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Students, scholars, or math enthusiasts should undoubtedly remember essential mathematical formulas when applying them to practical exercises. For instance, formulas for calculating the area of triangles, squares, parallelograms, etc. However, within each shape, especially the triangle, there are various formulas for different types of triangles, such as calculating the area of a regular triangle differing from that of a right-angled, isosceles, or equilateral triangle.
With these tips for calculating triangle areas, students and scholars can easily apply them in their studies for smoother completion of assignments.
Calculating the Area of Triangles: Regular, Right-Angled, Isosceles, Equilateral
1. What is a triangle?
2. Calculating the area of a triangle.
3. Advanced triangle area calculations.
4. Important notes.
5. Exercises.
Comprehensive Guide on Calculating Triangle Areas: Regular, Right-Angled, Isosceles, Equilateral
1. What is a triangle? Special cases of triangles?
To solve triangle area formula exercises, firstly, you need to identify the type of triangle and then find the most accurate area formula. Currently, common triangles are divided into 7 main types:
- Scalene triangle: A convex polygon with 3 sides and 3 vertices, with no sides being collinear. The sum of angles in a triangle is 180°.
- Right-angled triangle: A triangle with one angle equal to 90°.
- Isosceles triangle: A triangle with two equal sides, and two adjacent angles are equal.
- Equilateral triangle: A triangle with 3 equal sides, 3 equal angles, each measuring 60°
- Right-angled isosceles triangle: A triangle with one angle equal to 90°, two equal sides, and two base angles equal to 45°.
- Obtuse triangle: A triangle with one angle greater than 90°.
- Acute triangle: A triangle with all angles less than 90°
You can find more information on Wikipedia article about triangles to learn more about this shape.
Images of common triangles
- See more: How to calculate altitude in a triangle
2. Calculating the Area of a Triangle
For a clearer understanding, Mytour will guide you on how to calculate the area of a triangle in order, starting from a regular triangle to special cases such as right-angled, isosceles, equilateral triangles, and more...
* Calculating the Area of a Regular Triangle
- Explanation: The area of a regular triangle is calculated by multiplying the height by the length of the base, and then dividing the result by 2. In other words, the area is equal to half the product of the height and the length of the base of the triangle.
- Formula for calculating the area of a regular triangle
Where:
+ a: Length of the triangle base (the base is one of the 3 sides of the triangle depending on the convention of the calculator)
+ h: Height of the triangle, corresponding to the perpendicular segment from the vertex to the base (triangle height is the perpendicular line from the vertex to the base of a triangle).
- If you already have the area of the triangle, you can find the height or side of the triangle with the following formulas:
+ Height H= (Sx2)/ a
+ Formula for calculating the side of the triangle corresponding to the height: a= (Sx2)/ h
- Example: For a triangle ABC, where the height from vertex A to base BC is 3, and the length of base BC is 6. Calculate the area of triangle ABC? (Unit: cm)
Answer: Let a = 6 and h = 3.
Therefore, S = (a x h)/ 2 = (6x3)/2 or 1/2 x (6x3) = 9 cm
* Note: In cases where the base or height is not given but the area and the remaining side are provided, apply the formulas derived above for calculations.
* Calculating the Area of a Right-Angled Triangle
- Explanation: Calculating the area of a right-angled triangle is similar to calculating the area of a regular triangle. It is half the product of the height and the length of the base. However, a right-angled triangle differs by clearly representing the height and the length of the base. There is no need to draw extra lines to calculate the height of the triangle.
- Formula for calculating the area of a right-angled triangle: S = (a x h)/ 2
+ a: Length of the right-angled triangle base (the base is one of the 3 sides of the triangle and perpendicular to another side)
+ h: Height of the triangle, corresponding to the perpendicular segment from the vertex to the base (triangle height is the perpendicular line from the vertex to the base of a triangle).
From this, derive the formulas for height and the corresponding side: h=(Sx2)/ a or a= (Sx2)/ h
- Example: Consider a right-angled triangle ABC, right-angled at B, with the length of base BC being 5 cm, and the height being 2 cm. What is the area of the right-angled triangle ABC? Unit: cm.
Answer: Let a = 5 and h = 2.
Therefore, S = (a x h)/ 2 = (5x2)/2 or 1/2 x (5x2) = 5 cm
Similarly, if the data inquires about calculating the length of the base or height, you can utilize the derived formulas above.
* Calculating the Area of an Isosceles Triangle
An isosceles triangle is a triangle with two equal sides and two equal angles. To calculate the area, use the height drawn from the vertex to the base, then divide by 2.
- Explanation: The area of an isosceles triangle is the product of the height and the length of the base, divided by 2.
- Formula for calculating the area of an isosceles triangle: S = (a x h)/ 2
+ a: Length of the isosceles triangle base (the base is one of the 3 sides of the triangle)
+ h: Height of the triangle (the triangle height is the perpendicular line from the vertex to the base).
- Example: For an isosceles triangle ABC with the height from vertex A to base BC measuring 7 cm, and the length of the base being 6 cm. What is the area of isosceles triangle ABC?
Answer: Let a = 6 and h = 7.
Therefore, S = (a x h)/ 2 = (6x7)/2 or 1/2 x (6x7) = 21 cm
* Formula for calculating the area of an isosceles right-angled triangle
Example: For the right-angled isosceles triangle ABC, right-angled at A, with AB = AC = 6cm. Calculate the area of triangle ABC.
Solution: Since AB = AC = a = 6cm
Considering the right-angled isosceles triangle ABC at A, we have:
S = (a2) : 2 = 36 : 2 = 18 cm2
* Formula for calculating the area of an equilateral triangle
An equilateral triangle has three equal sides and each angle measures 60 degrees. Any triangle with three equal angles is considered an equilateral triangle.
- Formula for the area of an equilateral triangle: S = a2 X (√3)/4
Where:
+ a: length of any side in the equilateral triangle.
- Example: Consider an equilateral triangle ABC with equal side lengths of 9 cm and all angles measuring 60 degrees. What is the area of equilateral triangle ABC?
Answer: Since each side AB = AC = BC = 9, we have side length a = 9.
Substitute into the formula for the area of an equilateral triangle: S = a2 x (√3)/4 = S = 92 x (√3)/4 = 81 x (√3)/4 = 81 x (1.732/4) = 35.07 cm2
In addition to the various methods for calculating the area of a triangle mentioned earlier, mathematics also employs popular approaches such as the Heron's formula, calculating triangle area based on angles, and trigonometric functions. Specifically:
* Formula for the triangle area when one angle is known
The triangle area using Sine is:
* Formula for calculating the triangle area using Heron's formula
Triangle area when all three sides are known:
* Extended Triangle Area Calculation Method
Note: When using this formula, it is essential to provide a prior demonstration.
Formula 1:
Where:
- a, b, c: Lengths of the triangle sides
- R: Radius of the circumcircle of the triangle
Formula 2:
Triangle area formula can also be applied:
Where:
- p: Semi-perimeter of the triangle
- r: Radius of the incircle of the triangle
Depending on the formula, apply them to different grade levels. Typically, the triangle area formulas for 5th and 8th grades are straightforward. From 10th grade onwards, students can apply more advanced formulas if they have studied trigonometry, in-circle, and circumcircle concepts.
4. Points to note when calculating the triangle area
- During calculations, ensure that the measurement units are consistent.
- For area, the units are measured in square units, such as m2, cm2, etc.
Regardless of the triangle area formula used, students and learners should understand that the height may not always lie within the triangle. In such cases, it is necessary to draw an additional height and supplemental base. Also, when calculating the triangle area, ensure that the height corresponds to the base it projects onto.
5. Exercise: Calculating Triangle Area
Example 1: A triangle has a base of 15 cm and a height of 2.4cm. Calculate the area of the triangle?
Solution Guide:
The area of the triangle is:
15 x 2.4 : 2 = 18 (cm2)
Answer: 18cm2
=> Students can refer to more basic and advanced triangle problems for 5th grade to better understand how to solve and approach these problems with ease.
