Surface Area

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Surface Area: meaning, method and solved example (6th Grade) – Maths PDF

In short: Surface area measures the whole outside skin of a solid by adding up what each face covers, always in square units.

6th GradeGeometry6.G.A.4

What is Surface Area?

Wrapping a gift covers the outside of a box with paper, and the amount of paper is its surface area. Every face counts, including the bottom and the back.

For a box with sides \(\ell\), \(w\) and \(h\) the total is \(SA = 2\ell w + 2\ell h + 2wh\). For other solids with flat faces, add every face; that equals the area of the unfolded pattern.

How to spot it

  • A problem asks for the paper, paint, cloth or metal needed to cover the outside of a solid.
  • A net is given and the question asks for its total area.
  • A solid has identical opposite faces, so the areas come in pairs.
  • The answer must use square units, even though the object is three-dimensional.

The method

  1. List all faces and group them in matching pairs.
  2. Find the area of each different face, such as \(\ell w\), \(\ell h\) and \(wh\).
  3. Double each pair or add every face one at a time.
  4. Add everything and write the result in square units.
12 x 4 = 48top 12 x 9 = 10812 x 4 = 489 x 4= 369 x 4= 36+ bottom 1082(108 + 48 + 36)= 384 square units
Pairs of faces: 108, 48 and 36 each appear twice, giving 384.

Surface Area: solved example

Problem

A gift box at a shop in Savannah is 12 inches by 9 inches by 4 inches. How much wrapping paper covers it exactly, with no overlap?

Solution

Faces: \(12 \cdot 9 = 108\), \(12 \cdot 4 = 48\) and \(9 \cdot 4 = 36\), each twice.

\(2(108 + 48 + 36) = 2 \cdot 192 = 384\). The box needs 384 square inches of paper.

Try it

Question 1

How much glass covers all sides of a cube-shaped display case whose edges are 6 cm?

Show the answer

Six faces of \(6 \cdot 6 = 36\): \(6 \cdot 36 = 216\) square centimeters.

Question 2

A box is 10 ft by 3 ft by 2 ft. What area would you need to paint, bottom included?

Show the answer

\(2(30 + 20 + 6) = 2 \cdot 56 = 112\) square feet.

Surface Area vs. Volume

Surface area covers the outside in square units; volume fills the inside in cubic units. Doubling every edge of a cube quadruples its outside area but multiplies the space inside by eight.

Volume

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