Drawing Rectangles and Triangles on a Square Grid

1. Reading and drawing on a square grid

A square grid is made of equal squares. We agree that one small square has side 1 unit and so area 1 square unit. The unit may be 1 cm, 1 m or any length stated on the plan. On a dot grid the dots sit where the lines would cross, and we join the dots to make shapes.

Each corner of a shape is placed on a grid point (a point where two lines cross). To draw a shape with given dimensions, we count squares along the lines rather than measuring with a ruler. A line 5 units long runs across exactly 5 squares.

1. 1Use a sharp pencil and a ruler. Choose a starting corner on a grid point.
2. 2Count the required number of squares along one side, e.g. 5 to the right, and mark the corner.
3. 3From there count the next side, e.g. 3 squares up, then complete the shape with the ruler.
4. 4Label the sides with their measurements and count the squares inside to check the area.

For example, to draw a rectangle 5 units long and 3 units wide, we count 5 squares across and 3 squares up. It contains $5\times 3=15$ unit squares, so its area is 15 square units. Its perimeter is $2(5+3)=16$ units.

2. Changing a rectangle: double the area, scale the sides

The area of a rectangle is length × width. This means that if you multiply one side by a number, the area is multiplied by the same number. A 4 by 2 rectangle has area 8. If we double the width to 4, the area becomes $4\times 4=16$, which is twice as much.

There are many ways to get an area of 16 square units: 4 by 4, 8 by 2, 2 by 8, or 16 by 1. All of these are correct answers to 'draw a rectangle whose area is twice as large'. The simplest is to keep one side and double the other.

Sometimes we are told to scale both sides. Suppose the original rectangle is 4 long and 2 wide. A rectangle which is twice as wide and one and a half times as long has width $2\times 2=4$ and length $4\times 1\frac{1}{2}=6$. Its area is $6\times 4=24$. Compare: $24÷8=3$, which is exactly $2\times 1\frac{1}{2}$.

▶ Count the squares inside each rectangle. Notice how the area changes when each side changes.

3. Drawing triangles and finding their area

A triangle on a grid is drawn by marking three grid points and joining them. The base is one side (usually along a grid line) and the height is the straight, perpendicular distance from the opposite corner to the base. Count squares for both.

Look at a triangle with a base of 6 squares and a height of 4 squares. If you draw the rectangle 6 by 4 around it, the triangle covers exactly half of the rectangle. The rectangle has $6\times 4=24$ squares, so the triangle has 12 squares. This gives the rule you will prove in the next lesson:

$$\text{Area of triangle}=\frac{1}{2}\times \text{base}\times \text{height}$$

▶ Same base and same height always means the same area, wherever the top corner is.

A fun problem: how many different triangles have an area of 12 square units on this grid? Keep the base 6 and the height 4. The top corner can be on any grid point of the top line, so there are many triangles, and they all have the same area. Each is a different shape but covers the same amount of space. You can also change the base and height, e.g. a base of 8 and a height of 3 gives $\frac{1}{2}\times 8\times 3=12$ too.

4. Drawing a triangle with a given area

Suppose you are asked to draw, on a dot grid, a triangle whose area is 3 square units. Work backwards: $\frac{1}{2}\times b\times h=3$ means $b\times h=6$. So choose any pair of whole numbers whose product is 6:

To draw the triangle with base 3 and height 2, join two dots 3 units apart on a grid line for the base. Then pick a dot 2 units above that line (anywhere along the parallel line) as the third corner. Join the dots with a ruler.

Notice that a triangle can sit in a rectangle's corner (a right-angled triangle) or have its top corner anywhere along the line. In both cases the height is measured straight up, not along the sloping side.

Worked examples

Try each one yourself first, then open the solution and compare every step.