How do you solve a right triangle when given different combinations of sides and angles?

A right triangle has one 90° angle, and the three sides are: hypotenuse (c, the longest side opposite the right angle), and two legs (a and b). The Pythagorean theorem a² + b² = c² is the fundamental relationship. Given two sides, find the third: if you know both legs: c = √(a² + b²). If you know the hypotenuse and one leg: the other leg = √(c² - known_leg²). Given one side and one acute angle: use trigonometric ratios — sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. SOH-CAH-TOA is the standard mnemonic. Example: hypotenuse = 10, angle = 35°. Opposite side = 10 × sin(35°) = 5.74. Adjacent side = 10 × cos(35°) = 8.19. Finding angles from sides: θ = arctan(opposite/adjacent), or θ = arcsin(opposite/hypotenuse), or θ = arccos(adjacent/hypotenuse). The two acute angles always sum to 90° (since all three angles must sum to 180°). Common right triangle ratios worth memorizing: 3-4-5 (and multiples: 6-8-10, 9-12-15), 5-12-13, 8-15-17, 7-24-25. The 45-45-90 triangle has legs in ratio 1:1:√2. The 30-60-90 triangle has sides in ratio 1:√3:2.

What are the most common real-world applications of right triangle calculations?

Construction and building — calculating roof pitch and rafter length. A 6/12 pitch means the roof rises 6 inches for every 12 inches of horizontal run. Rafter length = √(rise² + run²). For a 6/12 pitch with 12-foot run: rafter = √(6² + 12²) = √180 = 13.42 feet. Also used for staircase design: if the total rise is 9 feet with an ideal 7-inch riser height, you need 15-16 risers. Tread depth is calculated to maintain proper angle (typically 30–35° for comfortable stairs). Surveying and navigation — measuring distances to inaccessible points. If you can't cross a river but need to know its width: measure a baseline along the bank, measure angles to a point across the river from both ends, and use trigonometry to calculate the distance. GPS works similarly — triangulation from known satellite positions. Physics and engineering — resolving forces into components. A 100 N force at 30° to the horizontal has a horizontal component of 100 × cos(30°) = 86.6 N and a vertical component of 100 × sin(30°) = 50 N. This is essential in structural engineering (resolving loads in trusses), physics problems (projectile motion, inclined planes), and electrical engineering (AC circuit analysis with phasors). Everyday applications: finding the distance to walk diagonally across a rectangular field, determining if furniture will fit through a doorway when tilted, and calculating screen sizes (a 55-inch TV is measured diagonally — the actual width for a 16:9 ratio is 48 inches).

What are the primary trigonometric ratios used in right triangles?

The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). For an acute angle θ in a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. For instance, if the side opposite θ is 3 units and the hypotenuse is 5 units, then sin(θ) = 3/5 or 0.6.

What are the characteristics of special right triangles?

Special right triangles are specific cases that have fixed angle and side ratios, simplifying calculations. A 45-45-90 triangle has two equal legs (x) and a hypotenuse of x√2. A 30-60-90 triangle has legs in the ratio x : x√3 and a hypotenuse of 2x, where x is the side opposite the 30° angle.

What is a Pythagorean triple, and can you provide examples?

A Pythagorean triple is a set of three positive integers (a, b, c) that perfectly satisfy the Pythagorean theorem, a² + b² = c². These integers represent the side lengths of a right triangle where all sides are whole numbers. Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Μαθηματικά

Υπολογιστής Ορθογωνίου Τριγώνου

Θεώρημα Πυθαγόρα — a² + b² = c²

Side a (cm)

Side b (cm)

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Variable Legend

a= legb= other legc= hypotenuseA= areaα,β= acute angles

Pythagorean theorem

Relationship between the three sides.

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Area

Half the product of the two legs.

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Trigonometric ratios

SOH-CAH-TOA for angle α (opposite side a).

sin α

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cos α

…

tan α

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Angle from sides

Find an angle when all sides are known.

Angle α

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Angle β

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a= leg

b= other leg

c= hypotenuse

A= area

α,β= acute angles

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Reviewed September 2026

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