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Composite shapes

Measurement · level 4, lesson 4 of 7

A question to start

A room is shaped like a capital L. It is what you get from a 10 m by 8 m rectangle when a 4 m by 3 m corner is taken out. You walk all the way round the walls of the L-shaped room, then all the way round the walls of the full 10 m by 8 m rectangle. Which walk is longer?

They are the same: 36 m each. But the L-shaped room has less floor: 68 m² instead of 80 m². The perimeter did not notice the missing corner, but the area did. This lesson is about shapes like this, made of simpler shapes, and how to find their area in more than one way.

The idea

Shapes made of shapes

A composite shape is a shape made by joining simpler shapes together, such as rectangles, triangles and trapezoids. Most real floors, gardens, walls and fields are composite shapes. There is no single formula for them, and you do not need one. You need two moves:

  • Add: cut the shape into simple pieces, find each piece's area, and add.
  • Subtract: start with a simple shape that is too big, and take away the area of the part that is missing.

Both moves rest on one basic fact about area: area adds up. If a shape is split into pieces that do not overlap, the area of the whole is the sum of the areas of the pieces. That is just counting squares in groups: every square of the whole is in exactly one piece.

Find the missing lengths first

Drawings of composite shapes often leave out some lengths, because they can be worked out. In a shape with only right angles, the trick is: the lengths along the top add up to the length along the bottom, and the lengths up the left add up to the lengths up the right. Whichever way you walk from one side to the other, you cover the same distance.

Three ways to find the L

10 m6 m4 m 853 6 × 84 × 5 10 m8 4 × 310 × 8 minus the corneradd two pieces

First the missing lengths. Along the top: \(6 + 4 = 10\) ✓ (matches the bottom). Up the sides: the right side is 5 m, the step up is 3 m, and \(5 + 3 = 8\) ✓ (matches the left side).

  • Add, cutting upright. A 6 m by 8 m piece and a 4 m by 5 m piece: \(48 + 20 = 68\) m².
  • Add, cutting across. Cut along the line 5 m up instead. The bottom piece is 10 m by 5 m, and the top piece is 6 m by 3 m: \(50 + 18 = 68\) m².
  • Subtract. The full rectangle is \(10 \times 8 = 80\) m², and the missing corner is \(4 \times 3 = 12\) m²: \(80 - 12 = 68\) m².

Three methods, one answer. When you can, do it two ways: if they agree, you can be confident.

Why the perimeter stayed the same

The full rectangle's perimeter is \(2 \times (10 + 8) = 36\) m. Walking round the L: \(10 + 5 + 4 + 3 + 6 + 8 = 36\) m. Taking out the corner removed two walls (4 m and 3 m) from the outside, but added two new walls of exactly the same lengths on the inside of the step. So the perimeter did not change, while the area lost 12 m². Area and perimeter measure different things, and they do not have to move together.

Holes and borders

Subtracting is the natural move when there is a hole: a lawn with a pond, a wall with a window, a picture inside a frame. The area you want is the outside shape minus the hole.

A border (a path round a pool, a frame round a picture) is the same idea: big rectangle minus small rectangle.

pool 10 m × 5 m outside 12 m 7 path 1 m wide

Watch the outside lengths: a path 1 m wide adds 1 m on both ends, so the outside is \(10 + 2 = 12\) m long and \(5 + 2 = 7\) m wide.

Mixing shapes

Composite shapes can include triangles and trapezoids as well as rectangles. The end wall of a house, for example, is often a rectangle with a triangle on top (the gable). Cut along the line where the roof starts, find both areas, and add.

Worked examples

1. The L-shaped room (above): 68 m², by three methods.

2. A gable wall. The end wall of a shed is a rectangle 8 m wide and 3 m tall, with a triangle on top whose peak is 2 m above the top of the rectangle. There is a door 2 m tall and 0.9 m wide. How much wall needs paint?

  • Rectangle: \(8 \times 3 = 24\) m².
  • Triangle: base 8 m, height 2 m: \(\tfrac12 \times 8 \times 2 = 8\) m².
  • Door (a hole, so subtract): \(2 \times 0.9 = 1.8\) m².
  • Paint: \(24 + 8 - 1.8 = 30.2\) m².

3. The pool path. Pool 10 m by 5 m, path 1 m wide all round.

Outside: \(12 \times 7 = 84\) m². Pool: \(10 \times 5 = 50\) m². Path: \(84 - 50 = 34\) m².

Check by adding: the path is two long strips 12 m by 1 m along the top and bottom (\(2 \times 12 = 24\) m²) and two short strips between them, 5 m by 1 m (\(2 \times 5 = 10\) m²). \(24 + 10 = 34\) m² ✓. (Careful not to count the corners twice.)

4. A T-shape. A T-shaped patio has a top bar 9 m by 2 m and a stem 3 m wide that goes 5 m down from the bar.

Bar: \(9 \times 2 = 18\) m². Stem: \(3 \times 5 = 15\) m². Total: \(18 + 15 = 33\) m².

5. A field with a trapezoid. A field is a rectangle 40 m by 30 m with a trapezoid-shaped strip added on one 30 m side. The strip's parallel sides are 30 m (along the rectangle) and 20 m, and it is 10 m wide.

Rectangle: \(40 \times 30 = 1200\) m². Trapezoid: \(\tfrac12 (30 + 20) \times 10 = 250\) m². Field: \(1450\) m².

6. Turf cost. Turf costs 8 dollars per square metre. How much to turf the field in example 5?

\(1450 \times 8 = 11\,600\) dollars.

Try it

  1. An L-shape is a 12 cm by 9 cm rectangle with a 5 cm by 4 cm corner removed. Find its area two ways, and its perimeter.
  2. A picture is 30 cm by 20 cm. Its frame is 5 cm wide all round. What is the area of the frame alone?
  3. A wall is 5 m long and 2.4 m high, with a window 1.5 m by 1.2 m. Find the area to be wallpapered.
  4. A house end wall is a rectangle 6 m wide and 2.5 m tall with a triangular gable 6 m wide and 1.5 m tall on top. Find its area.
  5. A U-shape is made from a 10 m by 6 m rectangle by cutting a 4 m by 4 m notch out of the middle of the top edge. Find its area and its perimeter.
  6. A square garden has sides of 8 m. A square pond with sides of 3 m is in the middle. Turf costs 6 dollars per square metre. How much for the turf?
  7. A shape is a rectangle 6 cm by 4 cm with a right triangle stuck on its right end. The triangle shares the 4 cm side and sticks out 3 cm. Find the area.
Answers
  1. Subtract: \(12 \times 9 - 5 \times 4 = 108 - 20 = 88\) cm². Add (one way): cut upright where the corner starts. One piece is \((12 - 5) \times 9 = 7 \times 9 = 63\) cm², the other \(5 \times (9 - 4) = 5 \times 5 = 25\) cm²; \(63 + 25 = 88\) cm² ✓. Perimeter: the same as the full rectangle, \(2 \times (12 + 9) = 42\) cm.
  2. Outside: \((30 + 10) \times (20 + 10) = 40 \times 30 = 1200\) cm². Picture: \(30 \times 20 = 600\) cm². Frame: \(1200 - 600 = 600\) cm². (The frame is exactly as big as the picture!)
  3. \(5 \times 2.4 - 1.5 \times 1.2 = 12 - 1.8 = 10.2\) m².
  4. \(6 \times 2.5 + \tfrac12 \times 6 \times 1.5 = 15 + 4.5 = 19.5\) m².
  5. Area: \(10 \times 6 - 4 \times 4 = 60 - 16 = 44\) m². Perimeter: the outside of the rectangle is \(2 \times (10 + 6) = 32\) m. The notch removes 4 m of the top edge but adds three inside edges, \(4 + 4 + 4 = 12\) m. So \(32 - 4 + 12 = 40\) m. (This time the perimeter grew: a notch in the middle of a side adds two walls that a corner cut does not.)
  6. Turf area: \(8 \times 8 - 3 \times 3 = 64 - 9 = 55\) m². Cost: \(55 \times 6 = 330\) dollars.
  7. Rectangle \(6 \times 4 = 24\) cm², triangle \(\tfrac12 \times 4 \times 3 = 6\) cm². Total \(30\) cm².

Common mistake

Counting a piece twice, or using a wrong missing length. When you cut a shape, make sure the pieces do not overlap, and work out every missing length from the ones given before you multiply. In the pool path, the corners belong to the long strips or the short strips, not both.

Thinking a smaller area means a smaller perimeter. Cutting a corner out of a rectangle keeps the perimeter the same, and cutting a notch out of a side makes it longer, even though the area goes down both times.

Where it leads