Explainer: Surface Areas of Prisms

In this explainer, we will learn how to find the lateral and total surface areas of different types of prisms using more than one formula.

Definition: Prism

A prism is a solid object with two parallel, congruent faces called bases. Its cross-section parallel to the bases is constant throughout its height (also called length).

The bases can be any polygon, while the lateral faces are parallelograms. If the prism is a right prism, then the lateral faces are rectangles.

Definition: Lateral and Total Surface Areas

The Lateral Surface Area of a prism is the total surface area of only the lateral faces of the prism, that is, of the faces that are not the bases.

The Total Surface Area of a prism is the total surface area of the prism, that is, the sum of the areas of its lateral faces plus those of the bases.

Let us look at some examples to see how the lateral area and the total area of a prism are worked out.

Example 1: Finding the Total Surface Area of a Hexagonal Prism

The face of the prism shown is a regular hexagon with sides of length 2 units and an area of 10.39 square units. Work out the surface area of the prism.

Answer

We have here a regular hexagonal prism. It means that its faces are two regular hexagons and six rectangles of length 5 and width 2. We can draw the net of this prism to visualize all the faces easily.

The surface area of the prism is the total area of all its faces.

The area of one lateral rectangle is given by multiplying its length by its width, that is, 5β‹…2=10unit2. The lateral area of the prism is the area of the six lateral rectangles, that is, 𝐴lateral=6β‹…10=60unit2.

The surface area of the prism is given by 𝐴total=2⋅𝐴base+𝐴lateral.

The area of the base is given in the question, which is 10.39unit2. Substituting in, we find 𝐴total=2β‹…10.39+60=20.78+60=80.78unit2.

Example 2: Finding the Total Surface Area of a Prism

Work out the surface area of the prism.

Answer

To solve this type of question, it is useful to draw a net of the prism to visualize all its faces. First, we need to identify the base of the prism. Remember that a prism has two bases, which are two parallel congruent faces. We see that the prism here has a pair of L-shaped bases. All the faces between the bases are rectangles. With the information given on the diagram, we can draw the net of the prism.

On the net, the rectangular faces between the two bases are clearly to be seen. We see that all the rectangles have the same length: it is the height of the prism, here 3 units. They form a large rectangle of length 3 and width π‘Ž+2+4+5+1+𝑏, where π‘Ž and 𝑏 are the two missing sides of the base of the prism. The width of the rectangle formed by all lateral faces is actually the perimeter of the base.

The missing lengths can be easily found given that all angles in the bases are right angles. Therefore, we have π‘Ž+1=4and𝑏+2=5.

We find that π‘Ž=3and𝑏=3.

Now, we can work out the area of the large rectangle formed by all the lateral faces of the prism, which is given by multiplying its length by its width: 𝐴lateral=3β‹…(3+2+4+5+1+3)=3β‹…18=54unit2.

We need to find the area of the two bases. The base can be seen as made of two rectangles, or as the rectangle of length 5 and width 4 from which the rectangle of length 𝑏 and width π‘Ž has been removed. Using the second method, we find 𝐴base=5β‹…4βˆ’3β‹…3=11unit2.

We can of course check that we find the same area with adding the area of two rectangles making the base, for instance, 𝐴base=2β‹…4+3β‹…1or𝐴base=2β‹…3+5β‹…1.

We do find the same area however we compose rectangles to make the base.

To find the total surface area of the prism, we simply need to add two times the area of the base (because there are two bases) to the lateral area. We find that 𝐴total=2⋅𝐴base+𝐴lateral=2β‹…11+54=76unit2.

The surface area of the prism is 76unit2.

In the previous example, we have found an important result that can be used when we work out the surface area of a prism: on the net of a prism, all its lateral faces form a large rectangle whose dimensions are the height of the prism and the perimeter of the prism’s base.

Lateral Surface Area of a Right Prism

The lateral surface area of a right prism is given by 𝐴lateral=heightoftheprismβ‹…perimeterofthebase.

We can of course always find the lateral surface area by summing up the area of each lateral face. The above formula is only useful to have a compacter calculation and save time.

Example 3: Finding the Lateral Area of a Triangular Prism

Find the lateral area of the given prism to the nearest square centimeter.

Answer

We want to find the lateral area of the given prism. The first stage is to identify the bases of the prism. The bases are the two congruent and parallel faces. We see it is the right triangles that are at the front and at the back.

The three lateral faces are all rectangles.

We have the two dimensions only for one of the faces, while there is an unknown side for the other two, as the diagram shows.

The length π‘₯ is the length of the hypotenuse of the base. We can apply the Pythagorean theorem in the right triangle to find it: 102+152=π‘₯2100+225=π‘₯2325=π‘₯2√325=√π‘₯2π‘₯=√325.

We can work out now the lateral surface area: 15Γ—16+10Γ—16+√325Γ—16=240+160+288.4β‰…688cm2.

Note that all the lateral faces share one dimension: the height (or length) of the prism, here 16 cm.

It means that the area of any lateral face is 16 cm times the length of one side of the bases. The lateral surface area is then the height of the prism times the sum of the sides of one base. For this prism, this is just writing 15Γ—16+10Γ—16+√325Γ—16 as 16Γ—ο€»15+10+√325.

Example 4: Finding the Total Surface Area of a Triangular Prism

Find, to the nearest tenth, the surface area of this prism.

Answer

We want to find the surface area of the given prism. The first stage is to identify the bases of the prism. The bases are the two congruent and parallel faces. We see it is the right triangles that are at the front and at the back.

The three lateral faces are all rectangles.

We have the two dimensions only for one of the faces, while there is an unknown side for the other two, as the diagram shows.

The length π‘₯ is the length of the hypotenuse of the base. We can apply the Pythagorean theorem in the right triangle to find it: 62+122=π‘₯236+144=π‘₯2180=π‘₯2√180=√π‘₯2π‘₯=√180.

We can work out now the surface area, which is the sum of the lateral surface area and the area of the two bases: 8Γ—ο€»12+6+√180+2Γ—ο€Ό6Γ—122οˆβ‰…323.3mm2.

Example 5: Finding the Total Surface Area of a Trapezoidal Prism

Work out the surface area of the prism.

Answer

We have here a trapezoidal prism, that is, a prism whose bases (its two parallel and congruent faces) are trapezoids and with four rectangular faces that join the two bases. We can draw the net of the prism to better visualize all its faces.

The lateral surface area of the prism is the area of all its rectangular faces that join the two bases. We clearly see on the net that they form a large rectangle of length the perimeter of the base and width the height of the prism, that is, 6 units. Its area is given by multiplying its length by its width. We find 𝐴lateral=(9+5+6+4)β‹…6=24β‹…6=144unit2.

We can of course work out the area of each rectangular face individually and sum up all together; we find the same result.

Let us work out the area of the base of the prism. The base is a trapezoid. The area of a trapezoid is given by π‘Ž+𝑏2β‹…β„Ž, where π‘Ž and 𝑏 are its two parallel sides and β„Ž its height. Here, π‘Ž=6, 𝑏=9 and β„Ž=4. Plugging in, we find that 𝐴base=6+92β‹…4=30unit2.

The total surface area of the prism is 𝐴total=2⋅𝐴base+𝐴lateral.

Substituting in, we find that 𝐴total=2β‹…30+144=204unit2.

The surface area of the prism is 204unit2.

Key Points

  1. A prism is a solid object with two parallel, congruent faces called bases. Its cross section parallel to the bases is constant throughout its height (also called length).
  2. The bases can be any polygon, while the lateral faces are parallelograms. If the prism is a right prism, then the lateral faces are rectangles.
  3. The lateral surface area of a prism is the total surface area of only the lateral faces of the prism, that is, of the faces that are not the bases.
  4. The total surface area of a prism is the total surface area of the prism, that is, the sum of the areas of its lateral faces plus those of the bases.
  5. A good strategy to visualize all the faces of a prism and not forget any face when working out its surface area is to draw its net.
  6. The lateral surface area of a right prism is given by 𝐴lateral=heightoftheprismβ‹…perimeterofthebase.

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