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The area of a triangle is a fundamental concept in geometry that represents the amount of space enclosed by a triangle’s three sides. In simpler terms, it’s a measure of the flat surface covered by the triangle. We express the area in square units, such as square centimeters (cm²) or square meters (m²). Understanding how to calculate the area of a triangle involves familiarizing oneself with several formulas, each suited to different types of triangles and given data.
From the basic 1/2 x base x height formula to Heron’s formula and trigonometric approaches, mastering these methods enables precise and efficient area calculations. In this guide, we will explore the definition of a triangle’s area, delve into the various formulas available for different problems, and provide solved examples to describe their properties.
Table of Contents
Definition of Area of Triangles
The area of a triangle is a measure of the region enclosed by its three sides. It quantifies the amount of space within the boundaries of the triangle. The most commonly used formula to calculate the area of a triangle is:
| Area = ½ x base x height |
In this formula, the “base” is the length of one side of the triangle, and the “height” is the perpendicular distance from the base to the opposite vertex.
There are different types of triangles, each with unique properties that can influence how the area is calculated:
Equilateral Triangle: All three sides are of equal length, and all three interior angles are equal (each measuring 60 degrees). The area can also be calculated using the formula:
| Area = √3/4 x side² |
Isosceles Triangle: Has two sides of equal length and two equal angles opposite those sides. The standard formula can be used by identifying the base and the height.
Scalene Triangle: All three sides and all three angles are different. For scalene triangles, Heron’s formula can be particularly useful:
| Area = √s(s-a)(s-b)(s-c) |
where s is the semi-perimeter of the triangle, given by s = a+b+c/2, and a, b, and c are the lengths of the sides.
Right Triangle: Has one angle measuring 90 degrees. The area can be easily calculated using the legs of the triangle (the two sides that form the right angle):
| Area = ½ x base x height |
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Properties of Area of Triangles with Formulas
The area of a triangle is a measure of the region enclosed by its three sides, and it can be calculated using various formulas depending on the available information and the type of triangle. Here are some key properties and corresponding formulas for calculating the area of different types of triangles.
Basic Formula for All Triangles
The most fundamental formula for the area of any triangle is:
| Area = ½ x base x height |
- Base (b): The length of one side of the triangle.
- Height (h): The perpendicular distance from the base to the opposite vertex.
Equilateral Triangle
For an equilateral triangle (all sides and angles are equal):
| Area = √3/4 x side² |
- Side (a): The length of any side of the triangle.
Isosceles Triangle
For an isosceles triangle (two sides and two angles are equal):
| Area = ½ x base x height |
- Base (b): The length of the unequal side.
- Height (h): The perpendicular distance from the base to the opposite vertex.
Scalene Triangle
For a scalene triangle (all sides and angles are different), Heron’s formula is useful:
| Area = √s(s-a)(s-b)(s-c) |
- a, b, c: The lengths of the sides of the triangle.
- s: The semi-perimeter of the triangle, calculated as s=a+b+c/2.
Right Triangle
For a right triangle (one angle is 90 degrees):
| Area = ½ x base x height |
- base and height: The lengths of the two sides that form the right angle.
Using Trigonometry
When the lengths of two sides and the included angle are known, the area can be calculated using the formula:
| Area = ½ x a x b x Sin (C) |
- a and b: The lengths of the two sides.
- C: The included angle between sides aaa and bbb, in degrees or radians.
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Properties of Area of Triangles With Solved Examples
The area of a triangle can be calculated using various formulas depending on the type of triangle and the available information. Here, we will explore the formulas and provide five solved examples.
Q1: Simple Triangle
Given: Base = 8 units, Height = 5 units
Area=1/2×8×5=20 square units
Q2: Equilateral Triangle
Given: Side = 6 units Find Area
Area=√3/4×6²
=√3/4×36=9√3 ≈ 15.59 square units
Q3: Scalene Triangle
Given: Sides a = 7 units, b = 8 units, c = 9 units find Area?
s=7+8+9/2=12
Area=√12(12−7)(12−8)(12−9) = √12×5×4×3 = 720 ≈ 26.83 square units
Q4: Right Triangle
Given: base = 6 units and heights 8 units find Area of Right angled triangle?
Area=1/2×6×8=24 square units
Q5: Triangle with Two Sides and Included Angle
Given: Sides a = 7 units, b = 10 units, Included angle C=45∘
Area=1/2×7×10×sin(45∘)
Area=/12×7×10×√2/2
=1/2×7×10×0.707 = 24.75 square units
FAQs
Triangles have three sides, and their area is the whole area that is surrounded by their three sides. In simple terms, it’s half of the base times the height, so A = 1/2 × b × h.
If you have three angles, two of them are 45 degrees and one is 90 degrees, you have a 45-45-90 triangle. The legs of a 45-45-90 triangle are the two sides that are the same length.
A polygon is any shape with two dimensions that is made up of straight lines. Polygons are shapes like triangles, quadrilaterals, pentagons, and hexagons.
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