On This Page
- What is a triangle?
- Right triangle
- 30-60-90 and 45-45-90 triangles
- Triangle area
- Triangle height and altitude
- Triangle square footage
- Isosceles and equilateral triangles
- Triangle perimeter
- SSS, SAS, ASA, AAS, and SSA methods
- Worked examples
- Frequently asked questions
What Is a Triangle?
In ordinary Euclidean geometry, the three interior angles of a triangle always add to 180 degrees, or π radians. The side lengths must also pass the triangle inequality rule: the sum of any two sides must be greater than the third side.
That rule is why not every set of three numbers can form a real triangle. For example, sides 2, 3, and 8 cannot close into a triangle because 2 + 3 is not greater than 8.
Right Triangle
To solve a right triangle, enter 90 for one of the angles and provide two other known values, two sides, or one side and one non-right angle. The calculator returns the hypotenuse, both legs, both acute angles, area, and perimeter.
The hypotenuse is always the longest side and always sits opposite the 90° angle. For right triangles, the Pythagorean theorem gives the fastest route to the missing side:
Where c is the hypotenuse. To find a missing leg when the hypotenuse and one leg are known:
To find the hypotenuse from two legs:
The Pythagorean theorem is also the foundation of the distance formula between two points in coordinate geometry, the straight-line distance between (x1, y1) and (x2, y2) is the hypotenuse of the right triangle those coordinates form.
When one side and one acute angle are known, use trigonometry to find the remaining sides. For angle A opposite side a:
- sin(A) = opposite / hypotenuse
- cos(A) = adjacent / hypotenuse
- tan(A) = opposite / adjacent
The calculator handles all of this automatically. Enter what you know, and it selects the correct method. The tangent ratio, opposite over adjacent, which is rise over run, is the same relationship that defines slope in coordinate geometry. If you have two points on a coordinate plane and need slope, angle, and the line equation, the Slope Calculator handles those directly from the same inputs.
30-60-90 and 45-45-90 Triangles
Special right triangles have fixed angle sets and fixed side ratios, which makes them faster to solve by hand. This calculator finds all sides, area, and perimeter for both types when you enter one side and the matching angles.
30-60-90 Triangle
A 30-60-90 triangle has angles of 30°, 60°, and 90°. If the short leg (opposite 30°) has length x, the other sides are always:
To use the calculator: enter 30 for angle A, 60 for angle B, and 90 for angle C, then enter any one known side.
45-45-90 Triangle
A 45-45-90 triangle (also called an isosceles right triangle) has two equal legs and angles of 45°, 45°, and 90°. If each leg has length x, the hypotenuse is:
To use the calculator: enter 45 for angle A, 45 for angle B, and 90 for angle C, then enter either leg length as one of the known sides.
Triangle Area
Area updates automatically in the result panel as you enter known values, using whichever method the available inputs support.
Area From Base and Height
When base and perpendicular height are both known, the area formula is:
The height here is the perpendicular distance from the base to the opposite vertex, not a side of the triangle unless it is a right triangle.
How to Calculate Triangle Area Without Height (Heron's Formula)
When all three sides are known but no height is given, use Heron's formula to calculate the triangle area without height:
For example, a triangle with sides 5, 6, and 7: s = 9, area = √(9 × 4 × 3 × 2) = √216 ≈ 14.7 square units.
Area From Two Sides and the Included Angle
When two sides and the angle between them are known:
Triangle Height and Altitude
The height of a triangle (also called the altitude) is the perpendicular distance from a vertex to the opposite side. Each triangle has three altitudes, one for each base, but problems usually ask for the one relative to the given base.
If area and base are already known, rearrange the area formula to find height:
For an equilateral triangle with side s, the altitude is:
For a right triangle, the two legs are their own altitudes relative to each other, so height equals the shorter leg when the hypotenuse is used as the base:
The calculator returns area automatically once the triangle is solved, so you can always find the altitude by dividing 2 × area by whichever base you need.
Triangle Square Footage
To calculate the square footage of a triangular area, a garden bed, a roof section, a plot of land, use the same area formula with measurements in feet:
If only the three side lengths are known in feet, use Heron's formula instead. The result is automatically in square feet when all inputs are in feet.
Example: A triangular garden with base 18 ft and height 11 ft has area = ½ × 18 × 11 = 99 sq ft.
For plots where three side lengths are known but no perpendicular height was measured, enter all three side lengths in the calculator and read the area from the result panel. Convert to square feet by keeping all inputs in feet.
Isosceles and Equilateral Triangles
Isosceles triangle
An isosceles triangle has two equal sides and two equal base angles. To solve it, enter the two equal sides as side a and side b, and the base as side c. The calculator finds all three angles, the height from apex to base, area, and perimeter.
The apex angle (angle C, between the two equal sides) and each base angle (A and B, which are equal) are related by:
To calculate the area of an isosceles triangle when you know the equal side length (s) and base (b):
Equilateral triangle
An equilateral triangle has three equal sides and three 60° angles. Enter the side length as sides a, b, and c (all the same value), or enter just one side and the three angles as 60, 60, 60.
For an equilateral triangle with side length s:
For example, an equilateral triangle with side 10 cm has area = (100 × 1.732) ÷ 4 ≈ 43.3 cm² and height ≈ 8.66 cm.
Triangle Perimeter
Perimeter is the sum of all three sides:
The result panel shows the perimeter automatically once any valid triangle is solved. To calculate the perimeter of a right triangle when only two sides are given, find the third side with the Pythagorean theorem first, then add all three.
Triangle Solving Methods
Different input patterns use different formulas. The calculator checks the values you entered and chooses the first valid method that can solve a real triangle.
SSS: Three Sides
Use SSS when all three side lengths are known. A side-side-side triangle calculator uses the Law of Cosines to find the angles, then Heron's formula to find the area.
SAS: Two Sides and the Included Angle
When two sides and the angle between them are known, the missing side can be found with the Law of Cosines. The remaining angles can then be found from the Law of Sines and the 180-degree angle sum.
ASA and AAS: Two Angles and One Side
When two angles are known, the third angle is found by subtraction. Then the Law of Sines can scale the known side to the missing sides.
SSA: Two Sides and a Non-Included Angle
SSA can sometimes be ambiguous because two different triangles may match the same information. This calculator returns one valid solution and notes when the SSA method was used.
Worked Examples
Example 1: SSS Triangle With Area and Perimeter
Problem: A math workbook gives a triangle with sides a = 5, b = 6, and c = 7. Find the missing angles, perimeter, and area.
- Use the Law of Cosines to solve the angles.
- Perimeter = 5 + 6 + 7 = 18.
- Semi-perimeter = 18 ÷ 2 = 9.
- Area = √(9(9 - 5)(9 - 6)(9 - 7)) = √216 = 14.7.
Answer: The area is about 14.7 square units, and the perimeter is 18 units.
Example 2: SAS Triangle
Problem: A survey sketch shows two sides of 10 and 14 units with the included angle of 38 degrees. Find the third side first, then solve the triangle.
- Enter side a = 10, side b = 14, and angle C = 38 degrees.
- The calculator uses c² = a² + b² - 2ab cos(C).
- After side c is found, the remaining angles and area are calculated.
Answer: This is an SAS problem because the known angle is between the two known sides.
Example 3: ASA or AAS Triangle
Problem: A geometry problem gives angle A = 42 degrees, angle B = 63 degrees, and side c = 12. Solve the triangle.
- Find angle C: 180 - 42 - 63 = 75 degrees.
- Use the Law of Sines to find side a and side b.
- Use the solved sides to calculate perimeter and area.
Answer: Two angles and one side are enough to solve a unique triangle.
Example 4: Right Triangle Hypotenuse Check
Problem: A ladder forms a right triangle with a wall. The horizontal distance from the wall is 6 ft and the vertical height is 8 ft. Find the ladder length.
- The ladder is the hypotenuse.
- Use c² = 6² + 8².
- c² = 36 + 64 = 100.
- c = 10 ft.
Answer: The ladder length is 10 ft. You can also enter the two legs and a 90 degree angle in the calculator to check the full triangle.
Example 5: Triangle Square Footage
Problem: A triangular garden bed has a base of 18 ft and a height of 11 ft. Find the square footage.
- Area = 1/2 × base × height.
- Area = 1/2 × 18 × 11.
- Area = 99 square feet.
Answer: The triangular area is 99 sq ft.
Frequently Asked Questions
How do you calculate the area of a triangle?
Three methods depending on what is known. Base and perpendicular height: area = ½ × base × height. All three sides (no height): use Heron's formula, s = (a + b + c) ÷ 2, then area = √(s(s−a)(s−b)(s−c)). Two sides and the included angle: area = ½ × a × b × sin(C). The calculator applies the correct method automatically.
How do you calculate the perimeter of a triangle?
Add all three side lengths: perimeter = a + b + c. If a side is missing, solve the triangle first, then add all three solved sides.
How do you calculate the height (altitude) of a triangle?
If area and base are known: height = (2 × area) ÷ base. For an equilateral triangle with side s: height = (s√3) ÷ 2. For a right triangle, each leg is the altitude relative to the other leg as base.
How do you find the hypotenuse of a right triangle?
Use the Pythagorean theorem: c = √(a² + b²), where a and b are the two legs. Legs 3 and 4: c = √(9 + 16) = √25 = 5.
How do you find the missing side of a triangle?
For a right triangle with two sides known, use the Pythagorean theorem. For two sides and the included angle, use the Law of Cosines: c² = a² + b² − 2ab cos(C). For two angles and one side, use the Law of Sines. Enter the known values above and the calculator selects the right method.
How do you calculate the angles of a right triangle?
If both legs are known: angle A = arctan(a ÷ b). If one leg and the hypotenuse are known, use arcsin or arccos. For example, a = 3, c = 5: angle A = arcsin(3/5) ≈ 36.87°.
How do you calculate the area of an isosceles triangle?
Find the altitude from apex to base: h = √(s² − (b/2)²) where s is the equal side length and b is the base. Then area = ½ × b × h. Or enter the two equal sides and the base in the calculator and read the area directly.
How do you calculate the area of an equilateral triangle?
area = (s² × √3) ÷ 4, where s is the side length. Side 10 cm: area = (100 × 1.7321) ÷ 4 ≈ 43.3 cm².
How do you calculate triangle square footage?
Use area = ½ × base (ft) × height (ft). If only side lengths are known in feet, enter them in the calculator, Heron's formula gives the result in square feet automatically.
How many values are needed to solve a triangle?
Three, with at least one being a side length. Three angles alone (AAA) define the shape but not the size, side lengths, area, and perimeter cannot be found without at least one known side.
Why is my triangle invalid?
Three common causes: a side length equals or exceeds the sum of the other two (triangle inequality), the angles do not sum to 180°, or in an SSA case the given side is too short to reach the opposite base.
References
- Triangle, Wolfram MathWorld: definitions, classification by side and angle type, and key properties of triangles.
- Law of Cosines, Wolfram MathWorld: formula used for SSS and SAS triangle solving.
- Heron's Formula, Wolfram MathWorld: area from three side lengths without height.