Using Area and Base
Using Hypotenuse and Base (Right Triangle)
Equilateral Triangle (Side Length)
function toggleTriangleMethod() {
var method = document.getElementById("calcMethod").value;
document.getElementById("method-area").classList.remove("active-section");
document.getElementById("method-right").classList.remove("active-section");
document.getElementById("method-equilateral").classList.remove("active-section");
if (method === "area") {
document.getElementById("method-area").classList.add("active-section");
} else if (method === "right") {
document.getElementById("method-right").classList.add("active-section");
} else if (method === "equilateral") {
document.getElementById("method-equilateral").classList.add("active-section");
}
}
function calculateTriangleHeight() {
var method = document.getElementById("calcMethod").value;
var resultDiv = document.getElementById("tri-result");
var height = 0;
var error = "";
if (method === "area") {
var area = parseFloat(document.getElementById("triArea").value);
var base = parseFloat(document.getElementById("triBase1").value);
if (area > 0 && base > 0) {
height = (2 * area) / base;
} else {
error = "Please enter positive values for Area and Base.";
}
} else if (method === "right") {
var hyp = parseFloat(document.getElementById("triHypotenuse").value);
var base = parseFloat(document.getElementById("triBase2").value);
if (hyp > base && base > 0) {
height = Math.sqrt(Math.pow(hyp, 2) – Math.pow(base, 2));
} else {
error = "Hypotenuse must be longer than the base.";
}
} else if (method === "equilateral") {
var side = parseFloat(document.getElementById("triSide").value);
if (side > 0) {
height = (side * Math.sqrt(3)) / 2;
} else {
error = "Please enter a positive side length.";
}
}
resultDiv.style.display = "block";
if (error) {
resultDiv.style.backgroundColor = "#fce4e4";
resultDiv.style.borderColor = "#f44336";
resultDiv.innerHTML = "Error: " + error;
} else {
resultDiv.style.backgroundColor = "#e8f6ef";
resultDiv.style.borderColor = "#27ae60";
resultDiv.innerHTML = "Result: The height (altitude) is " + height.toFixed(2) + " units.";
}
}
Understanding Triangle Height (Altitude)
The height of a triangle, often referred to as its altitude, is the perpendicular distance from a vertex to the opposite side (the base). Finding the height is a fundamental skill in geometry, essential for calculating area and solving complex architectural or engineering problems.
Primary Formulas for Triangle Height
Depending on the information you have available, there are three primary ways to calculate the height:
From Area and Base: If you know the total area of the triangle, use the formula h = (2 × Area) / b. This is derived from the standard area formula (Area = 1/2 × base × height).
Right Triangles (Pythagorean Theorem): In a right-angled triangle, if the height is one of the legs, you can find it using h = √(c² - b²), where c is the hypotenuse and b is the base.
Equilateral Triangles: Because all sides are equal, the height can be found with a specific shortcut: h = (s × √3) / 2, where s is the side length.
Real-World Examples
Example 1: Using Area
Imagine you have a triangular garden plot with an area of 120 square feet and a base length of 20 feet. To find the height: h = (2 × 120) / 20 h = 240 / 20 = 12 feet.
Example 2: A Right-Angled Roof
A construction worker is measuring a right-triangle roof truss. The hypotenuse is 13 meters and the base is 5 meters. What is the vertical height? h = √(13² - 5²) h = √(169 - 25) = √144 = 12 meters.
Why is Triangle Height Important?
Calculating the altitude is crucial in several fields:
Architecture: Determining the pitch and peak of roofs.
Navigation: Calculating distances and heights using trigonometry.
Physics: Analyzing forces on inclined planes or structural stability in bridges.
General Education: It serves as a building block for understanding sine, cosine, and tangent functions.
Frequently Asked Questions
Can a triangle have more than one height?
Yes. Every triangle has three possible heights, depending on which side you choose as the base. Each height connects a vertex to its opposite side at a 90-degree angle.
Can the height be outside the triangle?
Yes, in an obtuse triangle, the altitude from one of the acute angles will fall on the extension of the opposite side, meaning the height sits outside the body of the triangle.