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Right Angle Triangle is a type of triangle that has one angle that measures exactly 90 Degrees. 90o angle, referred to as the right angle, distinguishes the right triangle from other triangles. The sides of a right triangle are called the Hypotenuse and the Legs. Right Triangle Formula includes several formulas related to the area, perimeter, and hypotenuse of a right triangle.
Area of a Right Angle Triangle is calculated by multiplying half of the base times the height.
| Area of Right Angle Triangle = \(\frac{1}{2}\) x B x H |
Where, b is the base (adjacent side) and h is the height (perpendicular side).
Perimeter of a Right Angle Triangle is calculated by adding the lengths of all three sides of the triangle. In mathematical terms, it can be expressed as:
| Perimeter of Right Angle Triangle = Side 1 + Side 2 + Hypotenuse |
Where, Side 1 and Side 2 are the lengths of the two shorter sides, and Hypotenuse is the length of the longest side.
Read More: Right Angle Triangle Theorem
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Key Terms: Right Angle Triangle, Triangle, Hypotenuse, Pythagoras Theorem, Right Angle, Isosceles Triangle, Scalene Triangle
What is Triangle?
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Triangle is a geometric shape with three sides and three angles. The sum of the interior angles of a triangle always adds up to 180 degrees.
- Triangles can be classified based on their sides and angles, such as equilateral, isosceles, scalene, right, acute, and obtuse.
- Triangles are widely used in mathematics, engineering, and design due to their stability and symmetry.
- Triangles have many real-world applications, such as construction, mapping, and navigation.
- In trigonometry, triangles are used to calculate distances, angles, and slopes.
The video below explains this:
Pythagoras Theorem Detailed Video Explanation:
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What is Right Angle Triangle?
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A triangle is called a right-angled triangle or simply a right triangle if any of its angles is a right angle (measuring 90o). Right triangles are also important in trigonometry, where they are used to define and calculate sine, cosine, and tangent ratios.
- In the below figure, Triangle ABC is a right triangle.
- AB is the hypotenuse of a right triangle.
- AC and BC are the other two sides of the triangle.
Right Triangle
Right Angle Triangle Area
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Right Angle Triangle Area is the space contained within the right triangle's perimeter.
- The area inside the boundary is typically divided into squares of equal size.
- Therefore, the area of a right triangle is determined by counting the number of unit squares inside the triangle.
- Square units are used to measure the area.
- It is usually measured in square units like m2, cm2, in2, etc.
Right Triangle Area Formula
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Right Triangle Area Formula is given as:
| Area of Right Angle Triangle = \(\frac{1}{2}\) x B x H |
Where
- B = Base (Adjacent Side)
- H = Height (Perpendicular Side)
As a result, to find the area of a right triangle, multiply the base by the height, then divide the result by two.
Derivation of Area of Right Angle Triangle
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Let's take a rectangle with length "l" and width "w" into consideration in order to determine the formula for the area of a right triangle. Draw a diagonal now, as demonstrated in the figure below.
Derivation of Right Angle Triangle Area
It can be seen from the figure that a rectangle is divided into two right-angled triangles that are congruent to one another and overlap one another.
Now,
| Area of Rectangle = Length x Width |
Area of Rectangle = 2 x (Area of one Right Triangle)
= \(\frac{1}{2}\) x Area of Rectangle
= \(\frac{1}{2}\)x Length x Width
Because, Width = Height (h) and Length = Base (b)
Area of Right Triangle = \(\frac{1}{2}\) x b x h (square units)
| Area of Right Angle Triangle = \(\frac{1}{2}\) x b x h |
Read More: Area of a Sector
Pythagoras Theorem
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Hypotenuse of a right triangle can be found using the Pythagoras theorem. According to the theorem, the hypotenuse's square is equal to the sum of its squares with the other two sides. Pythagoras Theorem can be used to determine the third side of a right triangle if the lengths of any two of its sides are known.
Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. The theorem is expressed mathematically as
| c2 = a2 + b2 |
Where
- c is length of the hypotenuse.
- a and b are the lengths of the legs.
Thus,
| Hypotenuse2 = Perpendicular2 + Base2 |
Right Triangle Perimeter Formula
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Perimeter of a triangle is the total length of all its sides. In a right triangle, the perimeter is equal to the sum of the lengths of its three sides.
- The formula for calculating the perimeter of a right triangle is P = a + b + c, where a and b are the lengths of the legs and c is the length of the hypotenuse.
- The hypotenuse can be calculated using the Pythagorean theorem, c = a2 +b2
- By combining the Pythagorean theorem and the perimeter formula, it is possible to find the perimeter of a right triangle using only the lengths of its legs.
- The perimeter of a right triangle is an important measurement in many real-world applications, such as construction, mapping, and navigation.
Right Angle Triangle Formula Solved Examples
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Here are some solved examples on Right Triangle Formula:
| Example 1. Find the perimeter of a right triangle with sides measuring 6 cm and 8 cm. Solution: To find the perimeter of a right triangle, simply add the length of all sides. The perimeter is calculated as: Perimeter = a + b + c In this case, a = 6 cm and b = 8 cm. The hypotenuse is calculated using the Pythagorean theorem: c = √(a2 + b2) = √(62 + 82) = √(36 + 64) = √100 = 10 cm The perimeter of the right triangle is calculated as: Perimeter = 6 + 8 + 10 = 24 cm Example2. Find the area of a right triangle with sides measuring 4 cm and 5 cm. Solution: The formula to calculate the area of a right triangle is: Area = \(\frac{1}{2}\)ab In this case, a = 4 cm and b = 5 cm. The area of the right triangle is calculated as: Area = \(\frac{1}{2}\)x 4 x 5 = \(\frac{1}{2}\)x 20= 10 cm2 Example 3. Find the length of the hypotenuse of a right triangle with sides measuring 3 cm and 4 cm. Solution: The length of the hypotenuse can be calculated using the Pythagorean theorem: c = √(a2 + b2) = √(32 + 42) = √(9 + 16) = √25 = 5 cm Thus, the length of the hypotenuse is 5 cm. |
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Things to Remember
- Right Triangle is a type of triangle that has one angle that measures exactly 90 degrees.
- The side opposite the right angle is called the hypotenuse, and the two other sides are called the legs.
- The area of a right triangle can be calculated using the formula Area = \(\frac{1}{2}\) × Base × Height.
- The perimeter of a right triangle is the sum of the lengths of all its sides. It can be calculated using the formula Perimeter = a + b + c.
- The length of the hypotenuse of a right triangle can be calculated using the Pythagorean theorem.
- Pythagoras Theorem states that c2 = a2 + b2, where c is the length of the hypotenuse and a and b are the lengths of the legs.
Sample Questions
Ques. Two sides of a right-angled triangle are 6 cm and 8 cm long, respectively. Figure out the following:
(i) The hypotenuse's length.
(ii) The triangle's perimeter.
(iii) Dimensions of the triangle. (5 Marks)
Ans. The parameters listed are:
One side B = 6 cm
Other side P = 8 cm
(i) The length of the Hypotenuse is as follows:
H² = P² + B²
Substituting the values,
H² = 8² + 6²
H² = 64 + 36
H² = 100
H = √100
H = 10cm
(ii) The Perimeter of the right triangle is,
S= a + b+ c
S= 6 +8+10
S = 24 cm
(iii) Area of right triangle is,
A =\(\frac{1}{2}\)x b x h
A = \(\frac{1}{2}\) x 6 x 8
A = 24cm2
Ques. A right triangle has a height of 15 cm and a 17 cm long side. Calculate the right triangle's area. (5 Marks)
Ans. Given,
- A right triangle's hypotenuse, or its longest side = 17 cm
- Height = 15 cm
Finding the right triangle's base is the first step in determining its area.
Finding the Base of a Right Triangle:
The base can be determined using Pythagoras' Theorem in the manner shown below:
Hypotenuse2 = Base2 + Height2
(17)2 = (Base)2 + (15)2
(Base)2 = 172 – 152
(Base)2 =289 – 225
(Base)2 = 64
As a result, Base = √64 = 8 cm.
Hence, The right triangle's base is 8 cm.
Finding the Area of a Right Triangle:
Area of a right triangle = \(\frac{1}{2}\) x b x h square units.
When we replace the values in the formula, we obtain
A = \(\frac{1}{2}\)×8×15 cm2
A = 4×15 cm2
A = 60 cm2
Therefore, the area of the right triangle is 60 cm2.
Ques. 32 cm is the circumference, 10 cm is the height, and 13 cm is the hypotenuse of a right-angled triangle. Find the area of the triangle. (5 Marks)
Ans. Given that,
- Perimeter, S= 32 cm
- Hypotenuse, H = 13 cm
- Height, P = 10 cm
- Third side, B =?
Perimeter is calculated as follows:
S = H = P + B
So, B = 32 – (13 + 10)
B = 9 cm
Hence, Area = \(\frac{1}{2}\) x b x h
A= \(\frac{1}{2}\) x 9 x 10
A= 45 cm2
Ques. A right-angled triangle's height and hypotenuse are 12 cm and 13 cm, respectively. Determine its area. (3 Marks)
Ans. Given that,
- Height = 12 cm
- Hypotenuse = 13 cm
By applying Pythagoras' theorem,
(13)2 = (Base)2 + (12)2
Base2 = 132 – 122 = 25 cm
Base = √25 = 5 cm
Using the Area of a Triangle formula,
Area = \(\frac{1}{2}\) x b x h
Area = \(\frac{1}{2}\) × 5 × 12
A = 30 cm2
Ques. Determine the height of a right triangle with a base length of 60 m and an area of 420 m2. (3 Marks)
Ans. Given:
- Base = 60 m
- Area = 420 m2
Formula for area of right angle triangle is
A = \(\frac{1}{2}\)x b x h Square units
Substituting the values in the formula er get,
420 = \(\frac{1}{2}\) x 60 x h
420 = 30 x h
h = 420 / 30
h = 14 m
Ques. Calculate the area of a right-angled triangle with a 12 cm-long side and a 15 cm-long hypotenuse. (2 Marks)
Ans. Using Pythagoras Theorem,
AB2 = AC2 – BC2
AB2 = 152 – 122
AB2 = 225 – 144
AB2 = 81
As a result, AB = 9
Hence, Area of the triangle = \(\frac{1}{2}\) x base x height
= \(\frac{1}{2}\) x 12 x 9
= 54 cm2
Ques. The triangle's base and height have a 3:2 ratio. Determine the base and height of a triangle with an area of 243 cm2. (3 Marks)
Ans. Assume the common ratio is x.
- The height of the triangle = 2x
- The base of the triangle = 3x
- Area of triangle = 243cm2
A = \(\frac{1}{2}\) x b x h
243 = \(\frac{1}{2}\) x 3x x 2x
243 = 3x²
X2 = 243/3
X = √81
X = √(9 x 9)
X = 9
Hence, height of triangle = 2 x 9 = 18 cm
Base of triangle = 3x = 3 x 9 = 27 cm
Ques. Find the area of a triangle whose sides are 41cm, 28 cm, 15 cm. Also, find the length of the altitude corresponding to the largest side of the triangle. (3 Marks)
Ans. Semi-perimeter of the triangle = (a + b + c)/2
= (41 + 28 + 15)/2
= 84/2
= 42 cm
Therefore, area of the triangle = √(s(s – a) (s – b) (s – c))
= √(42 (42 – 41) (42 – 28) (42 – 15)) cm2
= √(42 × 1 × 27 × 14) cm2
= √(3 × 3 × 3 × 3 × 2 × 2 × 7 × 7) cm2
= 3 × 3 × 2 × 7 cm2
= 126 cm²
Now, area of triangle = \(\frac{1}{2}\)× b × h
Therefore, h = 2A/b
= (2 × 126)/41
= 252/41
= 6.1 cm
Ques. The sides of the triangular plot are in the ratio 2 : 3 : 4 and the perimeter is 180 m. Find its area. (5 Marks)
Ans. Let the common ratio be x, then the three sides of triangle are 2x, 3x, 4x
Now, perimeter = 180 m
Therefore, 2x + 3x + 4x = 180
⇒ 9x = 180
⇒ x = 180/9
⇒ x = 20
Therefore, 2x = 2 × 20 = 40
3x = 3 × 20 = 60
4x = 4 × 20 = 80
Area of triangle = √(s(s – a) (s – b) (s – c))
= √(90(90 – 80) (90 – 60) (90 – 40))
= √(90 × 10 × 30 × 50))
= √(3 × 3 × 2 × 5 × 2 × 5 × 3 × 2 × 5 × 5 × 5 × 2)
= 3 × 2 × 5 × 2 × 5 √(3 × 5)
= 300 √15 m²
= 300 × 3.872 m²
= 1161.600 m²
= 1161.6 m²
Ques. What is the area of a right-angled triangle with a 15 cm base and a 20 cm height? (1 Marks)
Ans. Area of triangle = \(\frac{1}{2}\)x base x height
When we put the base and height values into the equation above, we get:
A = \(\frac{1}{2}\) x 15 x 20 cm²
A = 150 cm²
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