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September 18, 2018

‘Numeracy’ is being able to read numbers, and apply math to real-life situations. It’s similar to ‘literacy’, which is being able to read and understand words. Numeracy has mathematics as its core.

For the Energy Advisor Foundation training, you are required to understand mainly quantity and number as well as space and shape. Problem solving is a good skill to have, as well as understanding patterns and relationships. We have boiled down the numeracy competency requirements to three categories:

Arithmetic

Geometry

Conversions

Geometry is the subject of this article. Geometry uses mathematics to describe various properties of two and three dimensional objects, it is the math of shapes and figures. There are angles, points, lines, lengths, perimeters, circumferences, faces, surface area, and volume. These are fundamental math skills for anyone working in construction.

Every trade, subtrade, consultant, and energy advisor will use basic geometry nearly every day of their working life. The great thing about math in general, and geometry in particular, is that using it in your everyday life is the best way to learn it.

Geometry in construction math is focussed on three specific properties of an object: the perimeter, the area, and the volume.

Here’s a simple evaluation form for Geometry in PDF format for downloading, so you can rate yourself on your existing understanding of geometry. The rating is from 0 (I know nothing about this topic) to 4 (I’m an expert in this topic and can handle complex tasks on the daily). You don’t have to share this with anyone else, so rate yourself honestly!

Once you’ve rated yourself, give yourself a pat on the back for the areas where you are strong, and roll up your sleeves to improve your game in the other areas.

If you find Geometry a challenge, read on. There are links to lots of free exercises and workbooks in this article. This week’s article looks at the first part of the evaluation form: Simple Geometry (the first four statements on the form).

If you’ve got this part of the competency down, tune in next week for Complex Geometry.

## Simple geometry

A basic knowledge of some key concepts in geometry is essential to fully understand and interpret floor plans, and it's even more important when you're actually building the house. Length, area, and volume are the focus of geometry in the construction industry, but the Pythagorean Theorem, which applies to right angle triangles, is used extensively in home construction, especially in roofs. For example, a gable is made up of two right triangles placed back to back.

## Shapes

### Shapes are two-dimensional objects; they exist in one plane. Shapes are created by combining a set of line segments or a series of curves, each of which has a certain length. A polygon is a shape that is closed and is made up of only lines (no curves). A polygon can be a square, rectangle, triangle, octagon. A circle is not a polygon because it is created by using curves.

Polygons and circles allow us to calculate things like the surface area of a wall so we can determine how much drywall is needed, or how big a hole can be safely cut into a solid floor joist to allow a pipe to go through.

While there is a wide variety of complex shapes that geometry can describe, like a balbis, a triquetra, or an enneagram, most people working on residential construction will only need to be able to manipulate a few basic shapes: Triangles, Quadrilaterals, and Circles. You can usually break down even the most complex of floor plans or shapes into a group of these three main shapes. Let's go over these basic shapes and their properties.

### Triangles

Triangles always have three sides, but there are different types that can be grouped together. All triangles also have three interior angles, one at each point formed where the sides meet. There are several types of triangles, but they all share one thing in common: the three angles of any triangle must always add up to 180 degrees.

A right angle triangle has two perpendicular sides that meet at a 90 degree (or "right") angle. The remaining side forms what's called the hypotenuse.

An equilateral triangle has three sides of equal length. In addition to the sides being equal, all of the interior angles are also equal.

An isosceles triangle has two sides of equal length, with the two angles opposite the equal sides being equal as well.

These first three types are the most commonly found triangles in the construction industry. Sometimes, however these other three show up:

A scalene triangle has three sides of all different lengths.

An acute triangle has three acute angles (less than 90 degrees).

An obtuse triangle has one angle that is obtuse (more than 90 degrees).

Here’s an illustrated lesson on triangles.

### Quadrilaterals

Quadrilaterals are polygons that have four sides and four interior angles (quadri is Latin for 'four'). A quadrilateral always has two pairs of parallel sides. A perfect square has four sides the same length. A rectangle is a quadrilateral that has opposite sides of equal length; this is like a stretched square. Squares and rectangles have 4 right angles. The length of the sides of a square are the same, while a rectangle has a length measurement and a height measurement.

Windows and doors are squares and rectangles, their sides are commonly noted as height and width, not height and length.

Other quadrilateral shapes include:

the parallelogram, a rectangle that has no right angles, the rhombus, a parallelogram square, with four equal sides, and

the trapezoid, which has only one pair of parallel sides.

To get the lengths, heights, and areas of these shapes, you can break them down into a series squares, rectangles, and triangles.

Here’s an illustrated lesson on quadrilaterals.

### Circles

Have you ever tried to explain what a circle is without saying the word "circle"? A circle is defined as the set of points that are all at a fixed distance from a given point. The "given point" is the center of the circle, and the "fixed distance" is called the radius. The diameter is the width of the the circle; it's double the radius. We use the radius to calculate the circumference, or the perimeter of the circle, and, the area of the circle.

Here’s an illustrated lesson on circles

## Figures

Three-dimensional shapes are called figures, or solids. Like shapes, they have a length, a height, and they also have a third dimension, width. With the third dimension, they exist in more than one plane. In a drawing, you can't always show the third dimension clearly, especially with a complicated figure.

Like two-dimensional shapes, there are extremely complicated 3 dimensional figures that geometry can describe, like a dodecahedron. The good news is that fairly simple geometric figures make up nearly all of the spaces we build and the equipment we install: rooms, vaulted ceilings, water storage tanks.

In the real world, three-dimensional objects have a volume, which is the amount of space that an object occupies, expressed in cubic units. This is what we measure. We use figures to calculate the volume of air, liquid, or solid from a drawing.

Nearly all the figures that we work with in the construction industry can be broken down into two main categories: prisms and cylinders. Prisms are the three dimensional equivalent of polygons; their bases are shapes that are made out of straight lines like squares, rectangles and triangles. Cylinders are the three dimensional equivalent of circles.

### Prisms

A prism has a base shape. The base can be a square, a rectangle or a triangle, or a more complicated shape. The width of a prism gives it its three dimensional aspect, with a base at either end. A square room, for instance, is a cube; a box made up of six square faces. A rectangular room or a square fence post is a rectangular prism.

### Cylinders

A cylinder is a three-dimensional figure that has two circle-shaped bases; one at either end. The bases are on parallel planes and the cylindrical surface occupies the space between. If the elements of a cylinder are perpendicular to the base, like a can of paint, a decorative round column or a hot water tank, this is called a right cylinder.

## Length

When you measure the length of something, you are measuring the distance between two points on a straight line. You might measure the length of one side of a house, or the dimensions of a piece of insulation, or the length of a driveway.

Length is one of the simplest dimensions to measure, and it can help us find a lot of other information about an object or material.

You might measure a straight line on objects oriented in a different way, like height, width, or rise.

Perimeter is the distance around the outside of an object or space.

The perimeter is measured on a flat, two-dimensional plane. It's often used it to calculate the footprint of a house, or the length of the boundaries of a property, like where you might put a fence.

Here’s an illustrated lesson on how to find the perimeter of a polygon. (We’ll cover triangles and circles next time)

Here’s an interactive set of perimeter problems to practice on.

## Area

Area is a measure of the two-dimensional space that is defined by a perimeter. A lot of the time we call this surface area. The area of a shape can be found by using its other properties - like length, width, radius and circumference.

Remember that measurements of area have an exponent of 2, which lets us know that the units are squared, like meters squared, or square feet, because when we multiply numbers with units, the operations are performed on the units as well.

You can measure the surface area of a flat shape, like a wall, or a floor plan.

Here are some illustrated lessons on finding the areas of flat or 2 dimensional quadrilateral shapes:

The surface area of a solid (a three-dimensional shape) is a measure of the total area that the surface of the solid occupies. For instance, to find the surface area of a cube, we'd find the area of one of the sides and multiply it by 6 (because a cube has 6 identical sides).

Here’s an illustrated lesson on finding the surface area of a quadrilateral solid:

## Volume

We can measure the volume of any three-dimensional object. Volume is found by multiplying the dimensions of length and width by height.

You'll need to know the volume of air in a house to calculate what size equipment you'll need to heat it. After you have determined the floor area from the floor plan, you can calculate the volume by looking at the section drawings and/or the elevations. If most of your rooms are square with flat ceilings, it'll be pretty straightforward. Likewise, if you're filling a foundation wall with concrete - you're going to need to know the volume of material that needs to go into that wall.

Here are some illustrated lessons about volume for quadrilaterals:

Cube (1)

Cube (2)

If you feel you need more tutoring in geometry, or more practice, go to udemy.com and search for ‘basic geometry’. These courses are not free, but they are very reasonable in price.