Area of a Triangle
- Introduction
- Different Methods for Calculating the Area of Triangle
- The formula for Finding the Area of a Triangle
- Heron's Formula for the Area of a Triangle
- Area of a Triangle with Two Sides and the Included Angle (SAS)
- Calculating Area of a Right-Angled Triangle
- Calculating the Area of an Equilateral Triangle
- Calculating the Area of an Isosceles Triangle
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
The area of a triangle represents the amount of space enclosed by its three sides on a flat surface. To compute this, we use a straightforward formula: half of the base multiplied by the height, expressed as `A = 1/2 × \text{base} × \text{height}`. This formula is universally applicable, whether the triangle is scalene, isosceles, or equilateral. It's important to note that the base and height of a triangle are always perpendicular to each other.
Throughout this content, we will explore various other formulas for finding the area of triangles of different types. This measurement of area varies among triangles, influenced by the lengths of the sides and the angles within. Typically, we express the area of a triangle in square units such as `\text{m}²`, `\text{cm}²`, \( \text{in}²\), and the like.
Different Methods for Calculating the Area of Triangle
- Using Base and Height
- Using Heron's Formula
- Using Two Sides and Including Angle
- Using Right-Angled Triangle Properties
- Specifically for Equilateral Triangles
- Specifically for Isosceles Triangles
The Formula for Finding the Area of a Triangle
The most commonly used formula for determining the area of a triangle is essential for various mathematical calculations. It's formulated as half of the product of the base and the height of the triangle. This formula is universally applicable and remains constant across all triangles, including scalene, isosceles, and equilateral. It's important to note that the base and height of the triangle must be perpendicular to each other for accurate calculations.
The formula is expressed as:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
This simple formula allows for easy calculation of the area of any triangle, providing a foundational concept in geometry and mathematics.
Example: Consider a triangle with a base of `10` units and a height of `6` units. Use the formula for finding the area of a triangle to solve for its area.
Solution:
Given:
Base (\( b \)) `= 10` units
Height (\( h \)) `= 6` units
Using the formula:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Substitute the given values:
\( \text{Area} = \frac{1}{2} \times 10 \times 6 \)
Simplify the expression:
\( \text{Area} = \frac{1}{2} \times 60 \)
\( \text{Area} = 30 \text{ square units} \)
Therefore, the area of the triangle is \( 30 \text{ square units} \).
Heron's Formula for the Area of a Triangle
Heron's Formula provides an alternative method for calculating the area of a triangle when the lengths of its three sides are known. It's beneficial when the height and base lengths are not readily available.
The formula is expressed as:
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
where
\( a \), \( b \), and \( c \) are the lengths of the three sides of the triangle.
\( s \) represents the semi-perimeter of the triangle, calculated as \( s = \frac{a + b + c}{2} \).
Example: We have a triangle with side lengths \(a = 5\) units, \(b = 7\) units, and \(c = 9\) units. Calculate the area of the triangle using the Heron’s formula.
Solution:
First, we need to calculate the semi-perimeter (\(s\)) of the triangle using the formula \(s = \frac{a + b + c}{2}\):
\( s = \frac{5 + 7 + 9}{2} = \frac{21}{2} = 10.5 \text{ units} \)
Now, we can use Heron's Formula to find the area (\(A\)) of the triangle:
\( A = \sqrt{s(s-a)(s-b)(s-c)} \)
\( A = \sqrt{10.5(10.5-5)(10.5-7)(10.5-9)} \)
\( A = \sqrt{10.5 \times 5.5 \times 3.5 \times 1.5} \)
\( A = \sqrt{303.1875} \)
Taking the square root gives us:
\( A \approx 17.41 \text{ square units} \)
Therefore, the area of the triangle is approximately \(17.41\) square units.
Area of a Triangle with Two Sides and the Included Angle (SAS)
When two sides and the included angle of a triangle are given, we can determine its area using a specific formula. This scenario is known as the SAS (Side-Angle-Side) case.
The formula for finding the area of a triangle with two sides and the included angle is:
\( \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \)
where:
\( a \) and \( b \) are the lengths of the two given sides.
\( C \) is the measure of the included angle between the two sides.
\( \sin(C) \) represents the sine value of the included angle.
Example: A triangle has its side lengths \( a = 8 \) units, \( b = 6 \) units, and an included angle \( C = 45^\circ \). Use the formula for finding the area of a triangle in the SAS case to solve for its area.
Solution:
Side \( a \) `= 8` units
Side \( b \) `= 6` units
Included Angle \( C \) `=` \( 45^\circ \)
Use the formula:
\( \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \)
Substitute the given values:
\( \text{Area} = \frac{1}{2} \times 8 \times 6 \times \sin(45^\circ) \)
Calculate the value of \( \sin45^\circ \) which is approximately \( 0.707 \):
\( \text{Area} = \frac{1}{2} \times 8 \times 6 \times 0.707 \)
Simplify the expression:
\( \text{Area} = 24 \times 0.707 \)
\( \text{Area} \approx 16.968 \text{ square units} \)
Therefore, the area of the triangle is approximately \( 16.968 \) square units.
Calculating Area of a Right-Angled Triangle
Right-angled triangles, also known as right triangles, possess unique properties that simplify the calculation of their area. In a right triangle, one angle measures `90` degrees, and the other two angles are acute, totaling `90` degrees altogether. The side opposite the right angle is called the hypotenuse, while the sides adjacent to the right angle are known as the base and height. As one of the triangle sides acts as the height of the triangle, it makes calculating the area of the triangle easier and faster.
To calculate the area of a right-angled triangle, you can utilize the formula:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Here, the base represents one of the sides adjacent to the right angle, and the height refers to the other side adjacent to the right angle.
Example: The sides adjacent to the right angle in a right-angled triangle measure `6` units and `8` units. What is the area of the triangle?
Solution:
Given:
Base (\( b \)) `= 6` units
Height (\( h \)) `= 8` units
Using the formula for the area of a right-angled triangle:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
\( \text{Area} = \frac{1}{2} \times 6 \times 8 \)
\( \text{Area} = \frac{1}{2} \times 48 \)
\( \text{Area} = 24 \text{ square units} \)
Therefore, the area of the right-angled triangle is `24` square units.
Calculating the Area of an Equilateral Triangle
Equilateral triangles are a special type of triangle where all three sides are of equal length, and each interior angle measures `60` degrees. Due to their symmetrical nature, calculating the area of an equilateral triangle follows a specific formula.
To find the area of an equilateral triangle, you can use the formula:
\( \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 \)
Here, the side length (\(s\)) represents the length of any side of the equilateral triangle.
Example: An equilateral triangle has a side length of `10` units. Find the area of the triangle.
Solution:
Given:
Side Length (\(s\)) `= 10` units
Using the formula for the area of an equilateral triangle:
\( \text{Area} = \frac{\sqrt{3}}{4} \times (\text{side})^2 \)
\( \text{Area} = \frac{\sqrt{3}}{4} \times (10)^2 \)
\( \text{Area} = \frac{\sqrt{3}}{4} \times 100 \)
\( \text{Area} = \frac{\sqrt{3} \times 100}{4} \)
\( \text{Area} = \frac{100\sqrt{3}}{4} \)
\( \text{Area} = 25\sqrt{3} \)
Therefore, the area of the equilateral triangle with a side length of `10` units is \( 25\sqrt{3} \) square units.
Calculating the Area of an Isosceles Triangle
Isosceles triangles are special triangles whose two sides are of equal length, making the two opposite angles also equal. To calculate the area of an isosceles triangle, we can use a specific formula tailored to its properties.
The formula for finding the area of an isosceles triangle is:
\( A = \frac{1}{2} \times b \times \sqrt{a^2 - \frac{b^2}{4}} \)
Here, \( a \) represents the length of one of the equal sides, and \( b \) represents the length of the base (which is the side that is not equal).
Example: Let's consider an isosceles triangle with two equal sides of length `6` units each and a base of length `8` units. Calculate the area of the triangle.
Solution:
Given:
Length of equal sides (\( a \)): `6` units
Length of base (\( b \)): `8` units
We can substitute these values into the formula:
\( A = \frac{1}{2} \times 8 \times \sqrt{6^2 - \frac{8^2}{4}} \)
First, let's simplify inside the square root:
\( A = \frac{1}{2} \times 8 \times \sqrt{36 - 16} \)
\( A = \frac{1}{2} \times 8 \times \sqrt{20} \)
Now, let's simplify the square root of `20`:
\( \sqrt{20} \approx 4.472 \)
Now, we can substitute this value back into our equation:
\( A = \frac{1}{2} \times 8 \times 4.472 \)
\( A = 4 \times 4.472 \)
\( A \approx 17.888 \)
So, the area of the given isosceles triangle is approximately \( 17.888 \) square units.
Solved Examples
Example `1`. Given a right-angled triangle with a base of `12` units and a height of `5` units, find its area.
Solution:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
\( \text{Area} = \frac{1}{2} \times 12 \times 5 = 30 \text{ square units} \)
Example `2`. Determine the area of an equilateral triangle with a side length of `9` units.
Solution:
\( \text{Area} = \frac{\sqrt{3}}{4} \times \text{Side}^2 \)
\( \text{Area} = \frac{\sqrt{3}}{4} \times 9^2 = \frac{81\sqrt{3}}{4} \text{ square units} \)
Example `3`. Given an isosceles triangle with sides \(a = 7\) units, \(b = 7\) units, and an included angle \(C = 45^\circ\), find its area.
Solution:
\( \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \)
\( \text{Area} = \frac{1}{2} \times 7 \times 7 \times \sin(45^\circ) \approx 17.32 \text{ square units} \)
Example `4`. Consider a scalene triangle with side lengths \(a = 5\), \(b = 6\), and \(c = 8\) units. Calculate its area using Heron's formula.
Solution:
\( s = \frac{a + b + c}{2} = \frac{5 + 6 + 8}{2} = 9.5 \)
\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)
\( \text{Area} = \sqrt{9.5 \times 4.5 \times 3.5 \times 1.5} \approx 14.98 \text{ square units} \)
Practice Problems
Q`1`. What is the area of a right-angled triangle with a base of `10` units and a height of `8` units?
- `32` square units
- `40` square units
- `48` square units
- `60` square units
Answer: b
Q`2`. Determine the area of an equilateral triangle with a side length of `12` units.
- \( 36\sqrt{3} \) square units
- \( 48\sqrt{3} \) square units
- \( 72\sqrt{3} \) square units
- \( 144\sqrt{3} \) square units
Answer: a
Q`3`. Given an isosceles triangle with side lengths \(a = 9\) units, \(b = 9\) units, and an included angle \(C = 60^\circ\), what is its area nearest to the whole number?
- `35` square units
- `36` square units
- `45` square units
- `42` square units
Answer: a
Q`4`. Calculate the area of a scalene triangle with side lengths \(a = 7\), \(b = 10\), and \(c = 12\) units, to the nearest whole number.
- `24` square units
- `35` square units
- `36` square units
- `40` square units
Answer: b
Q`5`. Find the area of an isosceles triangle with sides of equal length \(a = 5\) units and base \(b = 5\) units, and an altitude from the vertex to the base measuring `4` units.
- `10` square units
- \( 4\sqrt{3} \) square units
- `8` square units
- \( 16\sqrt{3} \) square units
Answer: a
Frequently Asked Questions
Q`1`. How do you find the area of a triangle?
Answer: The area of a triangle can be found using different formulas based on the information available. If you know the base and height, you can use the commonly used formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). If you have the lengths of all three sides, you can use Heron's Formula. There are also specific formulas for right-angled, equilateral, and isosceles triangles.
Q`2`. What is Heron's Formula, and when is it used?
Answer: Heron's Formula is used to find the area of a triangle when the lengths of all three sides are known. It is expressed as \( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \), where \( s \) is the semi-perimeter of the triangle and \( a \), \( b \), and \( c \) are the lengths of the sides.
Q`3`. Can you find the area of a triangle if only two sides and the included angle are given?
Answer: Yes, you can find the area of a triangle if you know two sides and the included angle using the formula \( \text{Area} = \frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\theta) \), where \( \theta \) is the angle between the given two sides.
Q`4`. How do you calculate the area of an equilateral triangle?
Answer: To find the area of an equilateral triangle, you can use the formula \( \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 \), where \( \text{side} \) represents the length of any side of the equilateral triangle.
Q`5`. What are the properties of a right-angled triangle for area calculation?
Answer: In a right-angled triangle, the area can be calculated using the formula \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \), where the base is one of the sides adjacent to the right angle, and the height is another side adjacent to the right angle.