Triangle Area (SAS) Calculator

Calculate the area of a triangle when you know two sides and the included angle (Side-Angle-Side).

Enter Two Sides & Included Angle

Result

Area
16.970563
square units
Side a 8
Side b 6
Included Angle C 45 degrees
sin(C) 0.707107
Third Side c (Law of Cosines) 5.675237
Perimeter 19.675237
Height from side a 4.242641

Step-by-Step Solution

A = 1/2 x a x b x sin(C)

Understanding the SAS Triangle Area Formula

The SAS (Side-Angle-Side) triangle area formula is used when you know two sides of a triangle and the angle between them (the included angle). This is one of the most practical triangle area formulas because these measurements are often the easiest to obtain in real-world scenarios.

The formula derives from the basic triangle area formula (A = 1/2 x base x height) by expressing the height in terms of the known side and the sine of the included angle.

The SAS Area Formula

Given two sides a and b and the included angle C, the area is:

A = 1/2 x a x b x sin(C)

This works because if you take side a as the base, the height from the opposite vertex is b x sin(C), giving us: A = 1/2 x a x (b x sin(C)) = 1/2 x a x b x sin(C).

Related Triangle Formulas

SAS Area

Two sides and included angle.

A = 1/2 x a x b x sin(C)

Base x Height

When base and perpendicular height are known.

A = 1/2 x base x height

Heron's Formula

When all three sides are known.

A = sqrt(s(s-a)(s-b)(s-c))

Law of Cosines

Find the third side from SAS data.

c^2 = a^2 + b^2 - 2ab cos(C)

Law of Sines

Relate sides and opposite angles.

a/sin(A) = b/sin(B) = c/sin(C)

Coordinate Formula

From vertex coordinates.

A = 1/2 |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)|

How to Use the SAS Formula

Step 1: Identify the Two Sides and Included Angle

The included angle is the angle formed between the two known sides. It is critical that you use the angle between the sides, not an angle opposite to one of them. If you have an angle opposite a known side, you need a different approach (such as the ASA or AAS method).

Step 2: Calculate the Sine of the Angle

Convert the angle to radians if necessary, then compute sin(C). Remember that sin(90 degrees) = 1, which gives the maximum area for fixed side lengths.

Step 3: Apply the Formula

Multiply the two sides together, multiply by the sine of the included angle, and divide by 2.

When Is the Area Maximized?

For fixed side lengths a and b, the area A = 1/2 x a x b x sin(C) is maximized when sin(C) = 1, i.e., when C = 90 degrees. This means a right triangle with legs a and b has the largest possible area among all triangles with those two side lengths.

Worked Example

Find the area of a triangle with sides a = 10, b = 7, and included angle C = 60 degrees:

  1. sin(60 degrees) = 0.866025
  2. A = 1/2 x 10 x 7 x 0.866025
  3. A = 1/2 x 70 x 0.866025
  4. A = 35 x 0.866025 = 30.310889 square units

Practical Applications

  • Surveying: Measuring land area when two boundary lengths and the corner angle are known.
  • Navigation: Computing the area of a triangular course from bearing angles and distances.
  • Engineering: Calculating cross-sectional areas of triangular structural elements.
  • Physics: Finding the magnitude of the cross product of two vectors (related to torque and angular momentum).
  • Architecture: Determining roof areas and triangular wall sections.

Tips for Accurate Calculations

  • Make sure the angle is the included angle (between the two sides), not an opposite angle.
  • Check whether your angle is in degrees or radians before computing the sine.
  • If the included angle is 0 or 180 degrees, the area is zero (degenerate triangle).
  • The maximum area for given side lengths occurs when the included angle is 90 degrees.