Learn on PengiBig Ideas Math, Course 1Chapter 4: Areas of Polygons

Lesson 2: Areas of Triangles

In this Grade 6 lesson from Big Ideas Math, Course 1, Chapter 4, students derive and apply the triangle area formula A = ½bh by cutting rectangles along a diagonal and rearranging triangles into known quadrilaterals. Students practice calculating the area of triangles given the base and height, and solve real-life problems such as comparing proportional areas when dimensions are scaled. The lesson aligns with standard 6.G.1 and builds geometric reasoning alongside accurate calculation skills.

Section 1

Area of a Triangle

Property

For a triangle with base $b$ and height $h$ , the area, $A$ , is given by the formula:

A = \frac{1}{2}bh

When solving geometry applications, it is helpful to draw the figure.

Examples

A triangle's area is 35 square feet. Its base is 3 feet more than its height. Let height be $h$ and base be $h+3$ . The equation is $35 = \frac{1}{2}(h+3)h$ , or $h^2+3h-70=0$ . Solving gives $h=7$ . The height is 7 ft and the base is 10 ft.
A triangular banner has an area of 80 square inches. The height is 4 inches less than twice the base. Let base be $b$ . The equation is $80 = \frac{1}{2}b(2b-4)$ , or $b^2-2b-80=0$ . Solving gives $b=10$ . The base is 10 in and the height is 16 in.
The area of a triangular garden plot is 50 square meters. The base is four times the height. Let height be $h$ . The equation is $50 = \frac{1}{2}(4h)h$ , or $2h^2=50$ . This gives $h^2=25$ , so $h=5$ . The height is 5 m and the base is 20 m.

Explanation

This formula connects a triangle's area, base, and height. When one dimension is described in terms of the other, substituting into the formula creates a quadratic equation you can solve to find the exact lengths of the base and height.

Section 2

Right Triangle Area Using Legs

Property

The area of a right triangle equals half the product of its two legs:

\text{Area} = \frac{ab}{2}

where

a

and

b

are the lengths of the legs (the two sides that form the right angle).

Examples

Section 3

Solving for a Missing Dimension

Property

If you know the area of a triangle and one of its dimensions (base or height), you can rearrange the area formula to solve for the missing dimension.

Given the area formula $A = \frac{1}{2}bh$ :

To find the height:
$h = \frac{2A}{b}$
To find the base:
$b = \frac{2A}{h}$

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Area of a Triangle

Property

For a triangle with base $b$ and height $h$ , the area, $A$ , is given by the formula:

A = \frac{1}{2}bh

When solving geometry applications, it is helpful to draw the figure.

Examples

A triangle's area is 35 square feet. Its base is 3 feet more than its height. Let height be $h$ and base be $h+3$ . The equation is $35 = \frac{1}{2}(h+3)h$ , or $h^2+3h-70=0$ . Solving gives $h=7$ . The height is 7 ft and the base is 10 ft.
A triangular banner has an area of 80 square inches. The height is 4 inches less than twice the base. Let base be $b$ . The equation is $80 = \frac{1}{2}b(2b-4)$ , or $b^2-2b-80=0$ . Solving gives $b=10$ . The base is 10 in and the height is 16 in.
The area of a triangular garden plot is 50 square meters. The base is four times the height. Let height be $h$ . The equation is $50 = \frac{1}{2}(4h)h$ , or $2h^2=50$ . This gives $h^2=25$ , so $h=5$ . The height is 5 m and the base is 20 m.

Explanation

Section 2

Right Triangle Area Using Legs

Property

The area of a right triangle equals half the product of its two legs:

\text{Area} = \frac{ab}{2}

where

a

and

b

are the lengths of the legs (the two sides that form the right angle).

Examples

Section 3

Solving for a Missing Dimension

Property

If you know the area of a triangle and one of its dimensions (base or height), you can rearrange the area formula to solve for the missing dimension.

Given the area formula $A = \frac{1}{2}bh$ :

To find the height:
$h = \frac{2A}{b}$
To find the base:
$b = \frac{2A}{h}$

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Area of a Triangle

Property

For a triangle with base $b$ and height $h$ , the area, $A$ , is given by the formula:

A = \frac{1}{2}bh

When solving geometry applications, it is helpful to draw the figure.

Examples

A triangle's area is 35 square feet. Its base is 3 feet more than its height. Let height be $h$ and base be $h+3$ . The equation is $35 = \frac{1}{2}(h+3)h$ , or $h^2+3h-70=0$ . Solving gives $h=7$ . The height is 7 ft and the base is 10 ft.
A triangular banner has an area of 80 square inches. The height is 4 inches less than twice the base. Let base be $b$ . The equation is $80 = \frac{1}{2}b(2b-4)$ , or $b^2-2b-80=0$ . Solving gives $b=10$ . The base is 10 in and the height is 16 in.
The area of a triangular garden plot is 50 square meters. The base is four times the height. Let height be $h$ . The equation is $50 = \frac{1}{2}(4h)h$ , or $2h^2=50$ . This gives $h^2=25$ , so $h=5$ . The height is 5 m and the base is 20 m.

Explanation

Section 2

Right Triangle Area Using Legs

Property

The area of a right triangle equals half the product of its two legs:

\text{Area} = \frac{ab}{2}

where

a

and

b

are the lengths of the legs (the two sides that form the right angle).

Examples

Section 3

Solving for a Missing Dimension

Property

If you know the area of a triangle and one of its dimensions (base or height), you can rearrange the area formula to solve for the missing dimension.

Given the area formula $A = \frac{1}{2}bh$ :

To find the height:
$h = \frac{2A}{b}$
To find the base:
$b = \frac{2A}{h}$