Real-Life Problems Solved by Mathematics: Triangle Area, Heights, Distances and More

Mathematics was born out of practical necessity. Over thousands of years, it has brought us to a world where nearly everything around us — technology, buildings, communication, navigation, finance — contains mathematics at its core: from simple calculations to number theory, analysis, geometry, and probability. And the most beautiful part: many familiar “engineering” problems actually have rigorous mathematical solutions.

How Mathematics Solves Real-Life Problems: A Brief Overview

This is part one of a series exploring real-life problems solved by mathematics. We’ll look at five classic problems — from finding a triangle’s area without measuring its height, to calculating the width of a river without crossing it. Each one demonstrates how abstract formulas translate directly into practical tools.

Solved Math Problem 1: Triangle Area from Three Sides

Problem: You have three sides a, b, c, but measuring the height is inconvenient. How do you find the area of a triangle knowing only the side lengths?

Solution: Introduce the semi-perimeter p = (a + b + c) / 2 and apply Heron’s formula:

S = √(p(p − a)(p − b)(p − c))

No height measurement required — just the three side lengths and a square root.

Solved Math Problem 2: Height of an Object from Its Shadow

Problem: How do you determine the height of a tree or a pole?

Solution 1: If a stick of height h₁ casts a shadow of length t₁, and the object casts a shadow of length t₂, then by triangle similarity:

h₂ = h₁ · (t₂ / t₁)

Solution 2: Stand at a known distance d from the object, measure the angle α to its top, and find the height. In a right triangle:

tan(α) = h / d, therefore h = d · tan(α)

Both approaches rely on the same geometric principle — similar triangles — applied in slightly different ways.

Solved Math Problem 3: Distance to an Inaccessible Point (River Width and More)

Problem: How do you measure the distance to a point you cannot reach directly — the opposite bank of a river, a fenced-off area?

Solution: Construct a triangle on accessible ground with measurable sides and angles, then recover the unknown distance using triangle similarity or the laws of sines and cosines. This technique was used by surveyors and military engineers for centuries, and remains the foundation of modern geodesy.

Solved Math Problems in Everyday Finance: Discounts and Taxes

Problem: Calculate the final price after several successive percentages — a discount, then another discount, then a tax.

Solution: Percentages act as multipliers. A 10% discount means multiplying by 0.9. Two consecutive 10% discounts give:

× 0.9 × 0.9 = × 0.81

That is an 19% total reduction, not 20% — a result that surprises many people but follows directly from how multiplicative factors work.

Real-Life Math: Mixtures and Solutions

Problem: Mix solutions of different concentrations to obtain a target concentration.

Solution: The amount of substance is conserved: (concentration) × (volume). For example, to mix x liters of a 10% solution with y liters of a 30% solution and obtain a 20% solution:

0.1x + 0.3y = 0.2(x + y)

Solve for the ratio x/y and you have your answer. This principle applies in chemistry, cooking, pharmaceuticals, and industrial production.

What Comes Next

All five problems above were known and solved thousands of years ago. Yet they remain in daily use — in construction, surveying, finance, and science. Mathematics did not stop there: the problems only grow in scale and complexity.

To be continued.