enter the two side lengths and the angle in degrees, and you get the area in square units. The formula used is Area = ½ × a × b × sin(C), which is the standard trigonometric area formula.

Formula

The formula for the area of a triangle using trigonometry is:

Area = ½ × a × b × sin(C)

Where:

• a and b are the lengths of two sides of the triangle (in any consistent unit, such as meters or feet).
• C is the included angle between those two sides, measured in degrees.
• sin(C) is the sine of that angle (a dimensionless number between 0 and 1).

The result is in square units of whatever unit you used for the side lengths (e.g., if sides are in meters, area is in square meters).

Worked Example

Using the default values from the calculator: side a = 5, side b = 7, and angle C = 30°.

1. First, find the sine of 30°. sin(30°) = 0.5.
2. Multiply the sides: 5 × 7 = 35.
3. Multiply by ½ and by the sine: ½ × 35 × 0.5 = 8.75.

The area is 8.75 square units, which matches the calculator output.

When to Use This Formula

Use the trigonometric area formula when you know two sides and the included angle (SAS). This is common in surveying, construction, and physics problems where you can measure two lengths and the angle between them but not the height of the triangle. It is also useful for finding the area of irregular shapes that can be divided into triangles.

If you instead know all three side lengths, use Heron’s formula. If you know base and height, use the simple ½ × base × height. The trigonometric formula is the one to pick when the height is not directly available but an angle is.

How to Choose the Right Sides and Angle

The angle C must be the angle between the two sides you choose. In a triangle, each angle is opposite one side. If you label the sides as a, b, and c, with angles A, B, and C opposite them respectively, then the included angle between sides a and b is angle C. The formula works with any pair of sides and their included angle, so you can also write:

• Area = ½ × a × c × sin(B)
• Area = ½ × b × c × sin(A)

All three give the same area. Just make sure the angle you use is the one between the two sides you pick.

Common Mistakes and Misconceptions

A frequent mistake is using an angle that is not between the two given sides. For example, if you know sides a and b but use angle A (which is opposite side a) instead of angle C, you will get a wrong answer. Another common error is mixing units: both side lengths must be in the same unit (e.g., both meters, not one meter and one foot) before calculating. Also, make sure your calculator is in degree mode when you enter the angle; if it is in radians, the result will be incorrect.

Some people think the formula only works for acute triangles, but it works for obtuse triangles too. The sine of an obtuse angle is still positive (for angles up to 180°), so the area comes out positive. For a triangle, the angle must be between 0° and 180°.

Understanding the Result

The area is always positive and is expressed in square units. For example, if you enter sides in meters, the area is in square meters. If you enter sides in feet, the area is in square feet. The maximum possible area for two fixed side lengths occurs when the included angle is 90°, because sin(90°) = 1. As the angle approaches 0° or 180°, the area approaches zero, because the triangle becomes very flat.

This formula is exact for any triangle, not an approximation. It is derived from the basic area formula (½ × base × height) by recognizing that the height of the triangle relative to one side equals the other side times the sine of the included angle.

Frequently Asked Questions

Can I use this formula if I only know one side and two angles?

No, you need two sides and the included angle. If you have one side and two angles, you can first find the other sides using the law of sines, then use this formula.

What if my angle is in radians?

Convert it to degrees first (multiply by 180/Ï€) or use the sine function in radian mode. The calculator above expects degrees, but you can adjust if you are doing it by hand.

Does this formula work for right triangles?

Yes. If the included angle is 90°, sin(90°) = 1, so the formula becomes ½ × a × b, which is the same as ½ × base × height for a right triangle.

Why is the area zero when the angle is 0° or 180°?

Because the two sides lie on the same line, so the triangle has no height and collapses into a line segment, giving zero area.

What if I have three sides but no angle?

Use Heron’s formula instead: Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2. The trigonometric formula requires an angle.

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in a | Side a | units | 5 | 0
in b | Side b | units | 7 | 0
in C | Included angle C | degrees | 30 | 0
out area | Area | sq units | 0.5 * a * b * sind(C) | 2