How do you find the hypotenuse of a right triangle?

Use the Pythagorean theorem: c² = a² + b², so c = √(a² + b²), where a and b are the two legs and c is the hypotenuse. The hypotenuse is always the longest side — opposite the right angle. Examples: • a=3, b=4: c = √(9+16) = √25 = 5 (the classic 3-4-5 right triangle) • a=5, b=12: c = √(25+144) = √169 = 13 • a=1, b=1: c = √2 ≈ 1.41421 (isosceles right triangle — 45-45-90) • a=1, b=√3: c = √(1+3) = 2 (30-60-90 triangle) Common Pythagorean triples to memorise: • 3-4-5, 5-12-13, 8-15-17, 7-24-25 • Any multiple works: 6-8-10, 9-12-15, 10-24-26 • This calculator finds the hypotenuse from two legs in the 'Two Legs' mode

What is SOHCAHTOA and how do you use it?

SOHCAHTOA is a mnemonic for the three basic trigonometric ratios in a right triangle, relative to angle A: • SOH: sin(A) = Opposite / Hypotenuse • CAH: cos(A) = Adjacent / Hypotenuse • TOA: tan(A) = Opposite / Adjacent For the 3-4-5 triangle with angle A opposite the side of length 3: • sin(A) = 3/5 = 0.6 → A = arcsin(0.6) = 36.87° • cos(A) = 4/5 = 0.8 • tan(A) = 3/4 = 0.75 Using SOHCAHTOA to find sides: • Know hypotenuse c=10 and angle A=30°: opposite = c×sin(A) = 10×0.5 = 5, adjacent = c×cos(A) = 10×0.866 = 8.66 • Know adjacent b=8 and angle A=45°: opposite = b×tan(A) = 8×1 = 8, hypotenuse = b/cos(A) = 8/0.707 = 11.31 • This calculator shows all six trig ratios and their SOHCAHTOA definitions for every triangle

What are the special right triangles?

Two special right triangles have exact angle measures and simple side ratios that appear throughout mathematics, physics, and engineering. 45-45-90 triangle (isosceles right triangle): • Angles: 45°, 45°, 90° • Side ratio: 1 : 1 : √2 • If legs = a: hypotenuse = a√2 • Example: legs = 7, hypotenuse = 7√2 ≈ 9.899 • sin(45°) = cos(45°) = 1/√2 ≈ 0.7071, tan(45°) = 1 30-60-90 triangle: • Angles: 30°, 60°, 90° • Side ratio: 1 : √3 : 2 • If shortest side (opposite 30°) = a: medium side = a√3, hypotenuse = 2a • Example: a=5, sides = 5, 8.660, 10 • sin(30°)=0.5, cos(30°)=√3/2≈0.866, tan(30°)=1/√3≈0.577 • sin(60°)=√3/2≈0.866, cos(60°)=0.5, tan(60°)=√3≈1.732

How do you find a missing side using trigonometry?

If you know one angle (other than the right angle) and one side, you can find all other sides using SOHCAHTOA or its inverse. Case 1 — Know angle A and hypotenuse c: • Opposite leg a = c × sin(A) • Adjacent leg b = c × cos(A) • Example: A=30°, c=10 → a=10×sin(30°)=5, b=10×cos(30°)=8.660 Case 2 — Know angle A and opposite leg a: • Hypotenuse c = a / sin(A) • Adjacent leg b = a / tan(A) • Example: A=30°, a=5 → c=5/0.5=10, b=5/tan(30°)=8.660 Case 3 — Know angle A and adjacent leg b: • Hypotenuse c = b / cos(A) • Opposite leg a = b × tan(A) • Example: A=45°, b=7 → a=7×tan(45°)=7, c=7/cos(45°)=9.899 This calculator handles all these cases — select the appropriate mode and enter your known values.

What is the relationship between the incircle and circumcircle of a right triangle?

Every triangle has an incircle (largest circle that fits inside) and a circumcircle (smallest circle that contains all vertices). For right triangles these have especially clean formulas. Inradius (r) — radius of incircle: • r = (a + b − c) / 2, where c is the hypotenuse • Equivalently: r = Area / s, where s = (a+b+c)/2 is the semi-perimeter • For 3-4-5 triangle: r = (3+4−5)/2 = 1 • The incircle touches all three sides Circumradius (R) — radius of circumcircle: • For ANY right triangle: R = c/2 (half the hypotenuse!) • The hypotenuse is always the diameter of the circumcircle • This is because the angle in a semicircle is always 90° (Thales' theorem) • For 3-4-5 triangle: R = 5/2 = 2.5 • The circumcircle passes through all three vertices, with the hypotenuse as diameter

Right Triangle Calculator — Sides, Angles & Trig Ratios

The Right Triangle Calculator solves any right triangle from any two known values. Four input modes: two legs (a and b), leg and hypotenuse, leg and angle, or hypotenuse and angle. Given any two values, the calculator instantly finds all three sides, both non-right angles, area, perimeter, inradius,...

RIGHT TRIANGLE

KNOWN VALUES

Leg a

shorter leg

Leg b

other leg

LEG a

LEG b

HYPOTENUSE c

ANGLE A

36.869898°

ANGLE B

53.130102°

ANGLE C

90°

Area24 units²

Perimeter24 units

Inradius2 units

Circumradius (c/2)5 units

Altitude to c4.8 units

TypeScalene right

✓ Pythagoras check: 6² + 8² = 100 = 10²

STEP-BY-STEP

Given: leg a = 6, leg b = 8

Hypotenuse: c = √(a²+b²) = √(36+64) = 10

Angle A = arctan(a/b) = arctan(6/8) = 36.869898°

Angle B = 90° − A = 53.130102°

QUICK EXAMPLES

LIVE DIAGRAM

TRIG RATIOS

Function	Angle A	Definition	Angle B
sin	0.6	a/c = 6/10	0.8
cos	0.8	b/c = 8/10	0.6
tan	0.75	a/b = 6/8	1.333333

SOHCAHTOA

SOH: sin(A) = Opposite/Hypotenuse

= 6/10 = 0.6

CAH: cos(A) = Adjacent/Hypotenuse

= 8/10 = 0.8

TOA: tan(A) = Opposite/Adjacent

= 6/8 = 0.75

Created with❤️byeaglecalculator.com

RIGHT TRIANGLE FORMULAS

FORMULAS REFERENCE

NAME	FORMULA	DESCRIPTION
Pythagorean theorem	c² = a² + b²	Find hypotenuse from two legs
Leg from hyp	a = √(c²−b²)	Find missing leg
Angle from legs	A = arctan(a/b)	Find angle from two legs
SOH	sin(A) = a/c (opposite/hypotenuse)	Sine trig ratio
CAH	cos(A) = b/c (adjacent/hypotenuse)	Cosine trig ratio
TOA	tan(A) = a/b (opposite/adjacent)	Tangent trig ratio
Area	A = ½ × a × b	Half the product of the two legs
Inradius	r = (a + b − c) / 2	Radius of inscribed circle
Circumradius	R = c / 2	Half the hypotenuse — Thales theorem
Altitude to c	h = (a × b) / c	Height dropped to the hypotenuse

HOW TO USE

1
Select your input mode: Two Legs if you know both legs a and b; Leg + Hypotenuse if you know one leg and the hypotenuse; Leg + Angle if you know one leg and angle A; Hypotenuse + Angle if you know the hypotenuse and angle A.
2
Enter your two known values. For angles, enter degrees between 0 and 90 (exclusive). The calculator validates that the values form a valid right triangle.
3
All six results appear instantly: the three sides (a, b, c), both angles (A and B), area, perimeter, inradius, circumradius, and altitude to the hypotenuse.
4
Check the SOHCAHTOA panel — sin, cos, and tan are shown for both angle A and angle B with their definitions (opposite/hypotenuse etc.) and computed values.
5
Use the quick example buttons to load the classic 3-4-5 triangle, the 5-12-13 triangle, and the special 45-45-90 and 30-60-90 cases.

WORKED EXAMPLE

Example 1 (two legs 3-4-5): c=√(9+16)=5. A=arctan(3/4)=36.87°, B=53.13°. Area=6. Inradius=1, Circumradius=2.5. sin(A)=0.6, cos(A)=0.8, tan(A)=0.75. Example 2 (angle+hypotenuse A=30°,c=10): a=10×sin30°=5, b=10×cos30°=8.66. Example 3 (isosceles 45-45-90 a=7): b=7, c=7√2=9.899.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.