Find the area of a triangle

What you will learn

• Find the area of a triangle

Models used: number sentences, stories, diagrams.

Before you start

• Identify a triangle base and perpendicular height

Two triangles explain the half

We want to find how much surface this blue triangle covers. Its base is 6 units and its height is 4 units.

1. Base = 6 units. Perpendicular height = 4 units. The little square marks a right angle.

   We want to find how much surface this blue triangle covers. Its base is 6 units and its height is 4 units.

   Area measures the inside surface. The right-angle mark shows that the height is perpendicular to the base.

2. Base = 6 units. Perpendicular height = 4 units. The little square marks a right angle.

   Make an identical copy of the blue triangle. Turn the copy to fill the empty half.

   The orange triangle has exactly the same area as the blue one. Together they fill a rectangle, with no gap or overlap.

3. Each little square is one square unit. There are 6 squares across and 4 rows.

   The rectangle is 6 units across and 4 units high. Its area is 24 square units.

   6 × 4 = 24 square units

   There are six squares in each of four equal rows. That is why we multiply the base and height.

   Is 24 the area of one triangle or of both triangles together?

   Both triangles together. The rectangle contains two equal copies of our triangle.

4. Base = 6 units. Perpendicular height = 4 units. The little square marks a right angle.

   Share that area equally between the two triangles. Each triangle covers 12 square units.

   triangle area = base × height ÷ 2 = 12

   We divide by two because our rectangle contains two matching triangles. The formula records what the picture showed us.

Worked example: A new example, together

Find the triangle area with base 8 cm and perpendicular height 7 cm.

1. Base = 8 units. Perpendicular height = 7 units. The little square marks a right angle.

   Find the right-angle relationship before using a dimension. For a composite surface, keep the pieces separate.

   Two matching triangles make a rectangle or parallelogram with the same base and height.

2. Base = 8 units. Perpendicular height = 7 units. The little square marks a right angle.

   Triangle area = half of base × perpendicular height.

   Find the triangle area with base 8 cm and perpendicular height 7 cm.

   28 cm²

3. Base = 8 units. Perpendicular height = 7 units. The little square marks a right angle.

   8 × 7 ÷ 2 = 28.

4. Base = 8 units. Perpendicular height = 7 units. The little square marks a right angle.

   Area = 28 cm².

5. Base = 8 units. Perpendicular height = 7 units. The little square marks a right angle.

   Check the result against the question we started with.

   Double the triangle area: 28 × 2 = 56.

   We found 28 cm². Two matching triangles make a rectangle or parallelogram with the same base and height.

Common mistakes to discuss

• Forgetting to halve. Two matching triangles fill the rectangle. With base 8 cm and height 7 cm the area is 56 ÷ 2 = 28 cm².

Check your understanding

• Find the area of a triangle
• connect triangle area to a matching base and perpendicular height
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.