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The area of an isosceles triangle is the amount of space enclosed by its sides. We can calculate the area of an isosceles triangle using the general formula for the area of a triangle, which is half the base length × height. However, sometimes the base length isn’t given, and instead, one of the angles is. In such cases, we resort to other formulas that we will learn in this article. Below is a summary of triangle types based on their sides:
- Equilateral triangle: An isosceles triangle where the base length is equal to one of its legs.
- Isosceles triangle: A triangle with two sides of equal length, and consequently, two equal angles.
- Scalene triangle: A triangle where all three side lengths are different.
What is the Area of an Isosceles Triangle?
The area of an isosceles triangle is the space enclosed by the three sides of the triangle in a two-dimensional plane. As we know, an isosceles triangle is the triangle that has two equal sides and therefore has two equal angles.
As we know, the sum of a triangle’s angles is constant at 180 degrees, so if we know one of the two equal angles, we can determine all the triangle’s angles.
Here are some properties of an isosceles triangle that distinguish it from other types of triangles:
- The two equal sides of the triangle are called the legs of the triangle, and the angle between them is called the vertex angle of the triangle.
- The side opposite the vertex angle is called the base, and the two adjacent angles to it are equal.
- The perpendicular drawn from the vertex angle to the base divides the base into two equal segments and the vertex angle into two equal angles (i.e., it is both an angle bisector and a median).
Area is measured in square units, where the area of a triangle can be estimated in square meters (m2), square centimeters (cm2), square inches, etc.
Formula for the Area of an Isosceles Triangle
As we know, the area of a triangle is the space enclosed by its three sides in a two-dimensional plane. It can be determined using a set of formulas (according to the given information).
The basic formula for calculating the area of an isosceles triangle is (½ Base × Height). This is a general formula that can be applied to all types of triangles.
Below is a summary of the most important rules or formulas used to calculate the area of an isosceles triangle according to the information given in the problem.
| Known from the triangle | Formula Used |
|---|---|
| Base and its corresponding height | Area of triangle = 0.5 × Base × Height |
| Leg of the triangle and the triangle is right-angled and isosceles | Area of triangle = 0.5 × Side length2 |
| All three sides | Area = 1/4 × b × √ |
| Two sides and the angle between them | Area = ½ × b × a × sin(α) |
| Two angles and the side between them | Area = [a2 × sin(β/2) × sin(α)] |
Calculating the Area of an Isosceles Triangle Given its Side Lengths
By knowing the length of one of the equal sides and the base, you can calculate the area of the triangle using the formula:
Where:
- b is the length of the triangle’s base
- a is the length of one of the equal sides
This is actually derived from the previous formula and can be easily proven, but that is not the scope of our discussion in this article.
Example: Calculate the area of an isosceles triangle if you know that BC = 4 and AB = 3
Solution: As we can see, the base length BC = 4 and the length of one of the equal sides is 3. We can apply the previous formula to calculate the area of the triangle:
Area = 1/4 × b × √(4× a2 – b2)
= 1/4 × 4 × √(4× 32 – 42
= 1 × √( 36 – 16) = √20 = √(4×5) = 2√5 = 4.472
Area of an Isosceles Triangle Using Heron’s Formula
Heron’s formula can be used to calculate the area of a triangle when the lengths of all three sides are known. There’s no difference here, as the previous formula could also be used if desired, since all sides are known in this and the previous case.
Where:
- s is the semi-perimeter of the triangle, i.e.,
(a+b+c)÷2: where b in the formula represents one of the sides, not necessarily the base. - a, b, and c are the lengths of the triangle’s sides.
Area of an Isosceles Triangle Using Trigonometric Formulas
The area of an isosceles triangle can be calculated by applying the basic trigonometric formulas. This is done when the measure of an angle and the two sides enclosing it are known, or when the length of a side and the two angles adjacent to it are known. (angle + two sides or two sides + angle)
The area formula when the lengths of two sides and the included angle are known is:
Area = ½ × b × a × sin(α)
Where α is the angle measure and a, b are the lengths of the sides adjacent to the angle.
And the formula for the area of an isosceles triangle when two angles and the included side are known is:
Area = [a2 × sin(β/2) × sin(α)]
Where a is the side included between angles α and β (alpha and beta).
Formula for Calculating the Area of a Right-Angled Isosceles Triangle
A right-angled isosceles triangle is a right triangle where its two legs are equal in length. Therefore, the vertex angle is 90°, and the two equal angles are each 45°. The base is the hypotenuse of the triangle.
The formula for calculating the area of a right-angled isosceles triangle = 0.5 × length of the leg2
Area = ½ × a2
Where a is the length of one of the legs.
The previous formula can be easily derived from any of the preceding formulas.
Isosceles Triangle Area Problems
Problem 1: Find the area of an isosceles triangle if you know that one of its legs is 5 cm long and the base is 11 cm long.
Solution: Let’s list the given information:
- Length of one leg is 5 cm, so
a = 6 - Length of the base is 11 cm, so
b = 4
We can apply the formula: Area = 1/4 × b × √(4× a2 – b2)
Area = 1/4 × 4 × √(4× 62 – 42)
Area = 1 × √(128)
Area = 1 × √(64×2)
Area = 8√2
Area of the triangle = 11.31 cm2
Problem 2: Let’s consider triangle ABC with a base length AC = 10 meters and a height corresponding to the base = 17.
Solution: We have the base = 10 m2, so b=10, and we have the height = 17 m2, so h = 17. We apply the general formula for calculating the area of a triangle, which is 0.5 × b × h
Area = 0.5. b. h
Area = 0.5 × 10 × 17
Area = 85 m2
Problem 3: Calculate the area of an isosceles triangle if you know that one of its legs is 10 cm and the vertex angle is90 degrees.
Solution: We notice that the triangle is an isosceles right-angled triangle, so we can directly apply the following formula: Area of the triangle = 0.5 × (length of the triangle’s side)2
Area = 0.5× a2
Area = 0.5 × 102
Area = 50 cm2
