6. Geometry & Measurement: Solids

Lesson

Surface area is the sum of the areas of all the faces of a three dimensional ($3D$3`D`) shape, and is measured in square units. Recall that square units may be written: square inches ($in^2$`i``n`2), square centimeters ($cm^2$`c``m`2), square feet ($ft^2$`f``t`2) or square meters as ($m^2$`m`2).

When we are calculating surface area we can imagine that we are literally covering the entire outside of the figure with squares. Here, we will look at how to calculate the surface area of a specific category of $3D$3`D` shapes called square based pyramids

A pyramid is a $3D$3`D` shape that has a polygon as a base and sloping sides that meet at a point called the apex. If the apex is directly above (perpendicular to) the center of the base, the pyramid is called a right pyramid. These following are all right pyramids. Pyramids are named by the shape of their base.

As we can see from these diagrams, the triangular shaped sides slope towards the apex. This introduces a new term we use in calculations with pyramids called the slant height** **(or slope height).

When using the following applet, we will begin by adjusting the length and width so that they are the same. This will create a square based prism.

Notice that the slant height corresponds to the height of the triangle that makes up each face. Sometimes we need to calculate the slant height using the Pythagorean theorem.

The surface area of a solid is calculated by adding the area of all faces. For square pyramids, we have the base and a number of triangular faces.

This results in:

Surface Area of Square Pyramid

$\text{Surface area of square pyramid }=\text{area of square}+\text{area of triangles }$Surface area of square pyramid =area of square+area of triangles

or expressed as a formula, where $l$`l` is the slant height, $p$`p` is the perimeter of the base, and $B$`B` is the area of the base.

$S.A.=\frac{1}{2}lp+B$`S`.`A`.=12`l``p`+`B`

Find the surface area of the square pyramid shown. Include all faces in your calculations.

Consider the following square pyramid:

Find the length of the slant height.

Round your answer to two decimal places.

Using the value for slant height found in part (a), find the surface area of the square pyramid. Make sure to include all faces in your calculations.

Round your answer to one decimal place.

Volume is a measure of the space inside a $3D$3`D` solid shape. It is measured using units such as cubic inches ($in^3$`i``n`3), cubic centimeters ($cm^3$`c``m`3), cubic feet ($ft^3$`f``t`3) and cubic meters ($m^3$`m`3).

Shown below is a cubic centimeter. This is a cube that is $1$1 cm long, $1$1 cm wide and $1$1 cm high. When we are calculating volume we can imagine filling the inside of the object with cubes like this.

The volume of a square pyramid with a base area, $A$`A`, and height, $h$`h`, is given by the formula:

Volume of a square pyramid

Area of square base | $=$= | $side\times side$side×side |

A | $=$= | $s^2$s2 |

Volume of pyramid | $=$= | $\frac{1}{3}\times\text{area of square base }\times\text{height }$13×area of square base ×height |

$V$V |
$=$= | $\frac{1}{3}Ah$13Ah |

Careful!

The height used in the formula for the volume of a square pyramid is the *vertical *height, which is *perpendicular *to the base. If we are given the *slant *height, we will need to use the Pythagorean theorem to find the find the *vertical *height before using the formula.

Find the volume of the square pyramid shown.

When reading practical problems involving square based pyramids, we need to be able to determine whether the question requires us to find the surface area of the figure or the volume of the figure.

To determine which formula to use, we will think to ourselves:

- Is the question asking us to calculate a value that requires covering the outside of the figure or is it referring to square units? If so, we will want to use surface area.
- Is the question asking us to calculate a value that has to do with filling up the inside of the figure or is it referring to cubic units? If so, we will want to use volume.

The Great Pyramid of Giza is a regular square pyramid. It is currently $138.8$138.8 m tall and its base has a side length of $230.3$230.3 m.

Find the slant height $y$

`y`of the Great Pyramid.Round your answer to two decimal places.

Suppose that each face of the Great Pyramid was covered with solar panels.

Using your answer from part (a), find how many square meters of solar panel would be needed.

Round your answer to two decimal places.

Suppose that each square meter of solar panel produces a power of $500$500 W/m

^{2}.Using your answer from part (b), find how much power could be produced by the Great Pyramid.

Round your answer to the nearest whole number.

A square pyramid made of iron with base edges of $8.7$8.7cm and height $11.9$11.9cm is melted down and recast in the shape of a cube.

Find the volume of the pyramid, correct to 2 decimal places.

Find the side length of the cube, correct to 1 decimal place.

Solve problems, including practical problems, involving volume and surface area of cones and square-based pyramids