Comparing Measurement of Shapes

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education

There are a wide variety of geometric shapes. There are equations to determine their perimeters and areas. In this lesson, we will explore different shapes and compare their perimeters and areas.


When you go to the doctor, it is common for them to measure your height and weight. Let's pretend you are 5 feet tall and weigh 95 pounds. Your friend is 4 feet 6 inches tall and when he gets on the scale you see he weighs the same amount! People come in all shapes and sizes, and so do geometric shapes. Let's look at a few geometric shapes and show that they can be different shapes, but with the same areas or perimeters.

Rectangles and Triangles

A rectangle is a four-sided shape with all of the angles equaling 90°. Their adjacent sides are not equal in length.

A rectangle

Let's compare a rectangle to a right triangle. A right triangle is a three-sided shape with one 90° angle. If you add all of the angle values you will get 180°.

A right triangle

The perimeter of a shape is the distance around the outside of the shape. The perimeter of a rectangle is the sum of the length of the four sides. The perimeter of a right triangle is the sum of the length of the three sides. Let's look at Diagram 1, which shows the lengths of each side of a rectangle and triangle.

Diagram 1

Let's determine both of their perimeters by adding all of the lengths of their sides. We will use the metric unit centimeters to represent length. (The diameter of a penny is roughly a centimeter.)

Perimeter of Rectangle:

7 cm + 5 cm + 7 cm + 5 cm = 24 cm

Perimeter of Right Triangle:

12.2 cm + 10 cm + 7 cm = 29.2 cm

We can see the perimeters are not equal. Now let's determine their areas. Area is the amount of two-dimensional space a shape occupies. The area of a rectangle is length multiplied by width. The area of a triangle is one-half of the base multiplied by the height. Both areas will be in the metric unit square centimeters (cm2).

Area of Rectangle:

7 cm × 5 cm = 35 cm2

Area of Right Triangle:

(1/2) × (10 cm) × (7 cm) = 35 cm2

The area of both of these shapes is the same! This proves that different shapes with different perimeters can have the same area. Let's do another example.

Circle and Sphere

A circle is a two-dimensional shape with no sides, and a continuous curve with one radius all the way around the circle such as a Frisbee.


A sphere is a three-dimensional shape with no sides, and a continuous curve with one radius all the way around the sphere (like a basketball).


The perimeter around either a circle and a sphere is called the circumference (C) and they both have the same equation! C = 2 × π × r. We will estimate π to be 3.14. Let's compare a circle and a sphere. They both have a 2 cm radius. Diagram 2 shows these shapes.

Diagram 2

If their radii are the same, and the equations for their circumferences are the same, their circumferences are the same. This means if a bug were to walk around the edge of a circle and around the outside of a sphere, they would walk the same distance. Let's prove it.

Circumference of Circle:

2 × 3.14 × 2 cm = 12.56 cm

Circumference of Sphere:

2 × 3.14 × 2 cm = 12.56 cm

We were correct with our prediction! Both circumferences are the same.

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