## Right Angled Triangle

A right angled triangle is a triangle with one of the angles as 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle. In this triangle, the relationship between the various sides can be easily understood with the help of the Pythagoras rule. The side opposite to the right angle is the largest side and is referred to as the hypotenuse. Further, based on the other angle values, the right triangles are classified as an isosceles right triangle and a scalene right triangle. Also, the lengths of the sides of the right triangle, such as 3, 4, 5 are referred to as Pythagorean triples.

Right angle is equal to 90 degrees. In a right angled triangle, the three sides are called: Perpendicular, Base(Adjacent) and Hypotenuse(Opposite). Perpendicular is the side that makes right angle with the base of the triangle.

The largest side side which is opposite to the right-angle(90 degree) is known as the Hypotenuse. The side that is adjacent to the right angle are called legs cathetus. Thus both base and Perpendicular are known as Cathetus.

To find the side of the triangle, we need the sides of other two triangle.

The** Right angled triangle formula** known as Pythagorean theorem (Pythagoras Theorem) is given by

In trigonometry, the values of trigonometric functions at 90 degrees is given by:

Sin 90° = 1

Cos 90° = 0

Tan 90° = Not defined

Cot 90° = 0

Sec 90° = Not defined

Cosec 90° = 1

## What is a Right Triangle?

The definition for a right triangle states that if one of the angles of a triangle is a right angle – 90º, the triangle is called a right-angled triangle or simply, a right triangle. In the given image, triangle ABC is a right triangle, where we have the base, the altitude, and the hypotenuse. Here AB is the base, AC is the altitude, and BC is the hypotenuse. The hypotenuse is the important side of a right triangle which is the largest side and is opposite to the right angle within the triangle.

Here we can have understood the distinct features of a right triangle. The features of triangle ABC are as follows:

- AC is the height, altitude, or perpendicular
- AB is the base
- AC ⊥ AB
- ∠A=90º
- The side BC opposite to the right angle is called the hypotenuse and it is the longest side of the right triangle.

Some of the examples of right triangles in our daily life are the triangular slice of bread, a square piece of paper folder across the diagonal, or the 30-60-90 triangular scale in a geometry box.

## Right Triangle Formula

The great Greek philosopher, Pythagoras, derived an important formula for a right triangle. The formula states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs. It was named after him as Pythagoras theorem. The right triangle formula can be represented in the following way:** The square of the hypotenuse is equal to the sum of the square of the base and the square of the altitude**.

In a right triangle we have: (Hypotenuse)^{2} = (Base)^{2} + (Altitude)^{2}

**Pythagorean Triplet**: The three numbers which satisfy the above equation are the Pythagorean triplets. For example, (3, 4, 5) is a Pythagorean triplet because we know that 3^{2}= 9, 4^{2} = 16, and 5^{2} = 25 and, 9 +16 = 25. Therefore, 3^{2 }+ 4^{2} = 5^{2} These three numbers satisfying this condition are called the Pythagorean triplet. Some of the other examples of Pythagorean triples are (6, 8, 10), and (12, 5, 13).

## Perimeter of a Right Triangle

The right triangle perimeter is the sum of the measures of all the 3 sides. It is the sum of the base, altitude, and hypotenuse of the right triangle. Here, for the below right triangle, the perimeter is equal to the sum of the sides BC + AC + AB = (a + b + c) units. The perimeter is a linear value and has a unit of length.

## Right Triangle Area

The area of a right triangle gives the spread or the space occupied by the triangle. It is equal to half of the product of the base and the height of the triangle. It is a two-dimensional quantity and therefore represented in square units. The only two sides needed to find the right-angled triangle area are the base and the altitude.

Applying the right triangle definition, the area of a right triangle is given by the formula: Area of a right triangle = (1/2 × base × height) square units.

## Properties of Right Triangle

The first property of a right triangle is that it has one of its angles as 90º. The 90º angle is a right angle and the largest angle of a right triangle. Also, the other two angles are lesser than 90º or are acute angles. The right triangle properties are listed below:

- The largest angle is always 90º.
- The largest side is called the hypotenuse which is always the side opposite to the right angle.
- The measurements of the sides follow the Pythagoras rule.
- It cannot have any obtuse angle.

## Types of Right Triangles

We have learned that one of the angles in a right triangle is 90º. This implies that the other two angles in the triangle will be acute angles. There are a few special right triangles namely the **isosceles right triangles** and the **scalene right triangles**. The triangle having both the other two angles equal is referred to as an isosceles right triangle, and the triangle with the other two angles having different values is called a scalene right triangle.

### Isosceles Right Triangle

An isosceles right triangle is called a 90º-45º- 45º triangle. In triangle ABC, angle A = 90º; so by right triangle definition, triangle ABC is a right triangle. Also AB = AC; since two sides are equal, the triangle is also an isosceles triangle. Since AB = AC, the base angles are equal. We know that the sum of the angles of a triangle is 180º. Hence, the base angles add up to 90º which implies that they are 45º each. So in an isosceles right triangle, angles will always be 90º-45º- 45º.

### Scalene Right Triangle

A scalene right triangle is a triangle where one angle is 90° and the other two angles that up to 90º are of different measurements. In the triangle PQR, ∠Q =90º, hence, it is a right triangle. PQ is not equal to QR, hence, it is a scalene triangle. There is also a special case of a scalene triangle 30º-60º-90º which is also a right triangle where the ratio of the triangle’s longest side to its shortest side is 2:1. The side opposite to the 30º angle is the shortest side.

**Tips & Tricks**

Listed here are some of the important tips and tricks relating to a right triangle.

- The measurements of the side lengths will always satisfy the Pythagoras theorem.
- In a right triangle, the hypotenuse is the side opposite to the right angle and it is the longest side of the triangle.
- The other two legs are perpendicular to each other; one is the base and the other is the height.

**Important Notes**

- In a right triangle, (Hypotenuse)
^{2 }= (Base)^{2}+ (Altitude)^{2} - The area of a right triangle is 1/2 × base × height.
- The perimeter of a right triangle is the sum of the measures of all three sides.
- Isosceles right triangles have 90º, 45º, 45º as their degree measures.

## Right Angled Triangle Examples

**Example 1:**Can a right triangle have 11 inches, 60 inches, and 61 inches as its dimensions?

**Solution:**

If 11, 60, and 61 are a Pythagorean triplet, they will form a right triangle. 11^{2} = 121; 60^{2} = 3600; 61^{2}= 3721. We can see that: 121 + 3600 = 3721. Hence, the given numbers are a Pythagorean triplet and can be the dimensions of a right triangle. Therefore, 11 inches, 60 inches, and 61 inches form a right triangle.

**Example 2:** Find the area of a right-angled triangle whose base is 12 units and height is 5 units.

**Solution:**

The area of a triangle formula is 1/2 × b × h. Substituting b = 12 units and h = 5 units, we have, Area =1/2 × 12 × 5 = 30 units^{2}. Therefore, the area of the right triangle is 30 square units.

**Example 3:** The perimeter of a right triangular swimming pool is 720 units. The three sides of the pool are in the ratio 3:4:5. Find the area of the pool.

**Solution:**

The right triangle perimeter is the sum of the measures of all the sides. Therefore, 3x+4x+5x = 720

12x = 720

x = 60

The sides of the triangle are 3x=180 units, 4x=240 units, and 5x=300 units. Since, 180^{2} + 240^{2} = 300^{2}, these sides form a right triangle with a hypotenuse of 300 units. Therefore, the area of the swimming pool is 1/2 × 180 × 240= 21600 units^{2}. Therefore, the area of the swimming pool is 21600 square units.

## FAQs on Right Angled Triangle

### What is a Right Angled Triangle in Geometry?

A triangle in which one of the measures of the angles is 90 degrees is called a right-angled triangle or right triangle.

### What are the Different Types of Right Triangles?

The triangles are classified based on the measurement of the sides and the angles. The three types of right triangles are as mentioned below.

- An isosceles right triangle is a triangle in which the angles are 90º, 45º, and 45º.
- A scalene right triangle is a triangle in which one angle is 90º and the other two acute angles are of different measurements.
- 30º – 60º – 90º triangle is another interesting right triangle where the ratio of the triangle’s longest side to its shortest side is 2:1.

### What is the Measure of the Angles in a Right Triangle?

A right triangle has one of its angles as 90º. The other two angles are acute angles. And all three angles of the right triangle add up to 180° like any other triangle.

### What is the Formula for a Right-Angled Triangle?

The formula used for a right-angled triangle is the Pythagoras formula. It states the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagoras formula is (Hypotenuse)^{2} = (Base)^{2}< + (Altitude)^{2}. This formula has given the Pythagoras triplets such as 3, 4, 5.

### How do you Find the Area of a Right-Angled Triangle?

The area of a right-angled triangle is the space occupied by the triangle and it is equal to half of the product of the base and altitude of the triangle. It is two-dimensional and represented in square units.

Area of a right triangle = 1/2 × Base × Altitude square units

### Can a Right Triangle have Two Equal Sides?

Yes, a right triangle can have two equal sides. The longest side is called the hypotenuse and the other two sides may or may not be equal to each other. A right triangle that has two equal sides is called an isosceles right triangle.

### How to Find the Missing Side of a Right Triangle?

The missing side of a right triangle can be found from the measure of the other two sides. The Pythagoras rule is helpful to find the value of the missing side. As per the Pythagoras rule, we have the square of the hypotenuse equal to the sum of the squares of the other two sides of a right triangle. For example, if a, b and c are the three sides of the right-angled triangle (a being the hypotenuse), then we have the relationship as a^{2} =b^{2} + c^{2}.

### How to Find the Angle of a Right Triangle?

The calculation of angles of a right triangle is very simple. One of the angles of a right triangle is a right angle or 90^{º}. Now if one other angle of the triangle is known, then the missing angle can be easily calculated by using the angle sum formula which states that the sum of the angles of a triangle is always equal to 180º.

## Right Triangle Side and Angle Calculator

Finding out the missing side or angle couldn’t be easier than with our great tool – right triangle side and angle calculator. Choose two given values, type them into the calculator and the remaining unknowns will be determined in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling and you’ll find the formulas behind our calculator.

## How to find the sides of a right triangle

There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:

**Given two sides**

If you know two other sides of the right triangle, it’s the easiest option; all you need to do is apply the Pythagorean theorem:

`a² + b² = c²`

- if leg
`a`

is the missing side, then transform the equation to the form when a is on one side, and take a square root:`a = √(c² - b²)`

- if leg
`b`

is unknown, then`b = √(c² - a²)`

- for hypotenuse c missing, the formula is
`c = √(a² + b²)`

**Given angle and hypotenuse**

**Given angle and one leg**

Find the missing leg using trigonometric functions:

`a = b * tan(α)`

`b = a * tan(β)`

**Given area and one leg**

As we remember from basic triangle area formula, we can calculate the area by multiplying triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:

`area = a * b / 2`

For example, if we know only the right triangle area and the length of the leg `a`

, we can derive the equation for other sides:

`b = 2 * area / a`

`c = √(a² + (2 * area / a)²)`

## How to find the angle of a right triangle

If you know one angle apart from the right angle, calculation of the third one is a piece of cake:

Given `β`

: `α = 90 - β`

Given `α`

: `β = 90 - α`

However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions:

for `α`

`sin(α) = a / c`

so`α = arcsin(a / c)`

(inverse sine)`cos(α) = b / c`

so`α = arccos(b / c)`

(inverse cosine)`tan(α) = a / b`

so`α = arctan(a / b)`

(inverse tangent)`cot(α) = b / a`

so`α = arccot(b / a)`

(inverse cotangent)

and for `β`

`sin(β) = b / c`

so`β = arcsin(b / c)`

(inverse sine)`cos(β) = a / c`

so`β = arccos(a / c)`

(inverse cosine)`tan(β) = b / a`

so`β = arctan(b / a)`

(inverse tangent)`cot(β) = a / b`

so`β = arccot(a / b)`

(inverse cotangent)

## How do you solve a right angle triangle with only one side?

To solve a triangle with one side, you also need **one of the non-right angled angles**. If not, it is impossible:

- If you have the
**hypotenuse**, multiply it by**sin(θ)**to get the length of the side**opposite**to the angle. - Alternatively, multiply the hypotenuse by cos(θ) to get the side adjacent to the angle.
- If you have the non-hypotenuse side
**adjacent**to the angle, divide it by**cos(θ)**to get the length of the**hypotenuse**. - Alternatively, multiply this length by tan(θ) to get the length of the side opposite to the angle.
- If you have an angle and the side
**opposite**to it, you can divide the side length by**sin(θ)**to get the**hypotenuse**. - Alternatively, divide the length by tan(θ) to get the length of the side adjacent to the angle.

## How to find the missing side of a right triangle? How to find the angle? Example

Let’s show how to find the sides of a right triangle with this tool:

- Assume we want to find the missing side given area and one side.
**Select the proper option from a drop-down list**. It’s the third one. **Type in the given values**. For example, an area of a right triangle is equal to 28 in² and b = 9 in.**Our right triangle side and angle calculator displays missing sides and angles!**Now we know that:

- a = 6.222 in
- c = 10.941 in
- α = 34.66°
- β = 55.34°

Now, let’s check how does finding angles of a right triangle work:

- Refresh the calculator.
**Pick the option you need**. Assume that we have two sides and we want to find all angles. The default option is the right one. **Enter the side lengths**. Our right triangle has a hypotenuse equal to 13 in and a leg a = 5 in.**Missing side and angles appear**. In our example, b = 12 in, α = 67.38° and β = 22.62°.

## FAQ

### How many lines of symmetry does a right triangle have?

If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length) it has **one line of symmetry**. Otherwise, the triangle will have **no lines of symmetry**.

### Can a right angled triangle have equal sides?

**No, a right triangle cannot have all 3 sides equal**, as all three angles cannot also be equal, **as one has to be 90°** by definition. A right triangle can, however, have its two non-hypotenuse sides be equal in length. This would also mean the two other angles are equal to 45°.

### Are all right triangles similar?

**Not all right angled triangles are similar**, although some can be. They are similar if all their angles are the same length, or if the ratio of 2 of their sides is the same.

### Solved Examples

**Question 1: **Find is the value of X, where the 15 cm and 20 cm are the sides of the right-angled triangle?

**Solution:**

Given:

Adjacent side = 20 cm

Opposite side = 15 cm

The right angled triangle formula is given by

(Hypotenuse)^{2} = (Adjacent side)^{2} + (Opposite side)^{2}

= (20)^{2} + (15)^{2}

= 400 + 225

= 625 cm

Hypotenuse =

## What is a Triangle?

A triangle is a regular polygon, with three sides and the sum of any two sides is always greater than the third side. This is a unique property of a triangle. In other words, it can be said that any closed figure with three sides and the sum of all the three internal angles is equal to 180°.

Being a closed figure, a triangle can have different types and each shape is described by the angle made by any two adjacent sides.

**Acute angle triangle:**When the angle between any 2 sides is less than 90 degrees it is called an acute angle triangle.**Right angle triangle:**When the angle between a pair of sides is equal to 90 degrees it is called a right-angle triangle.**Obtuse angle triangle:**When the angle between a pair of sides is greater than 90 degrees it is called an obtuse angle triangle.

The other three types of triangles are based on the sides of the triangle.

- Scalene triangle (All the three sides are unequal)
- Isosceles triangle (Two sides are equal)
- Equilateral triangle (All the three sides are equal)

**Note:** A scalene triangle and an isosceles triangle both can be a right triangle. A scalene right triangle will have all three sides unequal in length and any of the one angles will be a right angle. An isosceles right triangle will have its base and perpendicular sides equal in length, which includes the right angle. The third unequal side will be the hypotenuse.

## Right Angled Triangle

A right-angled triangle is a type of triangle that has one of its angles equal to 90 degrees. The other two angles sum up to 90 degrees. The sides that include the right angle are perpendicular and the base of the triangle. The third side is called the hypotenuse, which is the longest side of all three sides.

The three sides of the right triangle are related to each other. This relationship is explained by Pythagoras theorem. According to this theorem, in a right triangle,

Hypotenuse^{2} = Perpendicular^{2} + Base^{2}

See the figure below to understand better.

The area of the biggest square is equal to the sum of the square of the two other small square areas. We can generate the Pythagoras theorem as the square of the length of the hypotenuse is equal to the sum of the length of squares of base and height.

### Shape of Right Triangle

A right triangle is a three-sided closed shape, that has one perpendicular side called the leg or height of the triangle.

## Right Angle Triangle Properties

Let us discuss, the properties carried by a right-angle triangle.

- One angle is always 90° or right angle.
- The side opposite angle of 90° is the hypotenuse.
- The hypotenuse is always the longest side.
- The sum of the other two interior angles is equal to 90°.
- The other two sides adjacent to the right angle are called base and perpendicular.
- The area of the right-angle triangle is equal to half of the product of adjacent sides of the right angle, i.e.,

**Area of Right Angle Triangle = ½ (Base × Perpendicular)**

- If we drop a perpendicular from the right angle to the hypotenuse, we will get three similar triangles.
- If we draw a circumcircle that passes through all three vertices, then the radius of this circle is equal to half of the length of the hypotenuse.
- If one of the angles is 90° and the other two angles are equal to 45° each, then the triangle is called an Isosceles Right Angled Triangle, where the adjacent sides to 90° are equal in length.

Above were the general properties of the Right angle triangle. The construction of the right angle triangle is also very easy. Keep learning with BYJU’S to get more such study materials related to different topics of Geometry and other subjective topics.

## Area of Right Angled Triangle

The area is in the two-dimensional region and is measured in a square unit. It can be defined as the amount of space taken by the 2-dimensional object.

The area of a triangle can be calculated by 2 formulas:

Here, s is the semi perimeter and is calculated as:

Where, a, b, c are the sides of a triangle.

Let us calculate the area of a triangle using the figure given below.

**Fig 1:** Let us drop a perpendicular to the base b in the given triangle.

**Fig 2:** Now let us attach another triangle to a side of the triangle. It forms the shape of a parallelogram as shown in the figure.

**Fig 3:** Let us move the red coloured triangle to the other side of the parallelogram as shown in the above figure.

**Fig 4:** It takes up the shape of a rectangle now.

Now by the property of area, it is calculated as the multiplication of any two sides

Hence, area =b × h (for a rectangle)

Therefore, the area of a right angle triangle will be half i.e.

For a right-angled triangle, the base is always perpendicular to the height. When the sides of the triangle are not given and only angles are given, the area of a right-angled triangle can be calculated by the given formula:

Where a, b, c are respective angles of the right-angle triangle, with ∠b always being 90°.

## Perimeter

As we know, the three sides of the right triangle are Base, Perpendicular and Hypotenuse. Thus the perimeter of the right triangle is the sum of all its three sides.

Perimeter of right triangle = Length of (Base + Perpendicular + Hypotenuse)

Example: If Base =4cm, Perpendicular= 3cm and Hypotenuse = 5cm. What is the perimeter of right triangle?

Perimeter = 4 + 3 + 5 = 12 cm

## Solved Examples

**Q.1: In a right triangle, if perpendicular = 8 cm and base = 6 cm, then what is the value of hypotenuse?**

Solution: Given,

Perpendicular = 8 cm

Base = 6cm

We need to find the hypotenuse.

By Pythagoras theorem, we know that;

Hypotenuse = √(Perpendicular^{2} + Base^{2})

H = √(6^{2} + 8^{2})

= √36 + 64

= √100

= 10 cm

Therefore, the hypotenuse of the right triangle is 10 cm.

**Q.2: If the hypotenuse is 13 cm and the base is 12 cm, then find the length of perpendicular of the right triangle?**

Solution: Given,

Hypotenuse = 13 cm

Base = 12 cm

Perpendicular = ?

By Pythagoras theorem, we know that,

Hypotenuse^{2} = Perpendicular^{2} + Base^{2}

Perpendicular^{2} = Hypotenuse^{2} – Base^{2}

P = √(13^{2} – 12^{2})

P = √(169 – 144)

P = √25

P = 5 cm

Therefore, the value of perpendicular is 5cm.

### Practice Problems

- Find the perpendicular length if a right triangle has a base of 2 units and a hypotenuse of √8 units.
- What is the area of the right triangle with a base of 7 cm and a hypotenuse of 25 cm?
- Show that in a right-angled triangle, the hypotenuse is the longest side.

## Frequently Asked Questions From Right Angle Triangle

### What are Right Angled Triangles?

Right-angled triangles are those triangles in which one angle is 90 degrees. Since one angle is 90°, the sum of the other two angles will be 90°.

### How to Find the Missing Side of any Right Angled Triangle?

For a right-angled triangle, trigonometric functions or the Pythagoras theorem can be used to find its missing sides. If two sides are given, the Pythagoras theorem can be used and when the measurement of one side and an angle is given, trigonometric functions like sine, cos, and tan can be used to find the missing side.

### Can a Triangle have Two Right Angles? Explain.

No, a triangle can never have 2 right angles. A triangle has exactly 3 sides and the sum of interior angles sum up to 180°. So, if a triangle has two right angles, the third angle will have to be 0 degrees which means the third side will overlap with the other side. Thus, it is not possible to have a triangle with 2 right angles.

### What is the sum of all the interior angles of the right triangle?

For any triangle, the sum of all the interior angles is equal to 180 degrees.

### What are the three sides of the right triangle?

The three sides of a right triangle are base, perpendicular and hypotenuse.

What is the formula for a right-angled triangle?

We can use the Pythagoras theorem to find the sides of a right triangle.

C^{2} = A^{2} + B^{2}

Next (trust me for the moment) we can re-arrange that into this:

x = sin^{-1}(0.5)

And then get our calculator, key in 0.5 and use the sin^{-1} button to get the answer:

x = **30°**

And we have our answer!

But what is the meaning of **sin ^{-1}** … ?

Well, the Sine function * “sin”* takes an angle and gives us the

**ratio**“opposite/hypotenuse”,

But * sin^{-1}* (called “inverse sine”) goes the other way …

… it takes the

**ratio**“opposite/hypotenuse” and gives us an angle.

### Example:

- Sine Function: sin(
**30°**) =**0.5** - Inverse Sine Function: sin
^{-1}(**0.5**) =**30°**

On the calculator press one of the following (depending on your brand of calculator): either ‘2ndF sin’ or ‘shift sin’. |

On your calculator, try using **sin** and **sin ^{-1}** to see what results you get!

Also try **cos** and **cos ^{-1}**. And

**tan**and

**tan**.

^{-1}Go on, have a try now.

## Step By Step

These are the four steps we need to follow:

**Step 1**Find which two sides we know – out of Opposite, Adjacent and Hypotenuse.**Step 2**Use SOHCAHTOA to decide which one of Sine, Cosine**or**Tangent to use in this question.**Step 3**For Sine calculate Opposite/Hypotenuse, for Cosine calculate Adjacent/Hypotenuse**or**for Tangent calculate Opposite/Adjacent.**Step 4**Find the angle from your calculator, using one of sin^{-1}, cos^{-1}**or**tan^{-1}

## Examples

Let’s look at a couple more examples:

Example

Find the angle of elevation of the plane from point A on the ground.

**Step 1**The two sides we know are**O**pposite (300) and**A**djacent (400).**Step 2**SOHCAH**TOA**tells us we must use**T**angent.**Step 3**Calculate**Opposite/Adjacent**= 300/400 =**0.75****Step 4**Find the angle from your calculator using**tan**^{-1}

Tan x° = opposite/adjacent = 300/400 = 0.75

**tan ^{-1}** of 0.75 =

**36.9°**(correct to 1 decimal place)

Unless you’re told otherwise, angles are usually rounded to one place of decimals.

**Example**

Find the size of angle a°

**Step 1**The two sides we know are**A**djacent (6,750) and**H**ypotenuse (8,100).**Step 2**SOH**CAH**TOA tells us we must use**C**osine.**Step 3**Calculate Adjacent / Hypotenuse = 6,750/8,100 = 0.8333**Step 4**Find the angle from your calculator using**cos**of 0.8333:^{-1}

cos a° = 6,750/8,100 = 0.8333

**cos ^{-1}** of 0.8333 =

**33.6°**(to 1 decimal place)

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