Introduction to Triangles
Triangles are some of the most interesting shapes that you can ever get a chance to study. They not only have a lot of patterns and interesting formulas that you can get a lot of knowledge from but they are also super fun to study. Triangles can be found everywhere, and another thing that can be found everywhere are the patterns associated with them. They are all around us and need a good observation to be understood. We suggest that when you take a look at the objects around you and look at the symmetry of a triangle, try to associate the knowledge that you learn from this article with your everyday life.
Know About Isosceles Triangle Perimeter Formula
A triangle is called an isosceles triangle if it has any two sides equal. The angles opposite to these equal sides are also equal. The area of an isosceles triangle can be calculated using the length of its sides. In the diagram, triangle ABC here sides AB and AC are equal and also ∠B = ∠C. The theorem that describes the isosceles triangle is “if the two sides of a triangle are congruent, then the angle opposite to these sides are congruent”.
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The perimeter of the Isosceles Triangle
As we know the perimeter of any shape is given by the boundary of the shape. In a similar way, the perimeter of an isosceles triangle is defined as the sum of the three sides of an isosceles triangle. The perimeter of an isosceles triangle can be found if we know its base and side. The formula of isosceles triangle perimeter is given by:
The perimeter of an isosceles triangle formula, P = 2a + b units where ‘a’ is the length of the two equal sides of an isosceles triangle and ‘b’ is the base of the triangle.
Formula to Find the Area of Isosceles Triangle
The area of an isosceles triangle is defined as the region occupied by it in the two-dimensional space. Generally, the isosceles triangle is half the product of the base and height of an isosceles triangle. Formula to calculate the area of an isosceles triangle is given below:
The area of an isosceles triangle A = ½ × b × h Square units, where ‘b’ is the base and ‘h’ is the height of the isosceles triangle.
Isosceles Triangle Properties
As we know the two sides are equal in this triangle, and the unequal side is called the base of the triangle.
The angles opposite to the two equal sides of the triangle are always equal.
The height of an isosceles triangle is measured from the base to the vertex (topmost) of the triangle.
The third angle of a right isosceles triangle is equal to 90 degrees.
Generally, the isosceles triangle is classified into different types named as,
Isosceles acute triangle.
Isosceles right triangle.
Isosceles obtuse triangle.
Let’s discuss in detail these three different types of an isosceles triangle.
Isosceles Acute Triangle
As we know that the different dimensions of a triangle are legs, base, and height. The axis of symmetry of an isosceles triangle is along the perpendicular bisector of its base. Depending upon the angle between the two legs, the isosceles triangle is classified as acute, right, and obtuse. The isosceles triangle can be acute if the two angles opposite to the legs are equal and are less than 90 degrees ( or acute angle).
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Isosceles Right Triangle
A right isosceles triangle has two equal sides, in which one of the two equal sides act as perpendicular and another one as a base of the triangle. The third side, which is unequal, is known as the hypotenuse. Therefore, we can apply here the Pythagoras theorem, where the square of the hypotenuse is equal to the sum of the square of base and perpendicular. Suppose, the sides of the right isosceles triangle are a, a, and h, where a is the length of two equal sides and h is the length of the hypotenuse, then;
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\[h = \sqrt{(a^2 + a^2)} = \sqrt{2a^2} = a\sqrt{2}\] or \[h = \sqrt{2} a \]
Isosceles Obtuse Triangle
We know that the obtuse triangle is a triangle in which one of its angles is greater than 90 degrees (or right angle). Also, it is not possible to draw a triangle with more than two obtuse angles. We know that the obtuse triangle can be of two types, i.e., scalene triangle or isosceles triangle. Therefore, the isosceles obtuse triangle is a triangle, which has two equal sides having an obtuse angle.
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Solved Examples:
1. Find the perimeter of an isosceles triangle, with a side of 5 cm and a base of 4 cm.
Solution: Given that, length of the base is 4 cm.
The length of the two equal sides is 5 cm.
We know that the formula to calculate the perimeter of an isosceles triangle is P = 2a + b units.
Now, substitute the value of base and side in the perimeter formula, we get
P = 2(5) + 4 = 10 + 4 = 14 cm
Therefore, the perimeter of an isosceles triangle is 16 cm.
2. Find the area of an isosceles triangle given its height as 5 cm and base as 4 cm?
Solution: Given that, base = 4 cm and height = 5 cm
We know that the area of an isosceles triangle is ½ × b × h square units
Now, substitute the base and height value in the formula to calculate the area
Area of an isosceles triangle is ½ × b × h
A = ½ × 4 × 5 = 10 cm2
Therefore, the area of an isosceles triangle is 10 cm2
Conclusion
An isosceles triangle is a triangle that has any of its two sides of the same length, and also two angles opposite to the equal sides will be of equal measure. The perimeter and area of an isosceles triangle are calculated by using the lengths of its equal sides and its base. There are various applications of isosceles triangles in mathematics and constructions. Hence it is one of the most fundamental concepts of geometry. Isosceles triangles have unique properties and we hope that this article was successful in teaching you about the perimeter formula for the same.
FAQs on Isosceles Triangle Perimeter Formula
1. Can the Isosceles Triangle have an Obtuse Angle?
Yes, an isosceles triangle can have only one obtuse angle. An angle whose measure is greater than 90 degrees and lesser than 180 degrees is known as an obtuse angle. An isosceles triangle is one where two sides are equal. This is what makes this triangle unique in itself because it can also have an obtuse angle as long as the angle sum property of the triangle leads to the sum of the three angles being 180 degrees.
2. What is the Measure of the Two Equal Angles of a Right Isosceles Triangle?
As we know in an isosceles right triangle two sides are congruent and the corresponding angles will be 45 degrees each which sums to 90 degrees. Hence the measure of the two equal angles of an isosceles right triangle is 45 degrees each. An isosceles triangle has two equal sides and the angles opposite to the two equal sides are also equal. This is what leads to all the unique properties of the triangle.
3. How do I avoid confusion in the types of triangles?
The types of triangles that you need to be concerned about right now are an equilateral triangle, an isosceles triangle, and a scalene triangle. The most important thing is to not get confused between the three triangles. Therefore, we suggest that you make a graph or a table that highlights all the major differences between the three triangles, based on angles and the sides. This can give you a much better chance to remember everything properly.
4. Is the Pythagorean Theorem applicable in an isosceles Right angle triangle?
An isosceles right-angle triangle is one where two sides of the triangle are equal and one of the three angles has a measure of 90 degrees. Just like any other right triangle, an isosceles right-angled triangle also satisfies the Pythagorean theorem, which means that the sum of the area of the square of the sides of the triangle will be equal to the area of the square of the hypotenuse in this case as well.
5. What are some basic properties of an Isosceles Triangle?
An isosceles triangle is a very interesting type of triangle. Some of its very peculiar properties are that two of its sides are equal in length. Because of this, the angles opposite to these equal sides are also equal. An isosceles triangle can also be acute and it can also be a right-angled triangle as well as obtuse. Another property is that the height of the isosceles triangle is always measured from the base to the topmost vertex of the triangle.