Find areas and perimeters of composite figures with rectangles, squares and triangles

What you will learn

• Find areas and perimeters of composite figures with rectangles, squares and triangles

Models used: number sentences, stories, tables.

Before you start

• Find the area of a triangle

A rectangle with a triangular roof

Split the house shape at the dotted line. We have a rectangle and a triangle.

1. The dashed line separates the rectangular wall from the triangular roof; it is inside the house.

   Split the house shape at the dotted line. We have a rectangle and a triangle.

   Base and perpendicular height define the matching rectangle or parallelogram. The triangle’s area is half their product.

2. The dashed line separates the rectangular wall from the triangular roof; it is inside the house.

   The rectangle covers 6 × 4 = 24 square centimetres. The roof covers half of 6 × 2, which is 6.

3. The dashed line separates the rectangular wall from the triangular roof; it is inside the house.

   The two parts meet along an edge, without overlapping. Add their areas.

   24 + 6 = 30 cm²

Worked example: A new example, together

A house outline has a bottom edge 12 cm, two vertical walls 6 cm each and two roof slopes 10 cm each. Find its outer perimeter.

1. Only the five boundary edges are counted. Read each value with the heading above its column.

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

   Perimeter adds the outside edges once and excludes internal joins.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

2. Only the five boundary edges are counted. Read each value with the heading above its column.

   Trace the five outer edges; the line under the roof is an internal join.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

   A house outline has a bottom edge 12 cm, two vertical walls 6 cm each and two roof slopes 10 cm each. Find its outer perimeter.

   44 cm

3. Only the five boundary edges are counted. Read each value with the heading above its column.

   Trace the five outer edges; the line under the roof is an internal join.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

4. Only the five boundary edges are counted. Read each value with the heading above its column.

   Add 12 + 6 + 6 + 10 + 10.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

5. Only the five boundary edges are counted. Read each value with the heading above its column.

   Perimeter = 44 cm.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

6. Only the five boundary edges are counted. Read each value with the heading above its column.

   Check the result against the question we started with.

   Removing the two walls and two slopes from 44 leaves the bottom 12.

   We found 44 cm. Perimeter adds the outside edges once and excludes internal joins.

   Only the five boundary edges are counted

   Bottom	Each of 2 walls	Each of 2 roof slopes
   12	6	10

Common mistakes to discuss

• Using the outside measurements only. Split the figure into parts, find each part, then combine. A missing side is found from the sides facing it.

Check your understanding

• Find areas and perimeters of composite figures with rectangles, squares and triangles
• 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.