How to find the base of a right triangle given two sides

Aug 20, 2021 | Turito Team USA

Math is an essential subject for almost every competitive examination including the SAT and ACT. Getting the fundamentals right is quite important when it comes to solving complex problems in Math. A lot of students find it difficult to memorize the steps and recollect the Mathematical formulae during the test prep. With the additional pressure of competitive examinations, the student’s woes are doubled.  

Basic Terminologies relating to finding the base of a Triangle: 

What is a base?
Base means bottom. 

What is the height?
The height of any object is how much it measures from its top to its bottom. 

How do you identify the base of a triangle?
Any of the three sides of a triangle can be considered the base of the triangle. 

How do you identify the height of a triangle?
The height of a triangle is the perpendicular line dropped onto its base from the corner opposite the base. 

 As Turito understands this difficulty and addresses the issue, we bring you how one can find the base of a Triangle. 

1. Way to find the triangle base from the area of the triangle: 

            Area of the triangle= ½ bh  

            Where b- base 

             h- height 

Once the area of the triangle is found, the area formulae can be applied to A=1/2 be in the reverse approach to get the length of the base. So, the formula for the base of the triangle is 

             b=2A/h 

       2. Ways to find the triangle base from a right-angled triangle: 

One can apply the Pythagorean theorem, or use the formula which involves the length of two sides or the hypotenuse.  The reverse technique helps even in this regard. 

Source:https://tutorme.com/blog/post/how-to-find-the-base-of-a-triangle/ 

Even if the area value is not known,  but know the length of the triangle- this formula can be used. In the right-angled triangle, the height and the base have the same length.  

       3. Way to find the triangle base of an Isosceles Triangle: 

 It is well-known fact that an isosceles triangle has two sides that have an equal length. 

a is the length of the two known sides of the isosceles that are equivalent. 

       4. Way to find the base for the Equilateral Triangle: 

 An equilateral triangle has all sides of equal length.   The regular area of the triangle formula applies here 

Visualize the equilateral triangle as two smaller right triangles to find the height, where the hypotenuse is the same length as the side length b. By the 30-60-90 rule, a special case of a right triangle, we know that the base of this smaller right triangle is  and the height of this smaller right triangle is , assuming b to be the hypotenuse. 

After finding the height, use it in the following formula by adding area and height values: 

 And substitute  for the height: 

Tools to use for finding the base of a triangle: 

The Pythagorean Theorem: This equation shows the relationship between a right triangle’s three sides, finding the area is easier. 

A triangle that contains a 90-degree or right angle in one of its three corners is called a right triangle. A right triangle’s base is one of the sides that adjoins the 90-degree angle. 

All about Pythagoras-  the creator of this resourceful formula: 

Pythagoras, who hails from Greece, is often linked with the discovery of the mathematical theorem still used today to calculate the dimensions of a right triangle. To complete the calculations, you must know the dimensions of the longest side of the geometric shape, the hypotenuse, as well as another one of its sides. 

The legendary Mathematician migrated to Italy in about 532 BCE because of the political and social conditions of the country. Pythagoras also determined the significance of numbers in music. Unfortunately, most of his works are not documented, which is why scholars don’t know if it was Pythagoras himself who discovered the theorem or one of the many students or disciples who were members of the Pythagorean brotherhood, a religious or mystical group whose principles influenced the work of Plato and Aristotle. 

Effective tips for finding the base of the triangle: 

1. Know the types of the triangles first: It is important to know the classification for triangles- names Isosceles, Equivalent, Right-Angled, and Scalene Triangle.  

 Isosceles Triangle: Of the three sides, two sides are of equal length for this type of triangle. 

 Equivalent Triangle: The three sides of this triangle are equal in length. 

 Right-Angled: In this triangle, one side is the longest compared to the other two sides and it is termed the hypotenuse.  

 Scalene:  Here, all the sides of the triangle are unequal. Solving problems and finding the area and base of this triangle is tricky to find.  

1. Memorize the formulae: The student is expected to memorize all the formulae associated with finding the area and length of the triangle. This will save time to find the solution effortlessly.
2. Think out of the box: Though Math is a subject that depends on step by step sequence of solving a problem, at times- the idea of reverse engineering to get the answer. Especially when it comes to competitive examinations, time is a valuable resource. Hence, thinking out of the box will make finding solutions easier. 

The Final Word: 

Finding the base of the triangle depends on the type of the triangle for the length, and the student is expected to memorize the formulae to use them accordingly to get the solution.  Taking the right guidance from the tutor, and enrolling in a course can easily help the student to build strong fundamentals.  

Math is a subject that is heavily concept-dependent and having strong fundamentals will greatly help the student in the long run- whether in competitive exams or pursuing a career in the subject.

The lengths of the sides of a right triangle are related by the Pythagorean Theorem, which states $a^2 + b^2 = c^2$, where $a$ and $b$ are the lengths the two legs and $c$ is the length of the hypotenuse. Using the information you have, you want to solve $$ 40^2 + b^2 = 43^2 $$ to get $b = \sqrt{249} \approx 15.78$.

Once you have the lengths of all three sides, you can use the Law of Cosines to figure out the missing angles, which states $$ x^2 = y^2 + z^2 - 2yz\cos \alpha, $$ where $x,y,z$ are the lengths of the legs of the triangle and $\alpha$ is the angle opposite $x$. You'll need to plug in all the side lengths (making sure that $x$ is the length of the side opposite the angle you want to find) and solve for $\alpha$ (using $\arccos$ at the final step).

EDIT: As pointed out in Marty's answer, the Law of Cosines is not needed for a right triangle. Let's call the angle between the base and the hypotenuse $\alpha$. We can use the simpler relationship $\cos \alpha = \frac{40}{43}$, which rearranges to $\alpha = \arccos \frac{40}{43} \approx 21.53^\circ$. Since a triangle has $180^\circ$ in total, the remaining angle must have approximate measure $180^\circ - 90^\circ - 21.53^\circ = 68.47^\circ$.

The area of a right triangle is the portion that is covered inside the boundary of the triangle. A right-angled triangle is a triangle where one of the angles is a right angle (90 degrees). It is simply known as a right triangle. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called legs. The two legs can be interchangeably called base and height. The area of right-angle triangle formula is given in the image below.

What is Area of a Right Triangle?

The area of a right-angled triangle, as we discussed earlier, is the space that is inside it. This space is divided into squares of unit length and the number of unit squares that are inside the right triangle is its area. The area is measured in square units. Let us consider the following right triangle whose base is 4 units and height is 3 units.

Can you try counting the number of unit squares inside this triangle? There are 6 unit squares in total. So the area of the above triangle is 6 square units. But it is not possible to calculate the area of a right triangle always by counting the number of squares. There must be a formula to do this. Let us see what is the formula for finding the area of a right triangle.

Area of Right Triangle Formula

In the above example, if we multiply the base and height, we get 3 × 4 = 12 and if we divide it by 2, we get 6. So the area of a right triangle is obtained by multiplying its base and height and then making the product half.

Area of a right triangle = 1/2 × base × height

Examples:

• The area of a right triangle with base 6 ft and height 4 ft is 1/2 × 6 × 4 = 12 ft2.
• The area of a right triangle with base 10 m and height 5 m is 1/2 × 10 × 5 = 25 m2.
• The area of a right triangle with base 11 in and height 5 in is 1/2 × 11 × 5 = 27.5 in2.

How to Derive Area of Right Triangle Formula?

Consider a rectangle of length l and width w. Also, draw a diagonal. You can see that the rectangle is divided into two right triangles.

We know that the area of a rectangle is length × width. So the area of the above rectangle is l × w. We can see that the two right triangles are congruent as they can be arranged such that one overlaps the other. Thus, the area of the rectangle is equal to twice the area of one of the above right triangles. i.e.,

Area of rectangle = l × w = 2 × (Area of one right triangle)

This gives,

Area of one right triangle = 1/2 × l × w.

We usually represent the legs of the right-angled triangle as base and height.

Thus, the formula for the area of a right triangle is, Area of a right triangle = 1/2 × base × height.

Area of Right Triangle With Hypotenuse

Let us recollect the Pythagoras theorem which states that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides. i.e., (hypotenuse)2 = (base)2 + (height)2.

Though it is not possible to find the area of a right triangle just with the hypotenuse, it is possible to find its area if we know one of the base and height along with the hypotenuse. Let us see an example.

Example: Find the area of a right angle triangle whose base is 6 in and hypotenuse is 10 in.

Solution:

Substitute the given values in the Pythagoras theorem,

(hypotenuse)2 = (base)2 + (height)2

102 = 62 + (height)2

100 = 36 + (height)2

(height)2 = 64

height = √(64) = 8 in.

So, the area of the given triangle = 1/2 × base × height = 1/2 × 6 × 8 = 24 in2.

1. Example 1: The longest side of a bread slice that resembles a right triangle is 13 units. If its height is 12 units, find its area using the area of a right triangle formula.

   Solution:

   We know that the longest side of a right triangle is called the hypotenuse.

   So, it is given that hypotenuse = 13 units and height = 12 units.

   Substitute the given values in the Pythagoras theorem,

   (hypotenuse)2 = (base)2 + (height)2

   132 = (base)2 + (12)2

   169 = (base)2 + 144

   (base)2 = 25

   base = √(25) = 5 units.

   The area of the bread slice = 1/2 × base × height = 1/2 × 5 × 12 = 30 square units.

   Therefore, the area of the given bread slice = 30 square units.

2. Example 2: A swimming pool is in the shape of a right triangle. Its sides are in the ratio 3:4:5. Its perimeter is 720 units. Find its area.

   Solution:

   Let us assume that the sides of the swimming pool be 3x, 4x, and 5x.

   It is given that its perimeter = 720 units.

   3x + 4x + 5x = 720

   12x = 720

   x = 60

   So the sides of the triangle are,

   3x = 3(60) = 180 units

   4x = 4(60) = 240 units

   5x = 5(60) = 300 units

   Since 300 units is the longest side of the swimming pool (which is in the shape of a right triangle), it is the hypotenuse.

   So, 180 units and 240 units must be the base and the height of the swimming pool interchangeably.

   Using the area of right triangle formula,

   The area of the swimming pool = 1/2 × base × height = 1/2 × 180 × 240 = 21,600 units2.

   Therefore, the area of the given swimming pool = 21,600 units2.

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FAQs on Area of Right Triangle

The area of a right triangle is defined as the total space or region covered by a right-angled triangle. It is expressed in square units. Some common units used to represent area are m2, cm2, in2, yd2, etc.

What is the Formula for Finding the Area of a Right Triangle?

The area of a right triangle of base b and height h is 1/2 × base × height (or) 1/2 × b × h square units.

How Do You Find the Perimeter and Area of a Right Triangle?

The area of a right triangle of base b and height h is found using the formula 1/2 × b × h and its perimeter is obtained by just adding all the sides. In case only two of its sides are given, then we use the Pythagoras theorem to find the third side.

How Do You Find the Area of a Right Triangle Without the Base?

If only the height and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the base using the Pythagoras theorem. Then we can use the formula 1/2 × base × height to find its area. For example, to find the area of a right triangle with a height of 4 cm and hypotenuse 5 cm, we first find its base using the Pythagoras theorem. Then we get,

base = √[(hypotenuse)2 - (height)2] = √(52 - 42) = √9 = 3 cm.

Area of the right triangle = 1/2 × 3 × 4 = 6 cm2.

How Do You Find the Area of a Right Triangle Without the Height?

If only the base and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the height using the Pythagoras theorem. Then we can use the formula 1/2 × base x height to find its area.

For example, to find the area of a right triangle with a base of 4 cm and hypotenuse 5 cm, we first find its height using the Pythagoras theorem. Then we get

height = √[(hypotenuse)2 - (base)2] = √(52 - 42) = √9 = 3 cm.

Area of the triangle = 1/2 × 3 × 4 = 6 cm2.

How Do You Find the Area of a Right Triangle With a Hypotenuse?

In fact, it is not possible to find the area of a right triangle just with the hypotenuse. We need to know at least one of the base and height along with the hypotenuse to find the area.

• If we know the base and the hypotenuse, we find the height using the Pythagoras theorem.
• If we know the height and the hypotenuse, we find the base using the Pythagoras theorem.

Then, we can find the area of the right triangle using the formula 1/2 × base × height.