Calculating the area of a triangle and trapezoid
Whether you are a student or owner of land or plots, with our calculators to calculate the area of a triangle and trapezoid you will get the result quickly and easily.
Calculate the area of triangles is a fundamental skill, whether in school or in practical everyday projects. Understanding these formulas is essential for solving engineering, architecture and even design problems.
Furthermore, the practical applications of area calculations are vast and you may need to calculate the area of a triangle in school, work, arts, and many other situations.
From planning the arrangement of furniture in a triangular space to calculating the amount of material needed for construction, knowledge about the areas of triangles is essential and can make many everyday tasks easier.
Area of Right Triangle
A area of a right triangle can be easily calculated using the basic formula: Area = (base * height) / 2. In this type of triangle, the sides that form the 90° angle are known as legs. One of them is considered the base, while the other is the height.
Right Triangle
Let's see how to apply this formula step by step with a practical example:
- Identify the legs: Consider a right triangle with legs of 6 cm and 8 cm.
- Choose the base and height: Here, 6 cm will be the base and 8 cm will be the height.
- Apply the formula: Substitute in the formula: Area = (6 * 8) / 2.
- Calculate the product: (6 * 8) = 48.
- Divide by 2: 48 / 2 = 24.
- Result: The area of the triangle is 24 cm².
This method is straightforward and useful for any right triangle, as shown in practical examples in Brazil School. With practice, you will be able to perform these calculations quickly and apply them in different everyday situations.
Trapezoid Area
Before calculating the area of a trapezoid, it is important to understand the differences between triangles and trapezoids. A triangle is a geometric figure with three sides and three angles, while a trapezoid is a quadrilateral polygon with exactly two parallel sides and two oblique sides.
In the case of a trapezoid, its bases are the parallel sides, one being the larger base and the other the smaller base. The formula for calculating the area of a trapezoid is: Area = (larger base + smaller base) * height / 2. The height is the perpendicular distance between the bases.
Let's apply the formula with a practical example: Consider a trapezoid with a smaller base of 4 cm, a larger base of 12 cm and a height of 5 cm. To calculate the area, follow the steps:
- Substitute the values into the formula: Area = (12 + 4) * 5 / 2
- Calculate the sum of the bases: 12 + 4 = 16
- Multiply by height: 16 * 5 = 80
- Divide by 2: 80/2 = 40 cm²
Thus, the area of the trapezoid is 40 cm². This calculation is applicable to any type of trapezoid, as shown in the practical examples of Brazil School.
Area of Equilateral Triangle
An equilateral triangle has all sides and angles equal, with each angle measuring 60 degrees. To calculate the area of this type of triangle, we use a specific formula that does not require the height, only the length of the side.
The formula is: Area = (side2 * √3) / 4. Here, “side” represents the measurement of one of the sides of the triangle.
Let's understand how to apply this formula with a practical example:
Example: Calculate the area of an equilateral triangle with sides measuring 8 cm.
- Substitute “side” into the formula: Area = (82 * √3) / 4.
- Calculate the square of 8: Area = (64 * √3) / 4.
- Divide 64 by 4: Area = 16√3 cm².
This calculation shows that the area of an equilateral triangle with sides of 8 cm is 16√3 cm². For a more practical approach, you can use an approximation for √3, such as 1.7, to make the calculations easier.
With this formula, determining the area of an equilateral triangle becomes a simple and straightforward process, useful for both students and educators.
Area and Perimeter of a Triangle
Calculating the area and perimeter of a triangle is essential to understanding its geometric properties. The perimeter of a triangle is the sum of the measures of all its sides, while the area measures the internal surface of the figure.
The formulas are simple and apply to all types of triangles:
- Perimeter: P = L1 + L2 + L3
- Area: A = (base * height) / 2
To illustrate, let's look at a practical example with an isosceles triangle, which has two equal sides and one different side.
Example: Consider a triangle with equal sides of 5 cm, different sides of 6 cm and height of 8 cm.
- Area: A = (6 * 8) / 2 = 24 cm²
- Perimeter: P = 5 + 5 + 6 = 16 cm
These formulas are fundamental for various calculations in geometry and help to visualize the shape and size of triangles in a practical and effective way.
Area of a Triangle with Vertices – Heron’s Formula
To calculate the area of a triangle given by its vertices, we can use the Heron's formula, which is effective when we know the measures of the three sides of the triangle, but not the height. This formula calculates the area from the semiperimeter (p) and the sides (a, b, c):
- Calculate the semiperimeter: p = (a + b + c) / 2
- The area is then given by: A = √[p(p – a)(p – b)(p – c)]
An alternative method, using analytic geometry, involves calculating the area as half the modulus of the determinant of the vertex coordinates. Let's look at a practical example:
Consider a triangle with vertices A (4, 0), B (0, 0) and C (0, 6). The determination of the area is based on the calculation of the determinant:
- Area = 1/2 * |4(0 – 6) + 0(6 – 0) + 0(0 – 0)| = 12
This example illustrates how we can use coordinates to determine area, providing a practical alternative to situations where the lengths of the sides are not known directly. For more details on the application of Heron's formula and other examples, see this article.
Area of Scalene Triangle
A scalene triangle is unique because it has all of its sides and angles with different measures. This contrasts with equilateral triangles, where all of the sides and angles are equal, and isosceles triangles, which have two sides of equal length and two angles of equal length.
To calculate the area of a scalene triangle, we can use the Heron's formula, especially useful when we know all the sides but not the height. First, calculate the semiperimeter (p) as the sum of the sides divided by two: p = (a + b + c) / 2. Then, apply the area formula: A = √[p(p – a)(p – b)(p – c)].
For example, consider a triangle with sides of 8 cm, 7 cm and 5 cm. Calculate the semiperimeter: p = (8 + 7 + 5) / 2 = 10 cm. Then substitute into Heron's formula: A = √(10 * (10 – 8) * (10 – 7) * (10 – 5)) = √300 = 10√3 cm². This method shows how it is possible to determine the area even without the height, using only the measurements of the sides.
Common questions
- How to calculate the area of a triangle with different sides? The area of a triangle with unequal sides, such as a scalene triangle, can be calculated using the Heron's formula. First, calculate the semiperimeter, then apply the formula: A = √(p * (p – a) * (p – b) * (p – c)).
- What is needed to calculate the area of a right triangle? For a right triangle, just use the basic formula: Area = (base * height) / 2.
- What is the formula for the area of an equilateral triangle? For an equilateral triangle, the formula is: Area = (side^2 * √3) / 4.
- How do you know if a triangle is scalene? A scalene triangle has all its sides and angles different, unlike equilateral and isosceles triangles.
