Triangle Area Sas
Calculator
Results
- Area
- 24.130399
- Third side
- 7.000625
- Perimeter
- 23.000625
Results
| Area | 24.130399 |
| Third side | 7.000625 |
| Perimeter | 23.000625 |
formula-map diagram
- Area
- 24.130399
- Third side
- 7.000625
- Perimeter
- 23.000625
Formula map
Formula
A = ½ × a × b × sin(C)= 24.130399958248
Note
Simplified model: these are the exact results of the stated idealised geometric formulas, assuming perfect shapes, exact inputs and consistent units. Real measurements carry error, and results such as the cyclic-quadrilateral area only hold when the figure truly satisfies the stated condition. Verify independently before relying on these figures.
More in Advanced geometry
See all →Frequently asked questions
What does 'SAS' mean, and why does it determine a triangle's area uniquely?+
SAS stands for Side-Angle-Side: two side lengths and the angle directly between them. This combination fully and uniquely determines the triangle's shape and size (by the SAS congruence rule), which is why it's enough information to calculate a definite area.
What is the SAS area formula?+
Area = (1/2) × a × b × sin(C), where a and b are the two known side lengths and C is the angle enclosed between them. This is essentially half of the parallelogram side-angle-area formula, since a triangle is exactly half of the parallelogram formed by those same two sides.
Why must the angle be the one specifically between the two given sides?+
The formula only works correctly if angle C is the included angle, the one formed exactly where sides a and b meet. Using a different angle of the triangle (one not between the two known sides) would require a different approach, like the law of sines, since it wouldn't correspond to this formula's geometry.
What happens to the triangle's area as the included angle approaches 0° or 180°?+
As the angle approaches either extreme, sin(C) approaches zero, so the calculated area shrinks toward zero even with fixed side lengths. This matches the geometric reality that the triangle collapses into a flat line at those extreme angles, since the two sides would essentially lay on top of each other.
Is there a maximum possible area for a triangle with two given fixed side lengths?+
Yes, the maximum area occurs precisely when the included angle is 90°, since sin(90°) = 1, the largest possible value sine can take. This is a useful optimization fact: for two given side lengths, a right angle between them always produces the largest possible triangle.