Have you ever wondered what happens to the area of a rectangle when its length is doubled? Does rotating a triangle change its area? Why do we use square units for area but ordinary units for perimeter?

Use the Interactive 2D Shape Explorer below to investigate these questions visually.

This learning tool is suitable for Primary and Secondary Mathematics students, including G1, G2 and G3 learners. You can change the measurements of a figure, rotate it and follow the calculation step by step.

How to use the 2D Shape Explorer

Select a shape from the shape menu.
Enter the required measurements.
Select the unit of measurement: mm, cm, m or km.
Drag the diagram or use the rotation slider to turn the figure.
Show or hide the measurements, perpendicular height, diagonals and grid.
Select Area or Perimeter. For a circle, select Area or Circumference.
Follow the formula, substitution, calculation and final answer.

Remember to use the labelled measurements. Although the proportions of the diagram change, diagrams in Mathematics questions should never be assumed to be drawn exactly to scale.

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Interactive 2D Shape Explorer

Change the measurements, rotate the figure and see the mathematics update instantly.

Learning mode

STEP 1

Set the measurements

Unit

Diagram controls

Rotation 0°

Measurements Perpendicular height Diagonals Grid

STEP 2

Explore the figure

Drag the diagram or use the rotation slider.

STEP 3

Calculate

Round to

Shapes included in the explorer

You can investigate:

Square
Rectangle
Scalene triangle
Isosceles triangle
Equilateral triangle
Kite
Rhombus
Parallelogram
Circle
Trapezium

What is area?

Area is the amount of space enclosed inside a 2D figure.

Area is measured in square units, such as:

mm²
cm²
m²
km²

For example, a rectangle measuring 6 cm by 4 cm has an area of:

Area = length × width

Area = 6 × 4

Area = 24 cm²

The small raised 2 in cm² tells us that the measurement represents an area.

What is perimeter?

Perimeter is the total distance around the outside of a 2D figure.

Perimeter is measured using ordinary linear units, such as mm, cm, m or km.

To find the perimeter of a polygon, add the lengths of all its sides.

For example, the perimeter of a rectangle measuring 6 cm by 4 cm is:

Perimeter = 2(length + width)

Perimeter = 2(6 + 4)

Perimeter = 20 cm

What is circumference?

Circumference is the distance around a circle. It is the circular equivalent of perimeter.

It can be calculated using either the radius or diameter:

Circumference = 2πr

or

Circumference = πd

Since the diameter is twice the radius:

d = 2r

Area and perimeter formula reference

Square

Area: A = s²
Perimeter: P = 4s

Here, s represents the side length.

Rectangle

Area: A = lw
Perimeter: P = 2(l + w)

Here, l represents length and w represents width.

Triangle

Area: A = ½bh
Perimeter: P = a + b + c

The height must be perpendicular to the selected base. A sloping side should not be used as the perpendicular height.

Equilateral triangle

Area: A = (√3 ÷ 4)s²
Perimeter: P = 3s

Every side of an equilateral triangle has the same length.

Kite

Area: A = ½d₁d₂
Perimeter: P = 2(a + b)

The symbols d₁ and d₂ represent the two diagonals.

Rhombus

The area can be calculated using its base and perpendicular height:

Area: A = bh

It can also be calculated using its diagonals:

Area: A = ½d₁d₂

Perimeter: P = 4s

Parallelogram

Area: A = bh
Perimeter: P = 2(a + b)

The perpendicular height is the shortest distance between the two parallel sides. It is not usually the same as the sloping side.

Circle

Area: A = πr²
Circumference: C = 2πr or C = πd

Here, r represents the radius and d represents the diameter.

Trapezium

Area: A = ½(a + b)h

The measurements a and b are the lengths of the two parallel sides. The measurement h is the perpendicular distance between them.

To find its perimeter, add the lengths of all four sides.

Important measurements to recognise

Base

The base is the side selected for use in an area formula. A triangle or parallelogram can be turned in different directions, so more than one side may be used as a base.

Perpendicular height

The perpendicular height meets the selected base at an angle of 90°. It may be inside or outside a figure.

Radius

The radius is the distance from the centre of a circle to its circumference.

Diameter

The diameter passes through the centre of a circle and joins two points on its circumference.

The diameter is twice the radius:

d = 2r

Diagonal

A diagonal joins two non-adjacent vertices of a polygon. It is not normally part of the perimeter because it lies inside the figure.

Common mistakes to avoid

Using a sloping side instead of the perpendicular height.
Confusing the radius with the diameter.
Forgetting the ½ in the triangle or trapezium area formula.
Including a diagonal when calculating perimeter.
Writing cm instead of cm² for an area.
Writing cm² instead of cm for a perimeter.
Assuming that a diagram is drawn exactly to scale.
Rounding values too early during a calculation.

Keep unrounded values in your calculator and round only the final answer unless the question tells you otherwise.

Try these investigations

Use the explorer to answer the following questions:

What happens to the area of a square when its side length is doubled?
If the length of a rectangle is doubled but its width remains unchanged, what happens to its area?
Can two different rectangles have the same area but different perimeters?
What happens to the area of a triangle when its height is doubled?
Does rotating a figure change its area or perimeter?
What happens to the area of a circle when its radius is doubled?
Can you change the slant of a parallelogram without changing its base and perpendicular height? What happens to its area?
Can two triangles with different appearances have the same base, perpendicular height and area?

Make a prediction before changing the measurements. Then use the explorer to test it.

Questions students often ask

Does rotating a shape change its area or perimeter?

No. Rotation changes only the orientation of a figure. Its side lengths, area and perimeter remain unchanged.

Why is area measured in square units?

Area measures how many unit squares can fit inside a figure. This is why units such as cm² and m² are used.

Is a square also a rectangle?

Yes. A square satisfies the definition of a rectangle because it has four right angles and two pairs of equal opposite sides. A square has the additional property that all four sides are equal.

Is a square also a rhombus?

Yes. A rhombus has four equal sides, so a square is a special type of rhombus with four right angles.

Why does a circle have a circumference instead of a perimeter?

Circumference is the special term used for the distance around a circle. It has the same general meaning as perimeter.

Why might my answer be slightly different?

Answers involving π, square roots or calculated side lengths may need to be rounded. Keep the full calculator value during your working and round only the final answer.

For teachers and parents

This explorer can be used to:

Introduce a new area or perimeter formula.
Demonstrate the difference between vertical height and perpendicular height.
Compare related figures such as squares, rectangles and rhombuses.
Investigate how changing one measurement affects the result.
Check answers after students complete their own working.
Encourage students to make and test mathematical predictions.

Students should still write the formula, substitute the measurements and show their working. The explorer is designed to support mathematical understanding—not to make the calculator do all the thinking.

Ready to explore three-dimensional figures? Continue to the Interactive 3D Solid Explorer to investigate cross-sectional area, surface area and volume.

You can also return to the Geometry Explorer to choose another learning tool.

Interactive 2D Shape Explorer: Area and Perimeter