Triangle Calculator

How to Calculate a Missing Angle of a Triangle Manually?

You can find a missing angle when you know the other two angles. Use the triangle angle sum formula: $A+B+C=180°$.

Formula

$C=180°-(A+B)$

Step by step Guide

1. Add the two known angles.
2. Subtract their total from 180°.
3. The answer is the missing angle.

Practical Example

A triangle has angles of 65° and 45°. Add them: $65°+45°=110°$. Then subtract: $180°-110°=70°$. The missing angle is 70°.

This method works for every triangle because its three internal angles always total 180°.

How to Calculate a Missing Length of a Triangle Manually?

The method depends on the triangle and the values you know. For a right triangle, use the Pythagorean theorem: $c=\sqrt{a^2+b^2}$.

Formula

$c=\sqrt{a^2+b^2}$

Step by step Guide

1. Identify the known sides.
2. Choose the correct formula.
3. Substitute the values.
4. Solve for the missing length.

Practical Example

A right triangle has legs of 6 and 8. Use $c=\sqrt{6^2+8^2}=10$. The missing hypotenuse is 10 units.

For other triangles, the Law of Sines or Law of Cosines can find missing side lengths.

How to Calculate the Area of a Triangle Manually?

Use the base and height when you know both measurements. The basic triangle area formula is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is its height.

Formula

$A=\frac{1}{2}bh$

Step by step Guide

1. Measure the base.
2. Measure the matching height.
3. Multiply them.
4. Divide the result by 2.

Practical Example

If $b=10$ cm and $h=6$ cm, then $A=\frac{1}{2}(10)(6)=30$ cm². The triangle area is 30 cm².

You can also use Heron's formula when you know all three side lengths.

How to Calculate Perimeter of a Triangle Manually?

A triangle's perimeter equals the total length of its three sides. Use the formula $P=a+b+c$.

Formula

$P=a+b+c$

Step by step Guide

1. Find all three side lengths.
2. Check that they use the same unit.
3. Add the three sides.
4. Write the answer with the correct unit.

Practical Example

For sides of 5 cm, 7 cm, and 9 cm, use $P=5+7+9=21$ cm. The perimeter is 21 cm.

Always use the same length unit for every side before calculating the perimeter.

What Information Do You Need to Solve a Triangle?

You need enough known sides or angles to determine the triangle. Common cases include SSS, SAS, ASA, AAS, and SSA.

Three Known Sides (SSS)

With three side lengths, you can solve the triangle using SSS methods and the Law of Cosines.

Two Sides and Included Angle (SAS)

With two sides and their included angle, you can use the Law of Cosines to solve the triangle.

Two Angles and One Side (ASA)

With two angles and the side between them, use the angle sum and Law of Sines.

Two Angles and a Side (AAS)

With two angles and one nonincluded side, find the third angle first. Then use the Law of Sines.

Two Sides and a Nonincluded Angle (SSA)

SSA can produce zero, one, or two possible triangles. This case creates the ambiguous case.

Triangle Formulas Used by the Calculator

The calculator uses common triangle formulas to find unknown sides, angles, area, and perimeter. Your known values decide which formula works best. Right triangles use the Pythagorean theorem, while general triangles can use the Law of Sines or Law of Cosines. Area calculations can use base and height, two sides with an included angle, or Heron's formula.

Angle Sum Formula

Every triangle has three internal angles that add to 180°. Use $A+B+C=180°$ to find a missing angle.

Pythagorean Theorem

For a right triangle, use $a^2+b^2=c^2$. The hypotenuse $c$ sits opposite the 90° angle.

Law of Sines

The Law of Sines connects sides and opposite angles. Use $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$.

Law of Cosines

The Law of Cosines finds a missing side or angle. Use $c^2=a^2+b^2-2ab\cos C$ for a missing side.

Triangle Area Formulas

The basic formula is $A=\frac{1}{2}bh$. You can also use $A=\frac{1}{2}ab\sin C$ or Heron's formula.

Types of Triangles

Triangles fall into groups based on side lengths or angle sizes. Knowing the type helps you choose the right geometry formula.

By Side Length

Side lengths create three main triangle types: equilateral, isosceles, and scalene.

Equilateral Triangle

An equilateral triangle has three equal sides and three equal angles. Each angle measures 60°.

Isosceles Triangle

An isosceles triangle has at least two equal sides. Its two opposite angles also have equal measures.

Scalene Triangle

A scalene triangle has three different side lengths. Its three angles also have different measures.

By Angle

Angle sizes create three main types: acute, right, and obtuse triangles.

Acute Triangle

An acute triangle has three angles smaller than 90°. Its largest angle still stays below a right angle.

Right Triangle

A right triangle has one 90° angle. The side opposite that angle is the hypotenuse.

Obtuse Triangle

An obtuse triangle has one angle greater than 90°. Its other two angles must each measure less than 90°.