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Triangles: Area

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Triangles: Area

Study Guide

Key Definition

The area of a triangle can be calculated using the formula $A = \frac{1}{2} \times b \times h$, where $b$ is the base and $h$ is the height.

Important Notes

To find the area of a triangle, you need to know its base and height.
The base can be any side, but the height is the perpendicular distance from the base to the opposite vertex.
The area of a rectangle is used to derive the formula for the triangle's area.
A triangle can be split into two congruent triangles by drawing a median to the base.
The triangle inequality theorem states that the sum of any two sides must be greater than the third.
Heron's formula: For a triangle with side lengths $a$, $b$, $c$ and semiperimeter $s = \frac{a + b + c}{2}$, the area is $A = \sqrt{s(s - a)(s - b)(s - c)}$.

Mathematical Notation

$\triangle$ represents a triangle

$\angle$ represents an angle

$\frac{1}{2}$ represents one half

$x^2$ represents x squared

$\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

The formula $A = \frac{1}{2} \times b \times h$ is derived by dividing a rectangle into two congruent triangles, showing that the area of each triangle is half the area of the rectangle.

Remember

The key formula for the area of a triangle is $A = \frac{1}{2} \times b \times h$.

Quick Reference

Area of Triangle:$A = \frac{1}{2} \times b \times h$

Understanding Triangles: Area

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The area of a triangle is half the product of its base and height. For example, with base 6 and height 4, $A = \tfrac{1}{2} \times 6 \times 4 = 12$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the area of a triangle with base $6$ units and height $4$ units?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You want to calculate the area of a triangular park with a base of $10$ meters and a height of $5$ meters.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the base of a triangle is tripled and the height is halved, what happens to the area?

Challenge Quiz

Single Choice Quiz

Advanced

A triangle has sides $a = 7$, $b = 24$, and $c = 25$. What is the area?

Recap

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Review key concepts and takeaways