A rectangle is 6 cm long and 3 cm wide. Find its area and perimeter.

1. For the area, multiply the length by the width.

   $$Area=6\times 3=18{cm}^2$$

2. For the perimeter, add all four sides.

   $$Perimeter=6+3+6+3=18cm$$

3. Notice the units: the area is 18 cm², but the perimeter is 18 cm.

A square has area 81 cm². Find its perimeter.

1. A square has equal sides, so find the side length by square-rooting 81.

   $$s=\sqrt{81}=9$$

2. The side length is 9 cm.

3. Multiply the side length by 4 to find the perimeter.

   $$P=4\times 9=36cm$$

A square has perimeter 32 cm. Find its area.

1. A square has 4 equal sides, so divide the perimeter by 4.

   $$s=32÷4=8$$

2. The side length is 8 cm.

3. Square the side length to find the area.

   $$A=8^2=64{cm}^2$$

The length of a rectangle is twice its width. The perimeter is 30 cm. Find the dimensions so it can be drawn on a centimetre grid.

1. Let the width be $w$ cm.

2. The length is twice the width, so the length is $2w$ cm.

3. A rectangle has two widths and two lengths.

   $$w+w+2w+2w=30$$

4. Simplify the left-hand side.

   $$6w=30$$

5. Divide by 6.

   $$w=5$$

6. The width is 5 cm and the length is 10 cm, so draw a 10 cm by 5 cm rectangle.

The length of a rectangle is three times its width. The area is 75 cm². Find the dimensions.

1. Let the width be $w$ cm.

2. The length is $3w$ cm.

3. Use area equals length times width.

   $$3w\times w=75$$

4. This means $3w^2=75$, so divide by 3.

   $$w^2=25$$

5. The width is 5 cm, because 5 times 5 is 25.

6. The length is 15 cm, because it is three times the width.

A triangle has base 8 cm and perpendicular height 5 cm. Find its area.

1. Use the triangle area formula.

   $$A=\frac{1}{2}bh$$

2. Substitute the base and height.

   $$A=\frac{1}{2}\times 8\times 5$$

3. Work it out.

   $$A=20{cm}^2$$

A slanted four-sided shape has two pairs of parallel opposite sides. Its base is 6 cm and its perpendicular height is 4 cm. Name the shape and find its area.

1. A four-sided shape with two pairs of parallel opposite sides is a parallelogram.

2. Use the parallelogram area formula.

   $$A=bh$$

3. Substitute the base and perpendicular height.

   $$A=6\times 4=24{cm}^2$$

The base of a triangle is twice its height. Its area is 25 cm². Find a possible base and height.

1. Let the height be $h$ cm.

2. The base is twice the height, so the base is $2h$ cm.

3. Use the triangle area formula.

   $$A=\frac{1}{2}bh$$

4. Substitute the area and the base.

   $$25=\frac{1}{2}\times 2h\times h$$

5. This simplifies to $25=h^2$, so the height is 5 cm.

6. The base is twice the height, so the base is 10 cm.

Two identical rectangles, each 6 cm by 3 cm, are joined to make a step-shaped six-sided shape. The outside edges are 6 cm, 6 cm, 3 cm, 3 cm, 3 cm and 9 cm. Find the perimeter.

1. Perimeter means add the outside edges only.

2. Write down the outside lengths.

   $$6+6+3+3+3+9$$

3. Add them carefully.

   $$6+6+3+3+3+9=30cm$$

A garden is shaped like a trapezium. The parallel sides are 6 m and 12 m, and the perpendicular height is 5 m. Find the area.

1. Add the two parallel sides.

   $$6+12=18$$

2. Halve the total.

   $$18÷2=9$$

3. Multiply by the perpendicular height.

   $$9\times 5=45m^2$$

A trapezium has area 42 cm². Its parallel sides are 4 cm and 8 cm. Find the perpendicular height.

1. Start with the trapezium area formula.

   $$A=\frac{1}{2}(a+b)h$$

2. Substitute the known values.

   $$42=\frac{1}{2}(4+8)h$$

3. Add the parallel sides.

   $$42=\frac{1}{2}\times 12h$$

4. Half of 12 is 6.

   $$42=6h$$

5. Divide by 6.

   $$h=7$$