Learn on PengiIllustrative Mathematics, Grade 6Unit 1 Area and Surface Area

Lesson 3: Triangles

In this Grade 6 lesson from Illustrative Mathematics Unit 1, students explore the relationship between parallelograms and triangles by decomposing quadrilaterals into two identical triangles using a single diagonal cut. Students discover that any parallelogram can always be split into two congruent triangles, and conversely, two identical triangles can be joined along any of their three sides to form a parallelogram. This special connection between the two shapes builds the foundation for understanding how to calculate the area of any triangle.

Section 1

Visualizing the Area: Triangles and Parallelograms

Session 1. Visualizing the Area: Triangles and Parallelograms

Property

Two identical (congruent) triangles can always be joined together to form a parallelogram. Therefore, the area of a single triangle is exactly half the area of a parallelogram that shares the same base and height.

Examples

  • Right Triangles: Two identical right triangles can be joined along their longest side (the diagonal) to form a rectangle, which is a type of parallelogram.
  • General Triangles: Take any triangle. If you make an exact copy, rotate it 180°, and join it to the original, the combined shape is a parallelogram. If that parallelogram has an area of 40 square cm, one triangle has an area of 20 square cm.

Section 2

Deriving the Triangle Area Formula from Parallelograms

Property

A triangle is exactly half of a parallelogram with the same base and height. To find its area, choose any side of the triangle as its base (length bb), and let hh be the perpendicular distance from the base to its opposing vertex.
The formula is:

Area⁡=12bh\operatorname{Area} = \frac{1}{2} bh

Examples

  • A right triangle has legs (base and height) of 5 m and 8 m. Its area is 12×5×8=20\frac{1}{2} \times 5 \times 8 = 20 square meters.
  • A triangular sign has a base of 40 cm and a height of 25 cm. Its area is 12×40×25=500\frac{1}{2} \times 40 \times 25 = 500 square cm.
  • An obtuse triangle has a base of 10 inches and its corresponding height is 6 inches. The area is 12×10×6=30\frac{1}{2} \times 10 \times 6 = 30 square inches.

Explanation

Any triangle is exactly half of a parallelogram! If you clone a triangle, flip it, and join it to the original, you create a parallelogram. That's why the triangle's area formula is simply one-half of the base times the height.

Section 3

Identifying the Height of a Triangle

Property

For any triangle with area A=bh2A = \frac{bh}{2}, the altitude hh is drawn perpendicular to the base bb.
In acute triangles, the altitude falls inside the triangle.
In obtuse triangles, the altitude to the longest side falls outside the triangle and must be drawn to the extension of the base.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Visualizing the Area: Triangles and Parallelograms

Session 1. Visualizing the Area: Triangles and Parallelograms

Property

Two identical (congruent) triangles can always be joined together to form a parallelogram. Therefore, the area of a single triangle is exactly half the area of a parallelogram that shares the same base and height.

Examples

  • Right Triangles: Two identical right triangles can be joined along their longest side (the diagonal) to form a rectangle, which is a type of parallelogram.
  • General Triangles: Take any triangle. If you make an exact copy, rotate it 180°, and join it to the original, the combined shape is a parallelogram. If that parallelogram has an area of 40 square cm, one triangle has an area of 20 square cm.

Section 2

Deriving the Triangle Area Formula from Parallelograms

Property

A triangle is exactly half of a parallelogram with the same base and height. To find its area, choose any side of the triangle as its base (length bb), and let hh be the perpendicular distance from the base to its opposing vertex.
The formula is:

Area⁡=12bh\operatorname{Area} = \frac{1}{2} bh

Examples

  • A right triangle has legs (base and height) of 5 m and 8 m. Its area is 12×5×8=20\frac{1}{2} \times 5 \times 8 = 20 square meters.
  • A triangular sign has a base of 40 cm and a height of 25 cm. Its area is 12×40×25=500\frac{1}{2} \times 40 \times 25 = 500 square cm.
  • An obtuse triangle has a base of 10 inches and its corresponding height is 6 inches. The area is 12×10×6=30\frac{1}{2} \times 10 \times 6 = 30 square inches.

Explanation

Any triangle is exactly half of a parallelogram! If you clone a triangle, flip it, and join it to the original, you create a parallelogram. That's why the triangle's area formula is simply one-half of the base times the height.

Section 3

Identifying the Height of a Triangle

Property

For any triangle with area A=bh2A = \frac{bh}{2}, the altitude hh is drawn perpendicular to the base bb.
In acute triangles, the altitude falls inside the triangle.
In obtuse triangles, the altitude to the longest side falls outside the triangle and must be drawn to the extension of the base.

Examples