Isosceles triangle calculator

What is an isosceles triangle?

An isosceles triangle is a geometric figure characterized by having two equal sides known as legs. The third side, which is not equal to the other two, is referred to as the base. A noteworthy property of isosceles triangles is that the angles opposite the equal sides, known as base angles, are also equal. The angle between the two equal sides is called the vertex angle. Due to their symmetry, isosceles triangles are widely used in geometry and have numerous interesting properties and theorems associated with them.

What can this calculator compute?

This calculator can compute the area and perimeter of an isosceles triangle if the lengths of the legs and the base are known, or if the base and height are given. It can also calculate these metrics if one leg and the vertex angle are known. For calculating other parameters of an isosceles triangle, you might use additional calculators for the sides, base, height, and angles.

Key terms and notations

• Legs ($a$): The two equal sides of the triangle.
• Base ($b$): The side that is different from the legs, located opposite the vertex.
• Height from the vertex ($h_1$): A perpendicular dropped from the vertex to the base (also acts as the median and the angle bisector).
• Height to the legs ($h_2$): A perpendicular dropped from the base angle to the opposite leg.
• Vertex angle ($\beta$): The angle between the two equal legs.
• Base angles ($\alpha$): The angles located at the ends of the base.
• Perimeter ($P$): The sum of the lengths of all the triangle’s sides.
• Area ($A$): The space enclosed by the sides of the triangle.

Properties of an isosceles triangle

1. Equality of legs: The legs (denoted as $a$) are equal in length.
2. Equality of base angles: The base angles (denoted as $\alpha$) are equal.
3. Bearer of median, height, and bisector: From the vertex, the height, median, and bisector coincide and form a right angle with the base.
4. Equality of heights to the legs: The heights from the base angles to the opposite legs are equal.
5. Equality of base angle bisectors: The bisectors of the base angles are equal.

Formulas

Here are the basic formulas for calculating the area and perimeter of an isosceles triangle:

1. Area ($A$):

   Knowing the legs and base:

   $$A = \frac{1}{4} \cdot b \cdot \sqrt{4a^2 - b^2}$$

   Knowing the base and height:

   $$A = \frac{1}{2} \cdot b \cdot h_1$$

   Knowing the leg and vertex angle:

   $$A = \frac{1}{2} \cdot a^2 \cdot \sin(\beta)$$

2. Perimeter ($P$):

   $$P = 2a + b$$

   If the base $b$ and height $h_1$ are known, replace $a$ in the perimeter formula with:

   $$a = \sqrt{h_1^2 + \left(\frac{b}{2}\right)^2}$$

   If the leg $a$ and vertex angle $\beta$ are known, replace $b$ with:

   $$b = 2a \cdot \sin\left(\frac{\beta}{2}\right)$$

Examples

Example of calculating area

Example 1: Find the area of an isosceles triangle with a leg length of $a = 5$ cm and a base length of $b = 6$ cm.

Using the formula:

$$A = \frac{1}{4} \cdot b \cdot \sqrt{4a^2 - b^2}$$

Substitute the known values:

$$A = \frac{1}{4} \cdot 6 \cdot \sqrt{4 \times 5^2 - 6^2} = 12 \text{ cm}^2$$

Example 2: Find the area of an isosceles triangle with a base of $b = 8$ cm and height $h_1 = 5$ cm.

Using the formula:

$$A = \frac{1}{2} \cdot b \cdot h_1$$

Substitute the known values:

$$A = \frac{1}{2} \cdot 8 \cdot 5 = \frac{1}{2} \cdot 40 = 20 \text{ cm}^2$$

Example 3: Find the area of an isosceles triangle with a leg $a = 7$ cm and vertex angle $\beta = 45^\circ$.

Using the formula:

$$A = \frac{1}{2} \cdot a^2 \cdot \sin(\beta)$$

Substitute the known values:

$$A = \frac{1}{2} \cdot 7^2 \cdot \sin(45^\circ) \approx 17.32 \text{ cm}^2$$

Example of calculating perimeter

Example 1: If the base of an isosceles triangle is 8 cm and its height is 6 cm, find the perimeter.

1. Calculate the leg:

   $$a = \sqrt{6^2 + \left(\frac{8}{2}\right)^2} = \sqrt{36 + 16} = \sqrt{52} \approx 7.21 \text{ cm}$$

2. Perimeter ($P$):

   $$P = 2 \times 7.21 + 8 = 22.42 \text{ cm}$$

Example 2: If the leg of an isosceles triangle is 10 cm and the vertex angle is 60º, find the perimeter.

1. Calculate the base:

   $$b = 2 \times 10 \cdot \sin\left(30º\right) = 20 \times 0.5 = 10 \text{ cm}$$

2. Perimeter ($P$):

   $$P = 2 \times 10 + 10 = 30 \text{ cm}$$

Notes

• An isosceles triangle can be an equilateral triangle if all sides are equal.
• The height also acts as a median and a bisector due to its symmetry.
• Trigonometric functions are often used to calculate angles and heights.

Frequently asked questions

How is the area of an isosceles triangle calculated?

The area of an isosceles triangle can be calculated in several ways:

• Knowing the base and the height: $A = \frac{1}{2} \cdot b \cdot h_1$
• Knowing the leg and the vertex angle: $A = \frac{1}{2} \cdot a^2 \cdot \sin(\beta)$
• Knowing the base and one leg: $A = \frac{1}{4} \cdot b \cdot \sqrt{4a^2 - b^2}$

Are all heights in an isosceles triangle equal?

No, the height from the vertex is equal to the median and bisector to the base, while the heights from the base angles to the opposite legs are equal to each other.

How to find the perimeter of an isosceles triangle if the leg is 7 cm and the base is 10.5 cm?

Use the formula: $P = 2a + b$.

In this case, $a = 7$, $b = 10.5$; therefore, $P = 2 \times 7 + 10.5 = 24.5 \text{ cm}$.

What data is needed to calculate the perimeter of an isosceles triangle?

To calculate the perimeter, the length of the base and one leg is sufficient. The height or angles can also be used in combination calculations.

Can Heron’s formula be used to calculate the area of an isosceles triangle?

Heron’s formula can certainly be used to determine the area if all sides of the triangle are known. It is applicable to isosceles triangles as well as any other triangle.