Find The Height Of Triangle Calculator

Use this height of triangle calculator to find the altitude of a triangle using various known values. It's a versatile tool for students and professionals.

Calculate Triangle Height

Summary Table

Parameter	Value
Method	–
Base	–
Area	–
Side a	–
Side b	–
Side c	–
Side 1 (TSA)	–
Side 2 (TSA)	–
Angle (TSA)	–
Base (BA)	–
Angle B (BA)	–
Angle C (BA)	–
Height	–

Summary of inputs and calculated height.

What is a Height of Triangle Calculator?

A height of triangle calculator is a tool used to determine the altitude (height) of a triangle based on different known parameters. The height of a triangle is the perpendicular distance from a vertex to the opposite side (the base). Every triangle has three heights, one for each side considered as the base.

This height of triangle calculator is useful for students learning geometry, engineers, architects, and anyone needing to find the dimensions of a triangle. It simplifies calculations that might otherwise require manual application of various formulas.

Common misconceptions include thinking a triangle has only one height, or that the height always falls within the triangle (which is not true for obtuse triangles).

Height of Triangle Calculator Formula and Mathematical Explanation

The formula used by the height of triangle calculator depends on the known values:

1. Given Area and Base

If you know the area (A) and the base (b), the height (h) is found using:

h = (2 * A) / b

This is derived from the basic area formula: `A = 0.5 * b * h`.

2. Given Three Sides (Heron's Formula)

If you know the lengths of the three sides (a, b, c):

1. Calculate the semi-perimeter (s): `s = (a + b + c) / 2`
2. Calculate the Area (A) using Heron's formula: `A = sqrt(s * (s – a) * (s – b) * (s – c))`
3. Calculate the height relative to each base:
   • Height to base a (h_a): `h_a = (2 * A) / a`
   • Height to base b (h_b): `h_b = (2 * A) / b`
   • Height to base c (h_c): `h_c = (2 * A) / c`

Our height of triangle calculator will show all three heights if you input three sides.

3. Given Two Sides and the Included Angle

If you know two sides (say, a and b) and the angle C between them:

1. Calculate the Area (A): `A = 0.5 * a * b * sin(C)` (where C is in radians)
2. The height relative to base 'a' (h_a) is: `h_a = b * sin(C)`
3. The height relative to base 'b' (h_b) is: `h_b = a * sin(C)`

The height of triangle calculator finds the height relative to one of the given sides.

4. Given Base and Two Adjacent Angles

If you know the base (say, 'a') and the two angles B and C adjacent to it:

1. Find the third angle A: `A = 180° – B – C`
2. Using the Law of Sines (`a/sin(A) = b/sin(B)`), find side 'b': `b = (a * sin(B)) / sin(A)`
3. The height relative to base 'a' (h_a) is: `h_a = b * sin(C)`

This method allows the height of triangle calculator to find the altitude from the vertex opposite the known base.

Variables Table

Variable	Meaning	Unit	Typical Range
A	Area of the triangle	Square units (e.g., m^2, cm^2)	> 0
b, a, c	Lengths of the sides (bases)	Units (e.g., m, cm)	> 0
h, ha, hb, hc	Height (altitude) of the triangle	Units (e.g., m, cm)	> 0
s	Semi-perimeter of the triangle	Units (e.g., m, cm)	> 0
B, C	Angles of the triangle	Degrees	0° – 180°

Practical Examples (Real-World Use Cases)

Example 1: Finding Height from Area and Base

A triangular garden has an area of 50 square meters and one side (base) is 10 meters long. What is the height corresponding to this base?

• Area (A) = 50 m²
• Base (b) = 10 m
• Height (h) = (2 * 50) / 10 = 100 / 10 = 10 meters.

The height of the garden relative to the 10m base is 10 meters.

Example 2: Finding Heights from Three Sides

A plot of land is triangular with sides 13m, 14m, and 15m.

• a = 13, b = 14, c = 15
• s = (13 + 14 + 15) / 2 = 42 / 2 = 21
• Area = sqrt(21 * (21-13) * (21-14) * (21-15)) = sqrt(21 * 8 * 7 * 6) = sqrt(7056) = 84 m²
• Height to base a (13m): h_a = (2 * 84) / 13 ≈ 12.92 m
• Height to base b (14m): h_b = (2 * 84) / 14 = 12 m
• Height to base c (15m): h_c = (2 * 84) / 15 = 11.2 m

The height of triangle calculator would give these three heights.

How to Use This Height of Triangle Calculator

1. Select the Method: Choose the radio button corresponding to the values you know (Area and Base, Three Sides, Two Sides & Included Angle, or Base & Two Adjacent Angles).
2. Enter Known Values: Input the required lengths and/or angles into the appropriate fields that appear. Ensure the units are consistent.
3. View Results: The calculator will automatically update and display the calculated height(s), area (if calculated as an intermediate step), and the formula used. The primary result is highlighted.
4. Interpret Results: The "Height" displayed is the altitude of the triangle corresponding to the base you specified or implied through your inputs. If you used three sides, it might show three heights.
5. Use Chart and Table: The chart visually compares the base and height, while the table summarizes inputs and results.

This height of triangle calculator simplifies finding the altitude, which is crucial in various geometric and real-world problems.

Key Factors That Affect Height of Triangle Results

• Known Values: The accuracy of the calculated height directly depends on the precision of the input values (sides, area, angles). Small errors in input can lead to different height results.
• Chosen Base: A triangle has three potential heights, each corresponding to a different side chosen as the base. The height of triangle calculator finds the height relative to the base you directly or indirectly specify.
• Type of Triangle: For an equilateral triangle, all three heights are equal. For an isosceles triangle, heights to the equal sides are equal. For a scalene triangle, all three heights are different.
• Angles: The angles of a triangle determine its shape and thus the relative lengths of its heights. For obtuse triangles, some heights fall outside the triangle.
• Units: Ensure all length inputs use the same units. The height will be in the same unit. Area should be in square units corresponding to the length unit.
• Triangle Inequality Theorem: When providing three sides, they must satisfy the triangle inequality theorem (the sum of any two sides must be greater than the third side) for a valid triangle and hence a real height to be calculated by the height of triangle calculator.

Frequently Asked Questions (FAQ)

1. What is the height of a triangle?

The height (or altitude) of a triangle is the perpendicular distance from a vertex to the line containing the opposite side (the base).

2. How many heights does a triangle have?

Every triangle has three heights, one for each side considered as the base.

3. Can the height be outside the triangle?

Yes, for an obtuse triangle, the heights drawn from the vertices of the acute angles fall outside the triangle.

4. How do I use the "Three Sides" method in the height of triangle calculator?

Select "Three Sides", enter the lengths of sides a, b, and c. The calculator uses Heron's formula to find the area first, then calculates the height relative to each side.

5. What if I have a right-angled triangle?

For a right-angled triangle, the two legs are also two of the heights. The third height is from the right-angle vertex to the hypotenuse. You can use the "Area and Base" or "Three Sides" method with our height of triangle calculator, or use our specific right-triangle calculator.

6. Why does the calculator show three heights for the "Three Sides" method?

Because any of the three sides can be considered the base, and each has a corresponding height. The height of triangle calculator computes all three possibilities.

7. What units should I use?

Be consistent. If you enter sides in centimeters, the height will be in centimeters, and area (if used or calculated) in square centimeters.

8. Does this calculator work for equilateral triangles?

Yes, an equilateral triangle is a special case where all sides are equal. The height of triangle calculator will correctly find the height (all three heights will be the same).