Triangles: Area - Varsity Tutors

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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 xRemember 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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Beginner Explanation

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

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$12$ square unitsB$24$ square unitsC$10$ square unitsD$16$ square unitsCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

You want to calculate the area of a triangular park with a base of $10$ meters and a height of $5$ meters.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$84$ square unitsB$70$ square unitsC$80$ square unitsD$60$ square unitsCheck AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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