Acute Triangle Calculator

An acute triangle is defined by having three acute interior angles (all strictly under 90°). To determine its geometric properties, the calculator utilizes foundational trigonometric laws depending on your known inputs. For instance, if you know two sides and the included angle (SAS configuration), the Law of Cosines determines the third side: c = sqrt(a² + b² - 2ab \cos(\gamma)). The area is found using the formula Area = 0.5 × a × b \sin(\gamma), while the perimeter is simply the sum of all three sides (a + b + c). For side-side-side (SSS) configurations, Heron's formula is employed alongside the inverse cosine function to confirm that all resulting interior angles remain strictly below 90 degrees.

Let us evaluate an acute triangle using three known side lengths (SSS configuration): side a = 5 cm, side b = 7 cm, and side c = 8 cm. First, we calculate the semi-perimeter s = (5 + 7 + 8) / 2 = 10 cm. Next, we find the area using Heron's formula: Area = sqrt(10 × (10 - 5) × (10 - 7) × (10 - 8)) = sqrt(10 × 5 × 3 × 2) = sqrt(300) ≈ 17.32 cm². To find the interior angles, we apply the Law of Cosines. For angle opposite to side a: \alpha = \arccos((7² + 8² - 5²) / (2 × 7 × 8)) = \arccos(88 / 112) ≈ 38.21°. Similarly, angle \beta ≈ 58.41° and angle \gamma ≈ 83.38°. Because all three angles are below 90°, we verify this is indeed a valid acute triangle with a perimeter of 20 cm.

Always verify the triangle inequality theorem before running calculations: the sum of any two side lengths must strictly exceed the length of the remaining side. Double-check your angle units to ensure your calculator is set to degrees or radians consistently. Finally, remember that while equilateral triangles are always acute, scalene and isosceles triangles can also be acute provided their largest angle stays below 90 degrees.