## How to Find the Base of a Right Triangle

The Pythagorean Theorem, an equation that shows the relationship between a right triangle's three sides, can help you to find the length of its base. 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.

#### TL;DR (Too Long; Didn't Read)

The Pythagorean Theorem is essentially, *a*^{2} + *b*^{2} = *c*^{2}. Add side *a* times itself to side *b* times itself to arrive at the length of the hypotenuse, or side *c* times itself.

### The Pythagorean Theorem

The Pythagorean Theorem is a formula that gives the relationship between the lengths of a right triangle's three sides. The triangle's two legs, the base and height, intersect the triangle's right angle. The hypotenuse is the side of the triangle opposite the right angle. In the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:

a^2 + b^2 = c^2

In this formula, * a* and

** are the lengths of the two legs and

**b**** is the length of the hypotenuse. The

**c**^{2}signifies that

**,

**a****, and

**b**** are

**c***squared*. A number squared is equal to that number multiplied by itself – for example, 4

^{2}is equal to 4 times 4, or 16.

### Finding the Base

Using the Pythagorean theorem, you can find the base, *a*, of a right triangle if you know the lengths of the height, *b*, and the hypotenuse, *c*. Since the hypotenuse squared is equal to the height squared plus the base squared, then:

a^2 = c^2 - b^2

For a triangle with a hypotenuse of 5 inches and a height of 3 inches, find the base squared:

c^2 - b^2 = (5 × 5) - (3 × 3) = 25 - 9 = 16 \\ \implies a = 4

Since b^{2} equals 9 , then *a* equals the number that, when squared, makes 16. When you multiply 4 by 4, you get 16, so the square root of 16 is 4. The triangle has a base that is 4 inches long.

### A Man Called Pythagoras

The Greek philosopher and mathematician, Pythagoras, or one of his disciples, is attributed 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.

Pythagoras migrated to Italy in about 532 BCE because of the political climate in his own country. Besides being credited with this theorem, Pythagoras – or one of the members of his brotherhood – also determined the significance of numbers in music. None of his writings have survived, 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.

### Measurement: Triangle Base and Height

#### 1. Identifying the base and height of a triangle.

**What is a base?**

Base means bottom.

**What is 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.

**Notes:**

- The height of a triangle corresponds to its base. If the base changes, so does the height.
- The height of a triangle is the shortest line onto the base from its opposite corner.
- The height of a triangle may be outside the triangle.

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## TutorMe Blog

Depending on whether you're studying the area of a triangle, the Pythagorean Theorem, or advanced trigonometry in your high school math class, there are many ways to find the base of a triangle. Here are a couple of the most common scenarios and methods:

### 1. How To Find the Base from the Area of a Triangle

The formula for the area of a triangle is:

If you know the area of a triangle and also the height of the triangle, you can apply the area formula in reverse to calculate the length of the base, using algebra to isolate the variable that you care about:

### 2. How To Find the Base of a Right Triangle

When it comes to finding the base for a right triangle, you can apply either the Pythagorean Theorem or the area formula depending on whether you know the lengths of two sides or only the area and height. If you know the length of the hypotenuse and the other side length as well, you can apply the Pythagorean Theorem in reverse:

If you don't know the area but you know the length of the side of the triangle, you can safely use the area formula. The right angle means the height (an imaginary perpendicular line from the base) and the side of the triangle are one and the same. Consider the following triangle abc:

Here, we flip the area formula and solve:

### 3. How To Find the Base of an Isosceles Triangle

If you know the side length and height of a triangle that is isosceles, you can find the base of the triangle using this formula:

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

### 4. Base of an Equilateral Triangle

All three sides of a triangle that is equilateral are the same length. We can use this to our advantage by once again, substituting the area formula:

To find h, we visualize the equilateral triangle as two smaller right triangles, 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.

Now that we know the height, we can apply the area formula:

And substitute for the height:

### How To Find the Base of a Triangle

As you can see, the base of a triangle can be calculated through many different ways depending on the type of triangle and information you have. However, the length of the base is always related to the area of the triangle with the area formula.

The Pythagorean Theorem can help you out if you are working with right triangles. And don’t forget that the definitions of an isosceles or equilateral triangle also are tools to help you work through geometric problems like these!

## Area of Triangle – Explanation & Examples

In this article, you will learn **the area of a triangle and**** determine the area of different types of triangles**. The area of a triangle is the amount of space inside the triangle. It is measured in square units.

Before getting into the **topic of a triangle area**, let’s familiarize ourselves with terms such as the base and height of a triangle.

** The base** is the side of a triangle which is considered to be the bottom, while

*of a triangle is the perpendicular line dropped onto its base from the vertex opposite to the base.*

**t****he height**In the illustration above, the dotted lines are the possible heights of △*ABC. *Note that every triangle has, possibly, three heights or altitudes.

- The height of triangle △
*ABC*is equal to*h*when the base is a side._{1 } - The height of triangle △
*ABC*is equal to*h2*when the base is*AB.* - The height of triangle △
*ABC*is equal to*h*when the base is_{3} - The height of triangle △
*ABC*can be outside a triangle (*h*), which is the same as the height_{4}*h*._{1}

*From the illustrations above, we can make the following observations:*

- The height of a triangle depends on its base.
- The perpendicular to the base of a triangle is equal to the height of the triangle.
- The height of a triangle can be outside the triangle.

Having discussed the concept of the height and the base of a triangle, let’s now embark on how to calculate the area of a triangle.

### How to Find the Area of a Triangle?

**The area of a rectangle is well known to us, i.e., length * width**. What will happen if we bisect the rectangle diagonally (cut into half)? What will be its news area? For example, in a rectangle with a base and height of 6 units and 12 units, respectively, the rectangle area is 72 square units.

Now, if you divide it into **two equal halves** (after bisecting the rectangle diagonally), the area of two new shapes must be 36 square units each. The two news shapes are triangles. That means if the rectangle is diagonally cut into two equal halves, the two new shapes formed are triangles, where each triangle has an area equal to ½ of the area of the rectangle.

**The area of a triangle is the total space or region enclosed by a particular triangle.The area of a triangle is the product of the base and height divided by 2.**

The standard unit for measurement of the area is square meters (m^{2}).

*Other units include:*

- Square millimeters (mm
^{2}) - Square inches (in
^{2}) - Square kilometers (km
^{2}) - Square yards.

### Area of a Triangle Formula

The general formula for calculating the area of a triangle is;

Area (A) = ½ (b × h) square units, where; A is the area, b is the base, and h is the triangle’s height. The triangles might be different in nature, but it important to note that this formula applies to all the triangles. Different types of triangles have different area formulas.

Note: The base and height must be in the same units, i.e., meters, kilometers, centimeters, etc.

#### Area of a right triangle

The area of a triangle = (½ × Base × Height) square units.

*Example 1*

Find the area of the right-angled triangle whose base is 9 m and height is 12m.

__Solution__

A = ¹/₂ × base × height

= ¹/₂ × 12 × 9

= 54 cm²

*Example 2*

The base and height of a right triangle are 70 cm and 8 m, respectively. What is the area of the triangle?

__Solution__

A = ½ × base × height

Here, we have 70 cm and 8 m. You can choose to work with cm or m. Let’s work in meters by changing 70cm to meters.

Divide 70cm by 100.

70/100 = 0.7m.

⇒ A = (½ × 0.7 × 8) m^{2}

⇒ A = (½ x 5.6) m^{2}

⇒ A = 2.8m^{2}

#### Area of an isosceles triangle

An isosceles triangle is a triangle whose two sides are equal and also two angles are equal. The formula for the area of an isosceles triangle is;

⇒A = ½ (base × height).

When the height of an isosceles triangle is not given, then the following formula is used to find the height:

Height= √ (a^{2} − b^{2}/4)

Where;

b = base of the triangle

a = Side length of the two equal sides.

Therefore, the area of an isosceles triangle can be;

⇒A = ½ [√ (a^{2} − b^{2 }/4) × b]

Also, the area of an isosceles right triangle is given by:

A= ½ × a^{2}, where a = Side length of the two equal sides

*Example 3*

Calculate the area of an isosceles triangle whose base is 12 mm and height is 17 mm.

__Solution__

⇒A = ½ × base × height

⇒ 1/2 × 12 × 17

⇒ 1/2 × 204

= 102 mm^{2}

*Example 4*

Find the area of an isosceles triangle whose side lengths are 5m and 9m

__Solution__

Let the base, b = 9 m and a = 5m.

⇒ A = ½ [√ (a^{2} − b^{2 }/4) × b]

⇒ ½ [√ (5^{2} − 9^{2 }/4) × 9]

= 9.81m^{2}

#### Area of an equilateral triangle

An equilateral triangle is a triangle in which the three sides are equal and the three interior angles equal. The area of an equilateral triangle is:

A = (a^{2}√3)/4

Where a = length of the sides.

*Example 5*

Calculate the area of an equilateral triangle whose side is 4 cm.

__Solution__

⇒ A = (a^{2 }/4) √3

⇒ (4^{2}/4) √3

⇒ (16/4) √3

= 4√3 cm^{2}

*Example 6*

Find the area of an equilateral triangle whose perimeter is 84 mm.

__Solution__

The perimeter of an equilateral triangle = 3a.

⇒ 3a = 84 mm

⇒ a = 84/3

⇒ a = 28 mm

Area = (a^{2 }/4) √3

⇒ (28^{2}/4) √3

= 196√3 mm^{2}

#### Area of a scalene triangle

A scalene triangle is a triangle with 3 different side lengths and 3 different angles. The area of a scalene triangle can be calculated using Heron’s formula.

Heron’s formula is given by;

⇒ Area = √ {p (p – a) (p – b) (p – c)}

where ‘p’ is the semi-perimeter and a, b, c are the side lengths.

⇒ p = (a + b + c) / 2

*Example 7*

Calculate the area of a triangle whose side lengths are 18mm, 20mm, and 12mm.

__Solution__

⇒ p = (a + b + c) / 2

Substitute the values of a, b and c.

⇒ p = (12 + 18 + 20) / 2

⇒ p = 50/2

⇒ p = 25

⇒ Area = √ {p (p – a) (p – b) (p – c)}

= √ {25 x (25 – 12) x (25 – 18) x (25 – 20)}

= √ (25 x 13 x 7 x 5)

= 5√455 mm^{2}

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