Triangle Area Calculator (2 sides and the included angle)
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The area of a triangle whose two sides a and b and included angle C are known is calculated with A = ½ × a × b × sin(C). For sides of 5 and 8 cm and an angle of 30°, the area is exactly 10 cm².
Explanation
This calculator complements the two other triangle-area methods already on the site: the standard triangle area calculator (base × height ÷ 2, which requires knowing or constructing a perpendicular height) and the Heron's formula calculator (which requires the three sides, with no angle). The side-angle-side (SAS) formula used here applies in a third, equally common situation: you know two sides of a triangle and the angle between them, without having measured a height or the third side — a typical configuration in topographic triangulation, navigation, or civil engineering, where two distances and an angle are often easier to measure than a perpendicular height. The formula derives directly from basic trigonometry: sin(C) converts side b into an "effective height" relative to base a, which brings the calculation back to the usual base × height ÷ 2 formula. The consistency of this formula with Heron's can be checked on the same triangle: for a triangle with sides 3 and 4 cm forming an exact 90° right angle between them, this formula gives 0.5 × 3 × 4 × sin(90°) = 6 cm², exactly the area found independently by Heron's formula on the 3-4-5 triangle.
Example: sides of 5 cm and 8 cm, angle of 30° between them
Inputs
Side a: 5 cm. Side b: 8 cm. Angle C: 30°.
Calculation
Area = 0.5 × 5 × 8 × sin(30°) = 20 × 0.5 = 10.
Result
The area of this triangle is 10 cm².
Frequently asked questions
When should you use this formula rather than base × height or Heron's formula?
Use this formula whenever you know two sides of a triangle and the exact angle between them, with no height or third side available. If you instead have a base and its perpendicular height, the standard calculation (base × height ÷ 2) is more direct; if you know the three sides with no angle, Heron's formula is best suited.
Why does the formula use the sine and not the cosine?
Because the sine of angle C, multiplied by side b, gives exactly the height of the triangle measured perpendicular to side a — it is this effective height that recovers the usual area formula. The cosine, on the other hand, appears in a different formula (the law of cosines), used to recover a missing side rather than an area directly.
What happens if the angle is 90°?
When angle C is exactly 90°, its sine is 1 (its maximum value), and the formula then simplifies exactly to A = ½ × a × b — the usual formula for a right triangle, where a and b directly play the roles of base and height since they are perpendicular to each other.