What is an SAS triangle?

SAS stands for "side-angle-side". It is a triangle with two sides and the included angle known. With these three values, the entire triangle can be determined, and all other quantities such as area or the third side can be calculated from it.

What is the formula for calculating the area of a SAS triangle?

The area is half the product of two known sides multiplied by the sine of the included angle. The formula for this is: A = (1/2) x a x b x sin(C). Angle C must be the angle between sides a and b.

How do I find the area of an SAS triangle with sides of 4 and 5 and an included angle of 30 degrees?

These values are then substituted into the SAS triangle formula: half of the product of 4, 5 and the sine of 30 degrees. As the sine of 30 degrees is equal to 0.5, the area is 0.5 × 4 × 5 × 0.5, which equals 5 square centimeters.

How to Calculate a Missing Side of a SAS Triangle?

You use the law of cosines. The law of cosines combines two sides and the included angle to find the third side, i.e., the side opposite the angle. The full formula is given above. If the included angle is 90 degrees, then the formula becomes the Pythagorean theorem. So if the two sides are 3 and 4, then the third side is 5.

Are SAS triangles congruent?

Yes. Since a triangle is uniquely determined by two sides and the included angle, any two triangles with equal corresponding sides and angles are congruent. This is called Side-Angle-Side Congruence (SAS).

What is the area of a triangle with sides of length 3 and 4 and an included angle of 90 degrees?

Since one of the angles is 90 degrees and its sine is 1, this is a right triangle. The area is half of 3 times 4 times 1, or 6 square units, and the third side is the hypotenuse with length 5.

Area of a Triangle (SAS) Calculator

Two sides and the included angle

Enter the two sides and the angle wedged between them (the included angle). The area is half their product times the sine of that angle, and the calculator solves the rest of the triangle from there.

Side a

Side b

Included angle (C)

Area

Third side (c)

Please input values into the fields

SAS is short for "side-angle-side" and refers to the case where two sides and the included angle of a triangle are known. As this suffices to fully determine a triangle, this calculator first displays the area and then calculates all other results, such as the third side, the three angles and the perimeter.

Please enter two sides and the included angle. The area will be immediately displayed, and based on the law of cosines, the missing third side is also calculated. The entire solved triangle is shown in a panel.

Formula for area of a SAS triangle:

You might know the standard formula for finding the area of a triangle as one-half base times height.

A = \tfrac{1}{2} \times \text{base} \times \text{height}

In SAS triangle you do not directly measure the height but can calculate it. If side b is chosen as base then corresponding height is product of side a and sine of included angle.

h = a \times \sin(C)

When you substitute this height into the formula for area of a triangle, it becomes the formula for area of an SAS triangle. It uses two sides and the angle between them to find the area.

A = \tfrac{1}{2} \, a \, b \, \sin(C)

If two sides are of length 4 and 5 with an included angle of 30 degrees, then the area is half the product of 4, 5, and the sine of 30 degrees. Since the sine of 30 degrees is 0.5, the area is 5 square units.

Method for calculating area given two sides and an angle:

The process is simple: multiply the lengths of both sides, multiply that result by the sine of the included angle and divide that result by two.

A = \frac{a \times b \times \sin(C)}{2}

It is important to note which angle is used. It must be the angle that is between the two sides. Using any other angle will not uniquely determine the shape. For this reason, the included angle must be explicitly stated in the side-angle-side formula.

Determination of the missing third side:

If two sides and the included angle are known, then the third side can be determined. The third side is found by using the Law of Cosines to calculate the unknown length.

c = \sqrt{a^2 + b^2 - 2 \, a \, b \, \cos(C)}

If two sides have lengths 4 and 5 and the included angle is 30 degrees, then the third side is about 2.52. If the included angle is exactly 90 degrees, the cosine term disappears and the formula reduces to the Pythagorean theorem. Thus if two sides have lengths 3 and 4, then the third side has length exactly 5.

Solve for the remaining parts of the triangle.

If all three sides are known, then any of the angles can be computed using the Law of Cosines; for example, angle A is given by:

\cos(A) = \frac{b^2 + c^2 - a^2}{2 \, b \, c}

The third angle is the remaining one. This is because the sum of all interior angles in any triangle is always 180 degrees.

B = 180^\circ - A - C

The perimeter (the sum of the three sides) and half the perimeter (half of that sum) can also be calculated from these three sides. In addition, the area computed by Heron's formula is the same as the area computed using the SAS formula, which provides a convenient check.

s = \frac{a + b + c}{2} \qquad A = \sqrt{s\,(s - a)(s - b)(s - c)}

Congruence of triangles according to SAS.

When two sides and the included angle are known, there is no room for change in the shape of the figure. There is exactly one triangle with those given values. So if two triangles have corresponding sides and the included angle that are congruent, then they are congruent by SAS (side-angle-side) which is one of the standard methods to prove that two triangles are completely identical.

This explains why this calculation tool provides a unique solution. In contrast, with SSA (side-side-angle), the angle is not between the two sides but if given two sides and an angle they may fit into two different triangles. Because of this ambiguity, SSA is not a reliable condition for a unique solution while SAS allows it.

If the SAS triangle is a right-angled triangle:

If the included angle is a right angle, then the two given sides are the legs of the right triangle and the third side is the hypotenuse. Since sine of 90 degrees is 1, this formula simplifies to half the product of the two legs. This is a known formula for right triangles. If the two sides are 3 and 4 and the included angle is 90 degrees, then the area is 6 and the hypotenuse is 5.

Examples of calculations:

Each row represents a triangle with one side, an angle and another side. The area is rounded to two decimal places.

Side a	Side b	Included angle	Area
4	5	30 deg	5.00 sq units
4	5	40 deg	6.43 sq units
9	7	30 deg	15.75 sq units
3	4	90 deg (right)	6.00 sq units
6	6	60 deg (equilateral)	15.59 sq units

Applications on sides, corners and edges:

Surveyors and sailors can use this method when they can measure the distance between two points from a single point and determine the angle between two sight lines. In this way, they can find the area of land enclosed by a boundary or the length of a line crossing that boundary without physically moving within the area.

Construction workers use the SAS (side-angle-side) principle when building roof trusses and braces as well as connections where two pieces of known length need to be joined at a specific angle. Students often learn this shortly after learning the sine and cosine laws, and they use these tools to check their solutions.

For accurate results, please follow these tips:

Make sure that the unit of measure for your angle matches the input you're using. While calculators can switch between degrees and radians, using the wrong units when calculating sine is probably the most common cause of errors in area calculations.

Enter each side in the unit used for measurement. Before performing the calculation all numbers will be converted to a common base unit so you can mix centimeters and inches if desired. Be sure that the angle entered is the angle between the two sides entered.

This tool is for general education and everyday planning purposes. For surveying, construction or other high risk activities you should verify the data against the standards and tolerances required by your project.

Frequently asked questions

What is an SAS triangle?: SAS stands for "side-angle-side". It is a triangle with two sides and the included angle known. With these three values, the entire triangle can be determined, and all other quantities such as area or the third side can be calculated from it.
What is the formula for calculating the area of a SAS triangle?: The area is half the product of two known sides multiplied by the sine of the included angle. The formula for this is: A = (1/2) x a x b x sin(C). Angle C must be the angle between sides a and b.
How do I find the area of an SAS triangle with sides of 4 and 5 and an included angle of 30 degrees?: These values are then substituted into the SAS triangle formula: half of the product of 4, 5 and the sine of 30 degrees. As the sine of 30 degrees is equal to 0.5, the area is 0.5 × 4 × 5 × 0.5, which equals 5 square centimeters.
How to Calculate a Missing Side of a SAS Triangle?: You use the law of cosines. The law of cosines combines two sides and the included angle to find the third side, i.e., the side opposite the angle. The full formula is given above. If the included angle is 90 degrees, then the formula becomes the Pythagorean theorem. So if the two sides are 3 and 4, then the third side is 5.
Are SAS triangles congruent?: Yes. Since a triangle is uniquely determined by two sides and the included angle, any two triangles with equal corresponding sides and angles are congruent. This is called Side-Angle-Side Congruence (SAS).
What is the area of a triangle with sides of length 3 and 4 and an included angle of 90 degrees?: Since one of the angles is 90 degrees and its sine is 1, this is a right triangle. The area is half of 3 times 4 times 1, or 6 square units, and the third side is the hypotenuse with length 5.

Related calculators

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Triangle Perimeter CalculatorFind the perimeter of a triangle three ways: from one side of an equilateral triangle, from the equal sides and base of an isosceles triangle, or from the three sides of a scalene triangle. Also gives the area, semi-perimeter, angles, heights, medians, and circles, and estimates a fencing cost.

Area of an Oblique Triangle CalculatorFind the area of an oblique (non-right) triangle from two sides and the included angle, two angles and a side, or three sides. Solves the whole triangle with the Law of Sines and the Law of Cosines and reports the perimeter, angles, heights, medians, and circles.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Solution of triangles
How the side-angle-side case is solved with the Law of Cosines and the Law of Sines.
Wikipedia: Law of cosines
The relationship that recovers the third side of a triangle from two sides and the included angle.
Wikipedia: Congruence (geometry)
The side-angle-side criterion and the other tests for when two triangles are congruent.
Wikipedia: Heron's formula
The area of a triangle from its three sides, used here as a cross-check of the SAS area.