Area of a Right Triangle Calculator
Enter any two known values of a right triangle and get area, missing side, and perimeter instantly.
📐 What is the Area of a Right Triangle?
The area of a right triangle is the amount of flat space enclosed by the triangle. Because a right triangle has two sides (called legs) that meet at a 90-degree angle, the area formula is particularly simple: area = half times leg a times leg b, or A = (1/2) × a × b. The two legs are perpendicular, so one naturally acts as the base and the other as the height, eliminating the need to compute a separate altitude.
Right triangles appear everywhere in applied mathematics and engineering. Roof trusses form right triangles so that horizontal and vertical loads can be resolved independently. Surveyors lay out right-triangle baselines to measure inaccessible distances using trigonometry. In construction, the 3-4-5 triangle is used to verify square corners because its sides satisfy the Pythagorean theorem. Staircase design involves a right triangle where the horizontal run and vertical rise are the two legs. Any time a problem involves a perpendicular relationship between two lengths, a right triangle is the underlying geometry.
Not everyone starts with two legs. Sometimes you know the hypotenuse (the longest side, opposite the right angle) and one leg, or the hypotenuse and one of the acute angles. This calculator handles all three cases. In the Hyp + Leg mode, it uses the Pythagorean theorem (b = sqrt(c squared minus a squared)) to find the missing leg before computing area. In the Hyp + Angle mode, it uses trigonometry (a = c times sin(A), b = c times cos(A)) to derive both legs.
A common misconception is that the hypotenuse determines the area uniquely. It does not. Infinitely many right triangles share the same hypotenuse but have different leg ratios and different areas. The maximum area for a given hypotenuse occurs when the triangle is isosceles (45-45-90), where area = c squared divided by 4. The minimum area approaches zero as one leg shrinks toward zero. This calculator shows area, the computed missing dimension, and the perimeter so you have all the geometric information you need in one place.
📐 Formulas
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 - Classic 3-4-5 Right Triangle
A right triangle has legs of 3 m and 4 m. Find the area, hypotenuse, and perimeter.
Example 2 - Roof Rafter from Hypotenuse and Span
A roof rafter (hypotenuse) is 5.2 m long. The horizontal span (one leg) is 4.5 m. Find the triangular cross-section area.
Example 3 - Triangular Plot with Hypotenuse and Angle Known
A triangular land plot is right-angled. The diagonal boundary measures 100 m and makes a 40-degree angle with the baseline. Find the area.
Example 4 - 45-45-90 Maximum Area Triangle
A right triangle is isosceles with both legs equal to 7 cm. Confirm it gives the maximum area for this hypotenuse.
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for area of a right triangle?
The area of a right triangle is A = half times base times height (A = 0.5 times a times b), where a and b are the two legs (the sides that form the right angle). Since the legs are perpendicular, one naturally serves as the base and the other as the height. For a triangle with legs 6 and 8, A = 0.5 times 6 times 8 = 24 square units.
How do you find the area of a right triangle with hypotenuse and one leg?
Use the Pythagorean theorem to find the missing leg first: b = square root of (c squared minus a squared). Then apply A = 0.5 times a times b. Example: hypotenuse c = 10, leg a = 6, so b = sqrt(100 minus 36) = sqrt(64) = 8, and A = 0.5 times 6 times 8 = 24 square units. The Hyp + Leg mode does both steps automatically.
How do you calculate right triangle area from hypotenuse and angle?
If you know hypotenuse c and acute angle A, the legs are: a = c times sin(A) and b = c times cos(A). Area = 0.5 times a times b = 0.5 times c squared times sin(A) times cos(A), which equals 0.25 times c squared times sin(2A). Example: c = 10, A = 30 degrees, so area = 0.5 times 100 times sin(30) times cos(30) = 0.5 times 100 times 0.5 times 0.866 = 21.65 square units.
What is the area of a 3-4-5 right triangle?
The 3-4-5 right triangle has legs 3 and 4 and hypotenuse 5. Area = 0.5 times 3 times 4 = 6 square units. This is the simplest Pythagorean triple and is commonly used in construction to verify right angles: a triangle with sides in the 3:4:5 ratio always contains a right angle by the Pythagorean theorem (9 plus 16 equals 25).
What is the area of a 45-45-90 right triangle?
A 45-45-90 right triangle has two equal legs. If each leg has length a, then area = 0.5 times a squared. For a 45-45-90 triangle with legs of length 5, area = 0.5 times 25 = 12.5 square units. The hypotenuse = a times sqrt(2) = 5 times 1.414 = 7.07 units. This triangle appears in square diagonals and many design patterns.
What is the area of a 30-60-90 right triangle?
A 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2. If the short leg = a, then the long leg = a sqrt(3) and hypotenuse = 2a. Area = 0.5 times a times a sqrt(3) = (sqrt(3)/4) times a squared. For a = 4: area = (1.732/4) times 16 = 6.928 square units. This triangle is half an equilateral triangle with side 2a.
How does the area of a right triangle compare to a rectangle of the same dimensions?
A right triangle with legs a and b has exactly half the area of the rectangle with the same base a and height b. This is because the triangle is literally half of that rectangle, cut diagonally. Rectangle area = a times b; triangle area = 0.5 times a times b. So a right triangle always covers 50% of its bounding rectangle.
Can you find the area of a right triangle with only the hypotenuse?
No, the hypotenuse alone does not uniquely define a right triangle. Infinitely many right triangles share the same hypotenuse with different leg ratios and different areas. You need one more piece of information: one leg, one acute angle, or the ratio of the legs. For example, hypotenuse 10 could have legs 6 and 8 (area 24) or legs 7.07 and 7.07 (area 25), among infinite possibilities.
What units does the area come out in?
The area unit is the square of whatever unit you use for the sides. If you enter legs in centimetres, area is in square centimetres. If you enter in metres, area is in square metres. To convert: 1 m squared = 10,000 cm squared; 1 ft squared = 144 in squared; 1 m squared = 10.764 ft squared. The calculator does not perform unit conversion, so keep all inputs in the same unit.
How do you find the area of a right triangle given the perimeter?
Given perimeter P and the constraint a squared plus b squared = c squared, you cannot find a unique area from P alone. You need at least one more piece of information (a side or angle). However, if you also know the hypotenuse c, then a plus b = P minus c, and ab = ((a+b) squared minus (a squared + b squared)) / 2 = ((P-c) squared minus c squared) / 2. Then area = 0.5 times ab.
Is the formula for a right triangle area different from other triangles?
No, the general formula A = 0.5 times base times height applies to all triangles. For a right triangle, the two legs are perpendicular, so either leg naturally serves as both base and height without needing an altitude construction. For non-right triangles, you must find the perpendicular height, which may lie outside the triangle (obtuse case). The right triangle formula is simply the general formula applied directly to the perpendicular legs.
What is the largest area a right triangle can have for a given hypotenuse?
The maximum area occurs when the two legs are equal (isosceles right triangle, 45-45-90 case). For hypotenuse c, equal legs = c divided by sqrt(2), and maximum area = 0.5 times (c divided by sqrt(2)) squared = c squared divided by 4. For example, hypotenuse 10 gives maximum area 100 divided by 4 = 25 square units. Any other leg ratio with the same hypotenuse produces a smaller area.