TutorialKart.comWhat Is a Right Angle Triangle?
A right angle triangle, also called a right triangle or right-angled triangle, is a triangle in which one interior angle measures exactly 90 degrees.
The other two angles are acute angles. Because the sum of the interior angles of every triangle is 180 degrees, the two acute angles in a right angle triangle always add up to 90 degrees.
The following picture shows a right angle triangle.
In the diagram, the right angle is at vertex B. It is commonly marked using a small square inside the angle.
Sides of a Right Angle Triangle
A right angle triangle has three sides: two legs and one hypotenuse.
- The two sides that meet at the right angle are called the legs.
- When one leg is treated as the base, the other leg acts as the perpendicular height.
- The side opposite the right angle is called the hypotenuse.
Hypotenuse of a Right Angle Triangle
The hypotenuse is the side opposite the 90-degree angle. It is always the longest side of a right angle triangle.
If the right angle is at vertex B, then the side opposite B is the hypotenuse. The remaining two sides form the right angle.
Pythagorean Theorem for a Right Angle Triangle
The side lengths of a right angle triangle satisfy the Pythagorean theorem.
If the two legs have lengths a and b, and the hypotenuse has length c, then:
a² + b² = c²
The hypotenuse can therefore be calculated using:
c = √(a² + b²)
For example, if the legs measure 3 units and 4 units, the hypotenuse is:
c = √(3² + 4²) = √25 = 5
A triangle with side lengths 3, 4, and 5 is therefore a right angle triangle.
How to Identify a Right Angle Triangle from Side Lengths
When all three side lengths are known, arrange them so that c is the longest side. The triangle is right-angled when:
a² + b² = c²
For side lengths 5, 12, and 13:
5² + 12² = 25 + 144 = 169
13² = 169
Since both values are equal, the triangle is a right angle triangle.
Area and Perimeter of a Right Angle Triangle
Right Angle Triangle Area Formula
The two perpendicular legs can be used directly as the base and height. The area is:
Area = 1/2 × base × height
If the base is 8 cm and the height is 6 cm, then:
Area = 1/2 × 8 × 6 = 24 cm²
Right Angle Triangle Perimeter Formula
The perimeter is the sum of the lengths of the two legs and the hypotenuse.
Perimeter = a + b + c
For a triangle with sides 3 cm, 4 cm, and 5 cm, the perimeter is:
Perimeter = 3 + 4 + 5 = 12 cm
Measurements Needed to Construct a Right Angle Triangle
A right angle triangle can be uniquely constructed from several different sets of measurements. The measurements must provide enough information to determine the lengths or directions of all three sides.
The following diagram labels the sides and angles of a right angle triangle.
Common construction cases include two perpendicular side lengths, one leg and one acute angle, or the hypotenuse and one additional measurement.
Construct a Right Angle Triangle from Base and Height
When the base and perpendicular height are known, they determine the two legs of the right angle triangle.
Steps to draw a right angle triangle with a known base and height:
- Draw a line segment equal to the given base length.
- At one endpoint of the base, construct a line perpendicular to the base.
- On the perpendicular line, mark the given height.
- Join the marked point to the other endpoint of the base. This new side is the hypotenuse.
The base and height meet at 90 degrees, so the resulting triangle is right-angled.
Construct a Right Angle Triangle from a Base and an Acute Angle
A right angle triangle can also be constructed when the base length and the acute angle between the base and hypotenuse are known.
Steps to draw a right angle triangle with a known base and acute angle:
- Draw a line segment equal to the given base length.
- At one endpoint, use a protractor to draw a ray making the given acute angle with the base.
- At the other endpoint, construct a line perpendicular to the base.
- Extend the angled ray and perpendicular line until they intersect.
- Use the intersection point as the third vertex of the right angle triangle.
Construct a Right Angle Triangle from Height and an Acute Angle
The base and height are interchangeable names for the two perpendicular legs. Rotating the triangle does not change its side lengths or angle measures.
To construct the triangle from a known height and the angle between the height and hypotenuse, use the same process as the previous construction. Draw the known leg, construct a perpendicular line at one endpoint, and draw the given acute angle at the other endpoint. Their intersection gives the third vertex.
Other Measurement Sets That Define a Right Angle Triangle
A right angle triangle can also be determined using either of the following measurement sets:
- The hypotenuse and one leg.
- The hypotenuse and one acute angle.
- One leg and one acute angle.
- All three side lengths, provided they satisfy the Pythagorean theorem.
Knowing only the three angles is not enough to determine one unique triangle. Triangles with the same angles can have different sizes.
Special Right Angle Triangles
45-45-90 Right Angle Triangle
A 45-45-90 triangle has two equal acute angles. It is an isosceles right triangle, so its two legs are equal.
If each leg has length x, the hypotenuse has length x√2. Its side-length ratio is:
1 : 1 : √2
30-60-90 Right Angle Triangle
A 30-60-90 triangle has acute angles measuring 30 degrees and 60 degrees.
The side opposite the 30-degree angle is the shortest side, and the hypotenuse is twice its length. The side-length ratio is:
1 : √3 : 2
Right Angle Triangle Facts
- A right angle triangle has exactly one 90-degree angle.
- The two remaining angles are acute and add up to 90 degrees.
- The hypotenuse is opposite the right angle and is the longest side.
- The two legs are perpendicular to each other.
- The side lengths satisfy
a² + b² = c². - The midpoint of the hypotenuse is equally distant from all three vertices.
- The circumcenter of a right angle triangle lies at the midpoint of the hypotenuse.
Frequently Asked Questions About Right Angle Triangles
Can a right angle triangle have two equal sides?
Yes. An isosceles right triangle has two equal legs and two equal acute angles of 45 degrees each.
How do you find the missing side of a right angle triangle?
Use the Pythagorean theorem. To find the hypotenuse, use c = √(a² + b²). To find a missing leg, rearrange the formula as a = √(c² − b²).
Is the hypotenuse always the longest side?
Yes. The hypotenuse lies opposite the largest angle, which is the 90-degree angle, so it is always the longest side.
Can a right angle triangle also be scalene?
Yes. A right angle triangle is scalene when all three sides have different lengths. A 3-4-5 triangle is one example.
Can a triangle have more than one right angle?
No. Two right angles would already total 180 degrees, leaving no angle measure for the third vertex.
Right Angle Triangle Summary
A right angle triangle contains one 90-degree angle, two perpendicular legs, and a hypotenuse opposite the right angle. Its sides follow the Pythagorean theorem, its area is half the product of its two legs, and it can be constructed from several combinations of side lengths and acute angles.