Triangles, defined by three sides and three vertices, have interior angles that always sum to 180 degrees. Each triangle has three angles, measured in degrees, with the sum of the three angles always equaling 180. Understanding this property is crucial for various triangle applications, including determining unknown angle measures, classifying triangle types, and solving complex triangle problems.
Explain what a triangle is, including its three sides and three vertices.
Triangle Tales: An Informal Guide to Triangle Geometry
Imagine a triangle as a trio of lines that huddle together like friends. These lines, called sides, form three pointy corners known as vertices. Each vertex is like a traffic circle where two sides intersect, while the sides act like roads connecting these vertices.
The sum of the angles inside a triangle is like a gigantic pizza – always equal to 180 degrees. Think of it as a triangle-shaped pizza party where the angles are the slices. The interior angles are the pizza slices inside the triangle, each representing a specific fraction of that 180-degree pizza.
Types of Triangles:
There’s a triangle for every occasion! Based on the size of their angles, triangles come in acute, right, and obtuse flavors. Acute triangles have three sharp angles, like shy kittens. Right triangles are the cool kids with one 90-degree angle, like a perfect right angle. And obtuse triangles are the rebels with one angle wider than 90 degrees, like a mischievous grin.
Classification by Sides:
Triangles can also be classified by their sides. If all three sides are different lengths, it’s a scalene triangle. If two sides are equal, it’s isosceles, like having a friend with a twin. And if all three sides are the same length, it’s an equilateral triangle, the perfect triangle of perfection.
Define the concept of angles and how they are measured.
Triangle Time: Unlocking the Secrets of Triangles
Yo, math lovers! Welcome to the world of triangles, where shapes have three sides and three angles. These geometric wonders might seem simple, but trust me, they’re a treasure trove of fascinating properties!
Defining Angles: The Sharp and Not-So-Sharp
Angles, the darlings of triangles, are like the points where two lines meet. They’re measured in degrees, and they can be as acute as a needle or as obtuse as a lazy couch potato. Right angles, on the other hand, are the perfect 90 degrees, like that super serious square that always follows the rules. So, remember, angles come in all shapes and sizes, and they’re what make triangles so, well, triangular!
Describe the three different ways to divide a triangle: median, altitude, and bisector.
Section 2: Properties of Triangles
Dividing a Triangle: Meet the Median, Altitude, and Bisector
Imagine a triangle as a pizza. You can slice it in three different ways:
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Median: This is like cutting the pizza in half, from one vertex to the midpoint of the opposite side. It’s like a line going straight through the center of the pizza.
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Altitude: This is like cutting the pizza from a vertex straight down to the opposite side, forming a perpendicular line. It’s like a straight drop from the top of the pizza.
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Bisector: This is like cutting the pizza into two equal halves, by drawing a line that divides the angle at a vertex in half. It’s like slicing the pizza into two perfectly symmetrical pieces.
These lines not only divide the pizza (or triangle), but they also have some special properties that we’ll explore in more detail later on!
Explain the sum of interior angles of a triangle and how to calculate it.
Unveiling the Secrets of Triangles: Dive into the World of Angles
Hey there, triangle enthusiasts! Let’s get our geometry hats on and explore the captivating realm of angles. Just like our own homes have walls that meet at corners, triangles have those awesome pointy things called vertices where their sides come together. And guess what? These vertices form angles! Yes, folks, angles are the stars of our triangle show.
Now, prepare to be amazed! The total sum of these interior angles in a triangle is always 180 degrees. It’s like a magical triangle formula that never fails. Picture this: you have a triangle, right? With three vertices and hence three interior angles. Their friendly neighborhood angles decide to join forces and create a grand total of 180 degrees—like a harmonious angle choir singing in perfect unison!
So, next time you’re in a triangle-related situation, just remember this golden rule: The interior angles of a triangle always add up to 180 degrees. It’s as easy as pi (well, not quite that easy, but still!). So, let’s raise a toast to these extraordinary angles—the backbone of our triangle adventures!
Introduce various types of angles and their relationships, such as exterior angles, interior opposite angles, alternate exterior angles, and the converse of alternate exterior angles.
The Intricate World of Triangle Angles
Triangles, with their sharp angles and rigid sides, are more than just geometric shapes. They’re a fascinating world unto themselves, filled with intriguing relationships that can make even mathematicians chuckle.
Imagine a triangle basking in the spotlight, its angles basking in their own glory. There are exterior angles, like shy wallflowers peeking outside the triangle’s boundaries. They’re always equal to the sum of the interior opposite angles, those friendly neighbors facing each other across the triangle’s interior.
But wait, there’s more! Alternate exterior angles are like twins separated at birth. They’re equal to each other, just like they’re mirror images. And the converse of alternate exterior angles? It’s like a mischievous prankster, always switching the values of exterior and interior opposite angles.
It’s like a grand dance of angles, each one playing its own unique tune. They twist, twirl, and interact, creating a harmonious symphony of geometric relationships. So, next time you see a triangle, don’t just take it for granted. Marvel at the angular tapestry that unfolds within its confines.
Triangle Geometry: Unraveling the Tricky Trio of Sides, Angles, and Theorems
Hey there, math enthusiasts! Let’s embark on a geometric adventure that’ll make you look at triangles with a whole new perspective. Sorry, no jokes about Triangles being the shape of your pizza slice today, we’re going the knowledge-over-humor route. 🍕
Triangle Basics: Sides and Angles
Picture a triangle, the simplest polygon with three straight sides and three corners. These corners are where the sides meet, known as vertices. Now, let’s talk about angles, the spaces between those lines. Angles are measured in degrees, but we’ll dig into that unit-conversion fun later.
Triangle Properties: What Makes Them Tick
Triangles have some unique traits that set them apart from other shapes. You got your medians, the lines connecting vertices to midpoints of opposite sides. Altitudes are perpendicular lines from vertices to opposite sides. And bisectors split angles into equal parts.
Oh, and the big reveal: the sum of the interior angles of any triangle is always 180 degrees. Remember this one, kids!
Angle Measurement: Degrees, Radians, and Gradians
Time to talk units! We measure angles with degrees, marked by a little circle (°). A full circle is 360 degrees. But wait, there’s more! Math loves to give us options. We also have radians (marked by rad) and gradians (grad). They’re basically different ways of slicing that angle pie.
Triangle Classifications: Types Galore
Now, let’s categorize these triangles based on their angles and sides. Acute triangles have all angles less than 90 degrees. Right triangles have one 90-degree angle. And obtuse triangles have one overachieving angle greater than 90 degrees.
As for sides, scalene triangles have all three sides different. Isosceles triangles have two equal sides. And equilateral triangles have all three sides equal. Ta-da, triangle diversity!
Triangle Theorems: The Geometry Gems
Hold on tight for the grand finale: triangle theorems! The Triangle Inequality Theorem says the sum of two sides must be greater than the third side. The Angle Sum Theorem proves the sum of the three interior angles is 180 degrees. The Exterior Angle Theorem tells us an exterior angle equals the sum of the opposite interior angles.
And the Pythagorean Theorem? It’s the golden boy of right-triangle math, helping us find missing lengths. Don’t forget the Sine Law and Cosine Law for solving more complex triangle problems. Oh, and Heron’s Formula for calculating area without height. Whew!
So, there you have it, folks! Triangle geometry, the not-so-tricky trio of sides, angles, and theorems. Remember, math can be fun… even when it’s about shapes with pointy corners.
Triangle Tales: A Guide to the Realm of Triangles
So, you think you know triangles? Three sides and three angles, right? Well, buckle up, triangle enthusiasts, because we’re about to take you on a wild and wacky adventure into the fascinating world of triangle geometry!
Chapter 1: Triangle Basics
Imagine triangles as the building blocks of geometry. They’re made up of three lines that meet at three points called vertices. And just like little LEGO bricks, triangles come in all shapes and sizes.
But before we go triangle-hopping, let’s talk about angles. Angles are like little elbow bends that measure the space between two lines. We use degrees to measure these angles, and they can range from 0° (a straight line) to 180° (a complete turn).
Chapter 2: Triangle Properties
Triangles have special ways of dividing themselves that are totally awesome. We’ve got medians, altitudes, and bisectors that slice and dice triangles in all sorts of fun ways.
And here’s a juicy fact: the sum of the interior angles of a triangle is always 180°. That’s like a triangle secret code that you can use to solve all sorts of tricky problems.
Chapter 3: Angle Measurement Shenanigans
Talking about angles, let’s not forget about the different units we can use to measure them. We’ve got degrees, radians, and gradians, kind of like the metric system for angles. Don’t worry, converting between them is as easy as pie (especially with a handy-dandy calculator).
Chapter 4: Triangle Classification Fiesta
Get ready for the triangle classification party! We’ve got acute triangles (with all angles under 90°), right triangles (with one 90° angle), and obtuse triangles (with one angle over 90°).
But wait, there’s more! We also have scalene triangles (all sides different), isosceles triangles (two sides the same length), and equilateral triangles (all sides equal). It’s like a triangle fashion show!
Chapter 5: Triangle Theorem Trivia
Now for the pièce de résistance: triangle theorems! They’re the rules of the triangle world, and they’ll help you conquer any triangle challenge that comes your way.
We’ve got the Triangle Inequality Theorem (don’t put too many sides on a triangle), the Angle Sum Theorem (that 180° rule we talked about), and the Exterior Angle Theorem (the outside angle is equal to the sum of the opposite interior angles).
Plus, there’s the Pythagorean Theorem, the Sine Law, and the Cosine Law. These bad boys will help you find lengths, areas, and angles like a boss.
And finally, we have the awesome Heron’s Formula, which calculates the area of a triangle without even knowing its height. Triangles just love to give us formulas!
So there you have it, folks! The ultimate guide to triangle geometry. Now go out there and conquer all those triangle puzzles and problems with confidence and a sprinkle of triangle humor.
Describe the different types of triangles based on their angles: acute, right, and obtuse.
Angle Adventure: Types of Triangles Based on Their Angles
Triangles, the basic building blocks of geometry, are like the superheroes of the shape world. They come in different shapes and sizes, and classifying them based on their angles is an exciting puzzle. Let’s dive into the angle adventure!
Acute Triangles:
Imagine a triangle that’s shy and doesn’t like to show its sharp angles. That’s an acute triangle. It’s a triangle where all three interior angles are less than 90 degrees.
Right Triangles:
Now, let’s talk about the triangle that forms a perfect 90-degree angle. That’s a right triangle. It’s a triangle where one angle is 90 degrees, and the other two angles add up to 90 degrees. Think of it as a triangle that’s standing straight up!
Obtuse Triangles:
Finally, we have the triangle that’s a bit of a rebel. It has one interior angle that’s greater than 90 degrees. That’s called an obtuse triangle. It’s like a triangle that’s leaning back, taking it easy.
Explain the classifications based on their sides: scalene, isosceles, and equilateral.
Triangle Times: Exploring the Realm of Triangles
Triangle geometry: a world of triangles, angles, and intriguing relationships. Let’s dive in, shall we?
Triplets of Sides: Scalene, Isosceles, and Equilateral
Just like triplets in the human world, triangles can also come in different flavors based on their sides. Get ready to meet the scalene, isosceles, and equilateral triangle gang.
Scalene Triangle: A bit of a loner, the scalene triangle has no two sides of equal length. Each side has its own unique personality, just like your quirky best friend.
Isosceles Triangle: The social butterfly, the isosceles triangle boasts two sides that are like twins, sharing the same length. Imagine identical twins holding hands and forming a V-shape.
Equilateral Triangle: The perfectionist of the triangle world, the equilateral triangle has three sides that are all equal. Think of a perfectly symmetrical snowflake, with three identical arms.
Introduce specific triangle types with unique properties, such as 30-60-90 triangles, 45-45-90 triangles, and 60-60-60 triangles.
4. Classifications of Triangles
Specific Triangle Types with Unique Properties
Now, let’s dive into the world of specific triangle types that rock the geometry realm with their groovy properties!
First up, we have the 30-60-90 triangle. This triangle is like a perfect summer day: it’s hot (30°), sunny (60°), and long (90°)! The ratio of its sides is always the same: 1:√3:2. It’s like a triangle version of the golden ratio, dude!
Next, we have the 45-45-90 triangle. This triangle is the chillest of the bunch, with all its angles being super mellow (45°). It’s the Zen master of triangles, always bringing balance and harmony to the geometry world. Its side ratios are also a thing of beauty: 1:1:√2.
Last but not least, let’s give a round of applause for the 60-60-60 triangle. This triangle is the equilateral superstar of the triangle family, with all three sides and angles being equal (60°). It’s the triangle that always gets the perfect score on geometry tests! Its side ratios are the definition of equality: 1:1:1.
These specific triangle types are like the rock stars of the triangle world, each with their own unique vibe and set of rules. They’re indispensable tools for solving geometry problems, and they’ll make you look like a geometry wizard in no time!
State and explain the Triangle Inequality Theorem.
Triangle Geometry: Unraveling the Secrets of Shapes
Hello there, geometry enthusiasts! Today, we’re diving into the fascinating world of triangles. These three-sided wonders hold more secrets than you might think, so buckle up as we explore their basic concepts and delve into the magical world of triangle properties.
Triangle 101: The Basics
Imagine a triangle as a three-legged creature with three corners (called vertices) and three arms (called sides) that connect them. Each angle is where two sides meet, and these angles are measured in degrees. Get ready to discover the diverse world of triangle angles!
Triangle Properties: Unlocking Hidden Truths
Get ready to meet the medians, altitudes, and bisectors—the triangle’s magical helpers! These lines are like GPS guides, dividing a triangle into smaller parts and revealing hidden relationships.
But wait, there’s more! The interior angles of any triangle always add up to a special number: 180 degrees. It’s like a triangle’s secret code. We’ll also explore different types of angles in triangles, like exterior angles and those pesky opposite angles.
Angle Measurement: Units and Conversions
Time for a unit conversion party! We’ll learn about degrees, radians, and gradians—the different ways to measure angles like we’re geometry superheroes. And here’s the fun part: we’ll show you how to switch between these units like magic!
Triangle Classifications: From Acute to Equilateral
Triangles come in all shapes and sizes! We’ll categorize them based on their angles (acute, right, and obtuse) and their sides (scalene, isosceles, and equilateral). Plus, we’ll introduce special triangle types like the legendary 30-60-90 triangles!
Triangle Theorems: The Rulebook of Geometry
Behold the mighty Triangle Inequality Theorem! It’s like the “no cheating” rule of geometry, stating that any side of a triangle must be shorter than the sum of the other two sides. And hold on to your protractors because we’re unveiling the famous Angle Sum Theorem. It’s like the secret formula for finding the missing angle in any triangle!
Dive into Triangle Geometry: Unlocking the Secrets of Shapes
Hey there, triangle enthusiasts! Welcome to our geometric escapade where we’ll unravel the fascinating world of triangles. Let’s start by getting to know these wonderful shapes better.
Basic Triangle Basics:
Picture a triangle, a three-sided polygon with three vertices (corners) and three angles. Each side has a length, and each angle has a measure. The three angles of a triangle have a special relationship that we’ll explore in a bit.
Triangle Properties:
Triangles have some cool ways to divide themselves. You’ve got medians, altitudes, and bisectors, all connecting different points within the triangle. But wait, there’s more! The sum of the interior angles of a triangle is always the same, no matter what shape or size it is. Now that’s a triangle secret worth knowing!
Angle Measurement in Triangles:
Now let’s talk about measuring angles. We’ve got degrees, radians, and gradians, but let’s stick with degrees for now. Why? Because it’s what we use in our everyday lives. Just remember, 360 degrees makes a full circle, and 180 degrees is half a circle.
Triangle Classifications:
Triangles come in different flavors based on their angles. We have acute triangles with three angles less than 90 degrees, right triangles with one 90-degree angle, and obtuse triangles with one angle greater than 90 degrees. Oh, and don’t forget about their sides! Triangles can be scalene (no equal sides), isosceles (two equal sides), or equilateral (all three sides equal).
Triangle Theorems:
The Angle Sum Theorem is a game-changer. It states that the sum of the interior angles of a triangle is always 180 degrees. This magic rule helps us solve for unknown angles in triangles, like when you’re trying to find the measure of a missing angle. And don’t forget the Pythagorean Theorem, the backbone of right-triangle geometry, which lets us calculate missing side lengths using the relationship between the sides and the right angle.
Real-World Triangle Applications:
Triangles are everywhere! From architecture to construction, nature to art, they’re hidden in plain sight. Understanding triangle geometry can help you navigate the world around you, from building a sturdy bridge to appreciating the beauty of a snowflake.
So, there you have it, folks! Triangle geometry is not just a bunch of rules and formulas. It’s a doorway to understanding the world’s shapes and structures. Whether you’re a student, a builder, an artist, or just someone who loves puzzles, triangles have something to offer you. So, embrace the triangle, and let’s unravel its wonders together!
Triangle Geometry: The Ultimate Guide for Math Lovers
Hey there, math enthusiasts! Let’s dive into the fascinating world of triangle geometry. From basic definitions to mind-bending theorems, we’re about to break down everything you need to know about these three-sided wonders.
Basic Triangle Geometry
What’s a triangle, you ask? It’s like a triangle sandwich with three sides and three vertices (corners). Angles, the rock stars of geometry, are formed where the sides meet.
Properties of Triangles
Let’s get fancy with triangle properties!
- Dividing Divas: Meet medians, altitudes, and bisectors – they divide triangles in oh-so-stylish ways.
- Angle Sum Shenanigans: The angles inside a triangle always add up to a sweet 180 degrees, no ifs, ands, or triangles!
- Angles and their Besties: Interior angles have their buds, like exterior angles and alternate exterior angles. They’re like a geometric soap opera, but with less drama.
Angle Measurement in Triangles
Angles aren’t just about degrees. They’ve got other cool units like radians and gradians. Don’t get lost in translation! We’ll show you how to waltz between these units like a pro.
Classifications of Triangles
Triangles come in all shapes and sizes. Acute triangles have all angles less than 90 degrees, right triangles have one angle at 90 degrees, and obtuse triangles have one angle greater than 90 degrees.
Side Show Stories: Triangles can also be classified by their sides. Scalene triangles have no equal sides, isosceles triangles have two equal sides, and equilateral triangles are the A-listers of the triangle world, with all sides equal.
Triangle Theorems
Buckle up for some triangle theorem thrillers!
- Triangle Inequality Theorem: This rule says that the sum of two sides of a triangle must be greater than the third side. No sneaky shortcuts allowed!
- Exterior Angle Theorem: Get ready to flip triangles inside out! The exterior angle (outside angle) of a triangle is equal to the sum of the opposite two interior angles. Mind-blowing, right?
- Pythagorean Theorem: Ah, the legend! This theorem rocks the triangle world by connecting the lengths of the sides in right triangles. (a²+b²=c², baby!)
Stay tuned for more triangle theorem fun in our next post!
Discuss the Pythagorean Theorem and its significance in calculating lengths in right triangles.
Triangle Geometry: Unraveling the Secrets of Triangles
Embark on an exciting journey into the fascinating world of triangle geometry, where we’ll explore the basics, properties, and intriguing facts that make these geometric gems so captivating.
Chapter 1: Triangle Fundamentals
Meet triangles, the shapes with three sides and three vertices. Each side has its own length, and the angles they form, we call them interior angles, have special relationships we’ll uncover later.
Chapter 2: Triangle Properties
Triangles have a unique set of properties. Like dividing them up into medians (connecting a vertex to the midpoint of the opposite side), altitudes (perpendicular lines from a vertex to the opposite side), and bisectors (splitting an angle in half).
Oh, and the sum of those interior angles? Always 180 degrees! We call that the Angle Sum Theorem.
Chapter 3: Triangle Angles
Angles, angles, everywhere! Triangles have exterior angles, interior opposite angles, alternate exterior angles, and the Converse of Alternate Exterior Angles. It’s like a triangle party!
Chapter 4: Triangle Classifications
Triangles come in all shapes and sizes! We’ve got acute triangles with sharp angles, right triangles with that perfect 90-degree angle, and obtuse triangles where one angle is over 90 degrees.
And let’s not forget about the scalene, isosceles, and equilateral triangles based on their side lengths.
Chapter 5: The Mighty Pythagorean Theorem
Now, let’s talk about the star of the show: the Pythagorean Theorem. This theorem is a lifesaver for finding missing side lengths in right triangles. It says that in a right triangle, the square of the length of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
It’s a triangle-solving superhero!
Unraveling the Secrets of Triangle Geometry
Triangle geometry might sound like a daunting concept, but it’s not as intimidating as it seems. Let’s break it down into bite-sized chunks and explore this fascinating world together.
Triangular Basics
First things first, what even is a triangle? It’s a polygon with three straight sides that meet at three corners called vertices. Each side connects two vertices, and each vertex connects two sides. Easy peasy.
Then we have angles. Think of them as the spaces between two sides that meet at a vertex. We measure angles in degrees, and the sum of interior angles in a triangle is always 180 degrees. That’s right, the three angles inside a triangle always add up to 180, no exceptions!
Dividing Triangles
Triangles can be divided into smaller pieces in several ways. There’s the median, which is a line segment that connects a vertex to the midpoint of the opposite side. You also have altitudes, which are perpendicular lines from a vertex to the opposite side, and bisectors, which divide an angle into two equal parts.
Angle Adventures
Angles in triangles have their own little dramas going on. We have exterior angles, which are formed by one side of the triangle and an extension of the adjacent side. Then there are interior opposite angles, which are on opposite sides of a transversal that intersects two sides of the triangle. And get this, there’s the alternate exterior angle theorem, which states that an exterior angle is equal to the sum of the opposite interior angles.
Triangle Classifications
Triangles can be classified by their angles and sides. Based on angles, we have acute triangles (all angles less than 90 degrees), right triangles (one angle of 90 degrees), and obtuse triangles (one angle greater than 90 degrees). Based on sides, we have scalene triangles (no equal sides), isosceles triangles (two equal sides), and equilateral triangles (all three sides equal).
Tri-umphant Theorems
Here come the theorems! The Triangle Inequality Theorem tells us that the sum of two sides of a triangle must be greater than the third side. The Angle Sum Theorem we already know. The Exterior Angle Theorem states that an exterior angle is equal to the sum of the non-adjacent interior angles.
And then we have the star of the show, the Pythagorean Theorem. It’s all about right triangles, stating that the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.
Finally, we have the Sine Law and the Cosine Law. These two bad boys help us solve triangles when we don’t have all the lengths and angles. They’re like secret weapons for triangle warriors.
Triangle Geometry: The Basics and Beyond
What is a triangle? It’s like a geometrical sandwich, with three slices called sides and three corners called vertices. And just like a sandwich, triangles come in all shapes and sizes!
Triangle Anatomy
Triangles have a special way of dividing themselves up into medians, altitudes, and bisectors. Medians are like the middle lines, altitudes are the perpendicular lines, and bisectors split the angles into two equal parts.
Every triangle is a sum of its angles. The sum of interior angles in a triangle is always 180 degrees. Know why? It’s because if you were to pull apart a triangle and lay it flat, it would form a straight line!
Angle Measurements: The Degree Dilemma
We measure angles in degrees, radians, or gradians. Degrees are the most common, so let’s stick with them for now. Imagine a clock: the hour hand moves 30 degrees every hour, and the minute hand moves 6 degrees every minute. So, a full circle is 360 degrees, like a complete hour.
Triangle Classifications: Size Matters
Based on their angles, triangles are acute, right, or obtuse. Acute triangles have all sharp angles, right triangles have one right angle (90 degrees), and obtuse triangles have one angle that’s greater than 90 degrees.
As for their sides, triangles can be scalene, isosceles, or equilateral. Scalene triangles have no equal sides, isosceles triangles have two equal sides, and equilateral triangles have all three sides equal.
Triangle Theorems: The Rules of Engagement
Triangles have some cool rules they have to follow, like the Triangle Inequality Theorem. It says that the sum of any two sides of a triangle must be greater than the third side. Think of it as a weird friendship rule: if you’re friends with A and B, then A and B can’t be friends with someone who you’re not friends with!
Another important theorem is the Angle Sum Theorem. It states that the sum of the interior angles of a triangle is always 180 degrees. This is basically the geometry version of the saying “there’s no place like home.”
Heron’s Formula: The Area Avenger
Calculating the area of a triangle can be tricky, but Heron’s Formula has got our backs! This magical formula lets us find the area of a triangle without knowing its height:
Area = √s(s – a)(s – b)(s – c)
Where:
– s = the semiperimeter (half the perimeter)
– a, b, c = the three sides of the triangle
Just plug in the side lengths, and voilà! You’ve got the area of your triangle without breaking a sweat.
