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.

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.

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.

Area of a Right Triangle Calculator

Q: 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).

Q: 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.

Q: 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.

Q: 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.

Q: 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.

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

Two Legs A = ½ × a × b

a = first leg (one side forming the right angle)

b = second leg (the other side forming the right angle)

Hypotenuse: c = √(a² + b²) by the Pythagorean theorem

Example: a = 6, b = 8 → A = 0.5 × 6 × 8 = 24 sq units; c = 10

Hyp + Leg A = ½ × a × √(c² − a²)

c = hypotenuse (longest side, opposite the right angle)

a = known leg

Missing leg: b = √(c² − a²) via the Pythagorean theorem

Example: c = 10, a = 6 → b = √(100 − 36) = 8; A = 0.5 × 6 × 8 = 24 sq units

Hyp + Angle A = ½ × c² × sin(A) × cos(A)

c = hypotenuse

A = one acute angle (in degrees, between 0 and 90)

Leg a = c × sin(A); Leg b = c × cos(A)

Equivalent form: A = ¼ × c² × sin(2A)

Example: c = 10, A = 30° → a = 5, b = 8.66; Area = 0.5 × 5 × 8.66 = 21.65 sq units

📖 How to Use This Calculator

Steps

Select your input type - Click Two Legs, Hyp + Leg, or Hyp + Angle depending on which measurements you have. Each tab shows the correct input fields for that combination.

Enter the known values - Type the leg lengths, hypotenuse, or angle into the number fields. Use the sliders for quick visual exploration or type directly for precise decimal values.

Click Calculate - Press the Calculate button to compute the area, missing side or leg, and perimeter. Results appear immediately below the button.

Use the results - Area is in square units. The secondary box shows the hypotenuse or missing leg. Perimeter gives the full sum of all three sides. Use the share buttons to copy or link the result.

💡 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.

Apply the area formula: A = 0.5 × a × b = 0.5 × 3 × 4 = 6 square metres.

Find the hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25 = 5 m.

Perimeter = a + b + c = 3 + 4 + 5 = 12 m. This triangle has sides in the classic 3:4:5 ratio.

Area = 6 sq m | Hypotenuse = 5 m | Perimeter = 12 m

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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.

Hypotenuse c = 5.2 m, known leg a = 4.5 m. Check: a must be less than c. 4.5 < 5.2. Valid.

Find the rise (missing leg): b = √(5.2² − 4.5²) = √(27.04 − 20.25) = √6.79 = 2.605 m.

Area = 0.5 × 4.5 × 2.605 = 5.861 sq m. Perimeter = 4.5 + 2.605 + 5.2 = 12.305 m.

Area = 5.8611 sq m | Rise (b) = 2.6051 m

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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.

c = 100 m, angle A = 40 degrees. Leg a = 100 × sin(40°) = 100 × 0.6428 = 64.28 m.

Leg b = 100 × cos(40°) = 100 × 0.7660 = 76.60 m.

Area = 0.5 × 64.28 × 76.60 = 2,460.9 sq m = 0.246 hectares.

Area = 2,460.9 sq m | Leg a = 64.28 m | Perimeter = 240.88 m

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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.

Legs a = b = 7 cm. Area = 0.5 × 7 × 7 = 24.5 sq cm.

Hypotenuse = √(49 + 49) = 7√2 = 9.899 cm.

Maximum area for this hypotenuse = c² / 4 = 9.899² / 4 = 98 / 4 = 24.5 sq cm. Confirmed.

Area = 24.5 sq cm (maximum for hypotenuse 9.899 cm)

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❓ Frequently Asked Questions

What is the formula for the area of a right triangle?+

The area of a right triangle = half times leg a times leg b (A = 0.5 times a times b), where a and b are the two perpendicular sides (legs). Because the legs are already perpendicular, one is the base and the other is the height, so no additional altitude calculation is needed. For 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 only the hypotenuse and one leg?+

First find the missing leg using the Pythagorean theorem: b = square root of (c squared minus a squared). Then apply A = 0.5 times a times b. For hypotenuse 10 and leg 6: b = sqrt(100 minus 36) = 8; A = 0.5 times 6 times 8 = 24 square units. The Hyp + Leg mode above does both steps automatically.

How do you calculate right triangle area from the hypotenuse and an angle?+

With hypotenuse c and acute angle A: leg a = c times sin(A), leg b = c times cos(A), then area = 0.5 times a times b = 0.5 times c squared times sin(A) times cos(A). A shortcut: area = c squared times sin(2A) divided by 4. For c = 10 and A = 30 degrees: area = 100 times sin(60) divided by 4 = 100 times 0.866 divided by 4 = 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 smallest integer-sided right triangle and is widely used in construction to check right angles: a corner is square if the diagonal across a 3-unit by 4-unit rectangle measures exactly 5 units.

What is the area of a 45-45-90 right triangle?+

A 45-45-90 triangle has two equal legs. If each leg = a, then area = 0.5 times a squared. For legs of length 5: area = 0.5 times 25 = 12.5 square units. The hypotenuse = a times sqrt(2). This triangle gives the maximum area for any right triangle with a given hypotenuse. It appears in square diagonals and is the cross-section of square lumber cut diagonally.

What is the area of a 30-60-90 right triangle?+

A 30-60-90 triangle has sides in ratio 1 : sqrt(3) : 2. If the short leg = a, long leg = a times sqrt(3), 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 exactly half of an equilateral triangle with side 2a.

Is a right triangle half the area of a rectangle?+

Yes, always. A right triangle with legs a and b has area 0.5 times a times b, which is exactly half the area of the rectangle with the same base a and height b (area a times b). The triangle is literally the rectangle cut diagonally along the hypotenuse. This relationship makes right triangles convenient for calculating half-rectangle areas in floor plans, land plots, and material cutting.

Can you find right triangle area with only the hypotenuse?+

No. The hypotenuse alone does not uniquely determine the triangle or its area. Infinitely many right triangles share the same hypotenuse but have different leg ratios and different areas. You need one more piece of information: a leg, an acute angle, or the leg ratio. The maximum area for hypotenuse c is c squared divided by 4 (isosceles case), and the area can approach zero for very unequal legs.

What units does the area come out in?+

Area is in the square of the unit used for the leg inputs. Legs in centimetres give area in square centimetres. Legs in metres give area in square metres. The calculator is unit-agnostic and does not convert. To convert: 1 m squared = 10,000 cm squared; 1 ft squared = 144 in squared; 1 m squared = 10.764 ft squared.

How do I find a leg if I know the area and the other leg?+

Rearrange the formula: b = 2 times area divided by a. If area = 30 and leg a = 6, then b = 2 times 30 divided by 6 = 10. This is the algebraic inverse of the area formula A = 0.5 times a times b. For example, to cut a triangular piece of fabric with area 50 cm squared from a strip 10 cm wide, the required cut length is b = 2 times 50 divided by 10 = 10 cm.

What is the largest possible area of a right triangle with hypotenuse 10?+

The maximum area is c squared divided by 4 = 100 divided by 4 = 25 square units. This occurs when the triangle is a 45-45-90 isosceles right triangle with equal legs = 10 divided by sqrt(2) = 7.071 units. Any other leg combination with hypotenuse 10 produces a smaller area. Mathematically this follows from the AM-GM inequality applied to the product of the legs subject to a squared plus b squared = 100.

How does a right triangle area formula differ from the general triangle area formula?+

The general triangle formula is A = 0.5 times base times height, where height is the perpendicular distance from the base to the opposite vertex. For a right triangle, the two legs are already perpendicular to each other, so one leg is the base and the other is automatically the height. No separate altitude construction is needed. The formula is the same; the right angle simply makes the height trivial to identify.

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📌 Quick Tips

💡If you know only the hypotenuse and one leg, use the Hyp + Leg mode. The calculator applies the Pythagorean theorem to find the missing leg before computing area, so you do not need to calculate it manually.

💡The 3-4-5 right triangle is the simplest Pythagorean triple. Its area is 0.5 times 3 times 4 = 6 square units. Check your formula by entering legs 3 and 4 in the Two Legs mode.

💡For the Hyp + Angle mode, the angle must be one of the two acute angles (between 0 and 90 degrees). A right angle (90 degrees) is the right angle itself and cannot be entered here.

💡Doubling both legs quadruples the area because area scales with the square of the linear dimensions. A right triangle with legs 6 and 8 has area 24, while one with legs 12 and 16 has area 96, which is four times 24.

💡If you know the area and one leg, rearrange the formula: b = 2 times area divided by a. For example, area 30 and leg 6 gives leg b = 2 times 30 divided by 6 = 10.