Area of Triangle – Definition with Examples
Welcome to another exciting adventure with Brighterly, where we make mathematics bright, fun, and easy to understand. Today, we’re diving deep into the world of triangles. Triangles are more than just simple shapes. They are fundamental to many branches of mathematics and can be found in the world around us, from the Great Pyramids in Egypt to your favorite slice of pizza. One key characteristic of these three-sided figures is their area – the amount of space they occupy. But how do we measure this space? That’s precisely what we’ll uncover today as we explore the concept of the area of a triangle.
What is the Area of a Triangle?
The area of a triangle is the amount of space that the triangle occupies. Just like other two-dimensional shapes such as squares and rectangles, the area is measured in square units. The concept of area is an essential part of geometry, a branch of mathematics that studies the properties of different shapes. In learning about the area of a triangle, children gain an important tool for understanding and working with shapes and spaces in everyday life, from architecture to art to nature. We see triangles everywhere, from the Pyramids of Egypt to a slice of pizza, so it’s beneficial to know how to calculate their area.
Area of a Triangle Formula
The basic formula for the area of a triangle is 1/2 x base x height, often represented as A = 1/2bh. This means that the area of a triangle is equal to half of the product of the triangle’s base length and its height. This formula applies when you know the base and the height of the triangle. If a triangle were to be ‘opened up’ to form a rectangle, its area would be double that of the original triangle, hence the division by 2 in the formula.
Area of a Right Angled Triangle
A right angled triangle is one where one of the angles is 90 degrees. Because the height of the triangle aligns perfectly with one of the sides in a right angled triangle, the formula for the area simplifies to the same 1/2 x base x height (A = 1/2bh), where the base and height are the two sides that form the right angle.
At Brighterly, we believe that practice is the key to mastery. That’s why we invite you to explore our area of a triangle worksheets, where you can find an array of additional practice questions, complete with answers.
Area of an Equilateral Triangle
An equilateral triangle is a special kind of triangle where all sides are equal in length. The area of an equilateral triangle can be found using the formula A = (√3/4) * side², where side refers to the length of one side of the triangle. This formula stems from the unique properties of equilateral triangles.
Area of an Isosceles Triangle
An isosceles triangle is a triangle with two sides of equal length. Even though it has a unique shape, we can still use the formula A = 1/2bh to calculate the area of an isosceles triangle, as long as we know the base (the side that is not equal to the others) and the height (a line drawn from the base’s midpoint to the opposite vertex).
Area of Triangle Using Heron’s Formula
There’s also a method called Heron’s formula to calculate the area of a triangle when we know all three sides. This formula, named after the ancient Greek mathematician Hero of Alexandria, is A = √[s(s - a)(s - b)(s - c)], where a, b, and c are the sides of the triangle and s is the semi-perimeter (s = (a + b + c)/2).
Area of Triangle With 2 Sides and Included Angle
When two sides and the included angle are known, the area of a triangle can be calculated using the formula A = 1/2ab sin(C), where a and b are the sides and C is the included angle. This formula is particularly useful in trigonometry.
How to Find the Area of a Triangle?
To find the area of a triangle, you first need to identify what information you have. Is it the base and height? The length of all three sides? Two sides and the included angle? Depending on this, you would use the appropriate formula.
Area of Triangle when 3 Sides are Given
As mentioned above, when all three sides of a triangle are given, we can calculate the area using Heron’s formula. It uses the lengths of all three sides and the semi-perimeter of the triangle to find the area.
Perimeter of a Triangle
The perimeter of a triangle is the sum of the lengths of all its sides. It’s the total distance around the triangle. This is another important geometric concept related to triangles, although it’s different from the area, which measures the space enclosed by the triangle.
Practice Questions on Area of Triangle
Remember, the best way to learn mathematics is by applying the concepts in practical situations. Let’s walk through some practice problems that cover different types of triangles and the various ways we can calculate their areas.
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Right-Angled Triangle
- Problem: The base of a right-angled triangle is 8 cm and its height is 5 cm. Find the area.
- Solution: Use the formula
A = 1/2bh(Area = 1/2 x base x height). Here, base (b) = 8 cm, and height (h) = 5 cm. So, the area of the triangle is 1/2 x 8 x 5 = 20 cm².
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Equilateral Triangle
- Problem: Find the area of an equilateral triangle with a side length of 6 cm.
- Solution: Use the formula
A = (√3/4) * side². Here, side length = 6 cm. So, the area of the triangle is (√3/4) x 6² = 15.59 cm² (rounded to two decimal places).
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Isosceles Triangle
- Problem: An isosceles triangle has a base of 10 cm and a height of 12 cm. Calculate the area.
- Solution: Use the formula
A = 1/2bh. Here, base (b) = 10 cm, and height (h) = 12 cm. So, the area of the triangle is 1/2 x 10 x 12 = 60 cm².
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Triangle using Heron’s Formula
- Problem: Find the area of a triangle with sides of lengths 7 cm, 8 cm, and 9 cm.
- Solution: Use Heron’s formula
A = √[s(s - a)(s - b)(s - c)]. First calculate the semi-perimeter (s) = (7+8+9)/2 = 12 cm. Then, substitute these values into the formula to get: A = √[12(12 – 7)(12 – 8)(12 – 9)] = 26.83 cm² (rounded to two decimal places).
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Triangle with 2 Sides and Included Angle
- Problem: Two sides of a triangle measure 5 cm and 6 cm, and the included angle is 60 degrees. What is the area of the triangle?
- Solution: Use the formula
A = 1/2ab sin(C). Here, a = 5 cm, b = 6 cm, and C = 60 degrees. So, the area of the triangle is 1/2 x 5 x 6 x sin(60) = 13.0 cm² (rounded to one decimal place).
Conclusion
At Brighterly, we believe that understanding mathematics is like building a tower, with each new concept serving as a block that strengthens and elevates your knowledge. Today, we’ve added the block of ‘Area of a Triangle’ to your tower. We’ve traveled through various types of triangles, explored multiple formulas, and seen how these calculations apply in practical situations. But remember, just as a building needs regular maintenance, your knowledge also needs regular practice to stay strong. So keep revisiting these concepts and continue practicing the problems we’ve worked on together. After all, in the world of mathematics, practice doesn’t just make perfect, it makes permanent. Until our next mathematical journey, stay curious and keep learning!
Frequently Asked Questions on Area of a Triangle
We’ve gathered some of the most common questions we receive about the area of a triangle, and have provided comprehensive answers to ensure your understanding is as solid as a well-built triangle.
What is the formula for the area of a triangle?
- The basic formula for the area of a triangle is
A = 1/2bh, where A is the area, b is the length of the base, and h is the height of the triangle. However, different types of triangles or different available measurements can require the use of other formulas, such asA = (√3/4) * side²for equilateral triangles, or Heron’s formulaA = √[s(s - a)(s - b)(s - c)]when the lengths of all three sides are known.
Why is the area of a triangle half the base times height?
- Imagine doubling a triangle by copying it and flipping it over to form a rectangle. The rectangle’s area would be base times height. Since the triangle is half of this rectangle, its area is half of the base times the height.
What is Heron’s formula, and when is it used?
- Heron’s formula is used to calculate the area of a triangle when the lengths of all three sides are known. Named after Hero of Alexandria, an ancient Greek mathematician, it uses the semi-perimeter (half of the total of the sides) and the lengths of the sides in its calculation.
What is the semi-perimeter of a triangle?
- The semi-perimeter of a triangle is half of the sum of the lengths of its sides. It’s often represented by the letter s and is used in Heron’s formula.
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