Mensuration
148. Boundaries and Spaces · Understanding Perimeter and Area
The perimeter is the sum of all side lengths, and the area is the space inside.
Perimeter is the total length of a closed figure's boundary (measured in linear units like cm or m). Area is the amount of surface inside the boundary (measured in square units like cm² or m²). Two different things — don't mix them up.
What this lesson covers
Try to break it
Drag any vertex of the pentagon. The boundary length (perimeter) and the shaded inside (area) both change as you stretch and squash. Perimeter is always the sum of the five side lengths; area is always the space those sides enclose. Try to make one without changing the other — impossible.
How you build it
Construct a closed pentagon.
- Place point A, the first vertex of the pentagon.
- Place point B, the second vertex of the pentagon.
- Place point C, the third vertex of the pentagon.
- Place point D, the fourth vertex of the pentagon.
- Place point E, the fifth vertex of the pentagon.
- Draw segment AB, the first side of the pentagon.
- Draw segment BC, the second side of the pentagon.
- Draw segment CD, the third side of the pentagon.
- Draw segment DE, the fourth side of the pentagon.
- Draw segment EA to close the pentagon.
The proof, step by step
Prove that the perimeter is the total boundary length and the area is the space inside.
- The perimeter is the total length of the boundary.
- The area is the region bounded by the perimeter.
Worked example
A triangle has side lengths of 3 cm, 4 cm, and 5 cm. What is its perimeter?
Perimeter is the sum of the side lengths. 3 + 4 + 5 = 12 cm.
- 10 cm
- 12 cm — correct
- 15 cm
- 20 cm