Isosceles triangle base calculator

Properties of an isosceles triangle

An isosceles triangle is a special type of triangle with two sides of equal length. These equal sides are called the legs, while the third side is called the base. The uniqueness of an isosceles triangle lies in its symmetry. The angle opposite to the base is called the vertex angle, and the two angles adjacent to the base are called base angles.

The isosceles triangle has these fundamental properties:

1. Equal base angles: The angles adjacent to the base are equal.
2. Height: The height drawn from the vertex to the base is also the median and angle bisector.

Our calculator helps determine the base of an isosceles triangle using various known parameters, as commonly found in geometry problems. If you need to calculate the leg length, use our isosceles triangle leg calculator.

Two related sections

Height and median in an isosceles triangle

The height in an isosceles triangle is the perpendicular line drawn from the vertex to the base. In an isosceles triangle, this line serves three functions: it’s simultaneously the height, median, and angle bisector of the vertex angle. The median connects the vertex to the midpoint of the opposite side, while the angle bisector divides the vertex angle into two equal parts.

Angles in an isosceles triangle

The base angles of an isosceles triangle are always equal. If we denote the vertex angle as $\beta$ and the base angle as $\alpha$, then:

$$\beta = 180^\circ - 2\alpha$$

Thus, knowing one angle allows us to easily find the others.

Formulas

Our calculator offers several options based on available input data. Let’s examine the formulas for calculating the base $b$ depending on known parameters.

Known height and leg

With known height $h_1$ from the vertex and leg length $a$, the base is calculated as:

$$b = 2 \sqrt{a^2 - h_1^2}$$

Known leg and base angle

With known leg length $a$ and base angle $\alpha$, use the trigonometric formula:

$$b = 2a \cdot \cos(\alpha)$$

Known height and base angle

With given height $h_1$ and base angle $\alpha$, find the base using:

$$b = 2 h_1 \cdot \cot(\alpha)$$

Known area and height

With given area $A$ and height $h_1$, the base is determined by:

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

Known perimeter and leg

With known perimeter $P$ and leg length $a$:

$$b = P - 2a$$

Examples

Example 1: Base from height and leg

Given height $h_1 = 5$ inches and leg $a = 13$ inches. The base $b$ is:

$$b = 2 \sqrt{13^2 - 5^2} = 2 \sqrt{169 - 25} = 2 \sqrt{144} = 2 \times 12 = 24 \text{ inches}$$

Example 2: Base from leg and base angle

Given leg $a = 10$ inches and base angle $\alpha = 30^\circ$:

$$b = 2 \times 10 \times \cos(30^\circ) = 17.32 \text{ inches}$$

Example 3: Base from height and base angle

Given height $h_1 = 8$ inches and base angle $\alpha = 48^\circ$:

$$b = 2 h_1 \cdot \cot(\alpha) = 2 \times 8 \times \cot(48^\circ)$$

Since $\cot(48^\circ) = 0.9$:

$$b = 2 \times 8 \times 0.9 = 14.4 \text{ inches}$$

Example 4: Base from area and height

Given area $A = 36$ square inches and height $h_1 = 6$ inches:

$$b = \frac{2A}{h_1} = \frac{2 \times 36}{6} = 12 \text{ inches}$$

Example 5: Base from perimeter and leg

Given perimeter $P = 28$ inches and leg $a = 10$ inches:

$$b = P - 2a = 28 - 2 \times 10 = 8 \text{ inches}$$

Notes

• Calculation accuracy depends on the precision of input data.
• Ensure all measurements use consistent units before calculating.
• When using trigonometric functions, verify whether angles are in degrees or radians.

Frequently asked questions

How to find the base if the height is 4 inches and the leg is 5 inches?

Using the formula with height $h_1 = 4$ inches and leg $a = 5$ inches:

$$b = 2 \sqrt{5^2 - 4^2} = 2 \sqrt{25 - 16} = 2 \sqrt{9} = 6 \text{ inches}$$

Can the base be determined from perimeter and lateral height?

Yes, if you know the perimeter $P$ and leg length $a$, use:

$$b = P - 2a$$

How does the base angle affect the base length?

As the base angle increases, the base length decreases for a fixed leg length, following the relationship:

$$b = 2a \cdot \cos(\alpha)$$

Why are the base angles equal?

Base angles are equal because they are adjacent to equal legs. This is a fundamental property of isosceles triangles, verified through symmetry.

What other useful properties does an isosceles triangle have?

The height from the vertex divides the triangle into two congruent right triangles, and the median, angle bisector, and height from the vertex coincide.