Triangular Prism Architecture--Captions

The Flat Iron Building in New York City is one of its oldest skyscrapers.

Even though it's now dwarfed by taller buildings, when it was built in 1902, this 22-floor building was a one-of-a-kind.

Its triangular design gave it a distinctive look and over time it would become a New York landmark.

Unlike most skyscrapers that are rectangular in shape, the Flat Iron Building is an example of a triangular prism.

As you can see from this animation a triangular prism has two triangular faces, and three quadrilateral sides.

A right triangular prism has rectangular sides.

This is the type of prism that defines the Flat Iron Building.

The rectangular sides ensure that the prism has square corners.

A skyscraper, regardless of its shape, needs to stand straight and tall.

A triangular prism can stand on its triangular side or on its rectangular side.

The Flat Iron Building is an example of a triangular prism that rests on one of its triangular faces.

To get a better understanding of the geometry of triangular prisms, look at a two-dimensional net.

This net shows the three rectangular sides here and the triangular faces here.

When you fold up these rectangular sides along these lines, and the triangular sides along these two lines, you end up with the triangular prism shown here.

When you fold up these rectangular sides along these lines, and the triangular sides along these two lines, you end up with the triangular prism shown here.

The surface area of the triangular prism is the sum of the triangular faces and the rectangular faces.

These can be easily calculated once you know the dimensions of the prism.

As for the volume of a triangular prism, it is found by multiplying the area of the base times the height.

Let's apply these concepts to the Flat Iron Building.

When we look at the Flat Iron Building from above we see its triangular shape, but do you notice something?

This isn't just any triangle.

It's a right triangle.

Notice the square corner here?

This changes our analysis of the building because we can now use right triangle properties.

If we overlay a grid over this right triangle, we can see some additional properties.

The base of the triangle is five units, while the height of the right triangle is ten units.

Let's turn this into a right triangle whose legs are in a ratio of 2 to 1, as shown here.

Using the Pythagorean Theorem, we can find the length of the hypotenuse.

We get a value of square root of five.

The area of this triangle is one half the base times the height.

In the case of right triangles, the height is one of the legs, so the area of this triangle is one.

Finally, using trig ratios we can find the angle measures for the acute angles of this right triangle.

These angle measures are rounded to the nearest whole number.

To find the volume of this type of triangular prism we draw a prism with right triangular legs of x and 2x and a prism height of h.

To find the volume of this type of triangular prism we draw a prism with right triangular legs of x and 2x and a prism height of h.

The volume is the area of the triangular face times the height, which resolves to x-squared times h.

The volume is the area of the triangular face times the height, which resolves to x-squared times h.

For the Flat Iron Building, the value of x is 87 ft and the height is 307 ft.

Plugging these values into the volume formula, we get an astounding value of over two million cubic feet.

With our understanding of the geometry of the Flat Iron Building, let's now look at the architecture.

With our understanding of the geometry of the Flat Iron Building, let's now look at the architecture.

At the time the Flat Iron Building was constructed, it was built on a triangular tract of land.

This more than anything else accounts for the triangular shape of the building.

The reason for the triangular tract is seen here.

The building was located at the intersection of 23rd St., Fifth Avenue, and Broadway.

The architect, Daniel Burnham, was already well known and highly regarded for his work in Chicago.

With the Flat Iron Building, he worked within the confines of the triangular space, and he made the most of it.

Because triangular prism architecture is relatively rare, Burnham looked to give it a classical look.

The rounded corner of the building has a tall column-like appearance, similar to the columns of Greek temples.

Unlike Greek columns, which serve as structural support the column-like structure of the Flat Iron building is decorative.

What holds the building up is its steel framework.

In choosing a right triangular design, he was able to have this side of the building mimic the standard rectangular office space.

Here is the floorplan for the Flat Iron Building.

Along the rectangular side of the building you see office spaces that are also rectangular in shape.

But along the diagonal the offices have trapezoidal shapes.

The office space is progressively squeezed as you go from the wider base to the top of the triangle.

But what these offices give up in space they make up for in a wider view of Manhattan.

Finally, going back to the geometry of the building, what Burnham essentially did was take a rectangular prism and split it along the diagonal to get the basic shape of the Flat Iron Building.

Finally, going back to the geometry of the building, what Burnham essentially did was take a rectangular prism and split it along the diagonal to get the basic shape of the Flat Iron Building.

The volume of the Flat Iron Building is therefore half of its rectangular prism counterpart.

What it loses in volume in makes up with a stylish design that has stood the test of time.