Perimeter and Area of 2D Shapes

First, a word about units

Units are easy marks, so do not lose them. The pattern is simple:

• Perimeter and circumference are lengths, so they use plain units: cm, m, km.
• Area covers a surface, so it always uses square units: cm², m², km².

Rectangles

A rectangle has a length (l) and a width (w). To go around it, you add up all four sides; to fill it, you multiply length by width.

Worked example #1. A rectangular room is 8 m long and 5 m wide. Find its perimeter and its area.

Perimeter:
  P = 2l + 2w
  P = 2(8) + 2(5)
  P = 16 + 10
  P = 26 m

Area:
  A = l × w
  A = 8 × 5
  A = 40 m²

Triangles

For a triangle, the perimeter is just the three sides added together. The area uses the base (b) and the height (h). The height must be the straight-up distance from the base to the top point, the perpendicular height, not a slanted side.

Worked example #2. A triangle has a base of 12 cm and a perpendicular height of 6 cm. Find its area.

  A = ½ × b × h
  A = ½ × 12 × 6
  A = ½ × 72
  A = 36 cm²

Circles

A circle is measured from its centre. The radius (r) is the distance from the centre to the edge. The diameter (d) goes all the way across through the centre, so the diameter is always twice the radius (d = 2r). The distance around a circle has a special name: the circumference.

Worked example #3. A circular tabletop has a radius of 7 cm. Find its circumference and area. Use π ≈ 3.14.

Circumference:
  C = 2πr
  C = 2 × 3.14 × 7
  C = 43.96 cm

Area:
  A = πr²
  A = 3.14 × 7²
  A = 3.14 × 49
  A = 153.86 cm²

Composite shapes: break them into pieces

A composite shape is just two or more simple shapes joined together. The trick is to split it into shapes you already know, find each area separately, then add them up. (Sometimes you subtract a piece, like a window cut out of a wall, but adding is the most common case.)

Worked example #4. The shape above is like a house outline: a rectangle 10 m wide and 6 m tall, with a triangular roof on top whose height is 4 m. The roof shares the 10 m top edge as its base. Find the total area.

• Rectangle: A = l × w = 10 × 6 = 60 m².
• Triangle: base = 10, height = 4, so A = ½ × 10 × 4 = 20 m².
• Add them: 60 + 20 = 80 m².

Rectangle:  A = 10 × 6      = 60 m²
Triangle:   A = ½ × 10 × 4  = 20 m²
                              ------
Total area:                   80 m²

Tips that make these problems easier

• Read the question twice. Does it want the distance around (perimeter or circumference) or the space inside (area)? Underline the word that tells you.
• Use the formula sheet on purpose. The rectangle, triangle, and circle formulas are all provided. Copy the right one down before you plug in numbers.
• For circles, square the radius before multiplying by π. r² means r × r, not 2 × r.
• Label your units every time. Length stays single (m); area becomes squared (m²). It is an easy mark to keep or to lose.
• For composite shapes, sketch the split. Draw a line breaking the shape into a rectangle, triangle, or circle piece, solve each, then add.

Your turn: practice problems

A calculator is allowed. Use π ≈ 3.14 where needed, write out each formula, and remember your units. Try them all before you peek.

1. A rectangular yard is 15 m long and 9 m wide. Find its perimeter and its area.
2. A triangle has a base of 14 cm and a perpendicular height of 9 cm. Find its area.
3. A circle has a radius of 5 m. Find its circumference and its area.
4. A shape is a 12 m by 8 m rectangle with a semicircle (radius 4 m) attached to one short end. Find the total area. (Hint: a semicircle is half a circle.)