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September 26, 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:
Simple geometry was the subject of last week’s article. Geometry uses mathematics to describe various properties of two and three dimensional objects, it is the math of shapes and figures. Last week we looked at simple polygons calle quadrilaterals - four sided shapes. This week, we look at more complex shapes - triangles and circles - and how to calculate areas and volumes in the real world.
Here’s the evaluation form for Geometry again, 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!
This week’s article looks at the first part of the evaluation form: Complex Geometry (the last five statements on the form). If you find this part of Geometry a challenge, read on. There are links to lots of free exercises and workbooks in this article.
If you’ve got this part of the competency down, tune in next week for Conversions.
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. Finding the length, area and volume of rectangles and squares is straightforward. But houses don’t always consist of easy-to-measure boxes.
In this article, we'll look at triangles and bring in some basic trigonometry (from Greek trigōnon, "triangle" and metron, "measure") to be able to determine lengths and angles of triangles that are common to construction. For example, 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.
Then we'll go over circles and cylinders and how to find their dimensions.
Knowing how to measure rectangles, squares, right triangles, and circles allows you to break down any complex shape that you might commonly find on a construction site or in a set of plans.
Shapes & Figures
Recall that shapes are two-dimensional objects; they exist in one plane, with a width and a length. 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.
Right Triangles and Pythagoras’ Theorem
Right triangles have two shorter sides and a longer side that is always called the hypotenuse. The other two sides are called opposite or adjacent, depending on which angle inside the triangle is being referenced.
Some triangles have special properties. For example, an isosceles right triangle has one 90 degree angle and 2 45 degree angles, with the length and height being the same.
The 30-60-90 triangle has a right angle and the other two angles are 30 degrees and 60 degrees.
If you know the lengths of all sides of a triangle, the perimeter is calculated the same way as for a rectangle: add up all the lengths of all the sides. If you don't know the length of one of the sides of a right angle triangle, you can use Pythagoras' Theorem to find it.
Pythagorus' theorem tells us that, in a right angle triangle, the length of the hypotenuse (call that ‘C’) is equal to the square root of the sum of the squares of the remaining sides (call those ‘A’ and ‘B’).
Pythagoras’ Theorem is: A² + B² = C².
Side C, the hypotenuse, is always the longest side, the side opposite the 90 degree angle. Sides A and B are the other two sides (it doesn't matter which one is labelled A or B).
The formula is rearranged depending on what side of the triangle you're looking for, A, B, or C. To solve the formula for any given side, you need to divide by the square root of the other two sides. Finding the square root of a number is the inverse operation of squaring that number (the square of a number is that number times itself, for example, 72 is 7 x 7). The square root symbol looks like this: √
Here is the original formula: A² + B² = C²
Solve for A: A = √(C²- B²)
Solve for B: B = √(C²- A²)
Solve for C: C = √(A²+ B²)
Square roots are easier to understand with examples:
√25 = 5, i.e. 5 is the square root of 25 since 52 = 5 x 5 =25
√4 = 2, i.e. 2 is the square root of 4 since 22 = 2 x 2 =4
The challenge is, not all numbers have a whole square root. For example, √13 is 3.60555. The easiest way to find the square root of a number is to use the √ function on your calculator.
Here’s an illustrated lesson with some interactive exercises (scroll to the bottom) on Pythagoras’ Theorem.
Here is an illustrated lesson with some exercises on squares and square roots.
Area of a Triangle
The area of a triangle (A) is equal to one half of the base (B) times the height (H).
This can be written as:
A = (B*H)/2, or A = 0.5 x (B x H).
Here's what's going on: a right triangle is basically half of a rectangle or square. So you are multiplying the two sides together to get the total area of the 'rectangle', and then dividing that area in two. In a right triangle, the height is going to be one of the sides, but for triangles without right angles, the height must first be determined.
Here’s an illustrated lesson on finding both the perimeter and the area of triangles.
Volume of Triangular Figures
To find the volume of a triangle, such as a sloped ceiling area, or a livable attic space, or a bay window, you calculate triangular prisms.
The volume of a triangular prism (V) is calculated using the area of the triangle that forms its base (that’s the formula A=B*H/2) multiplying that by the height (H) of the prism.
The formula is: V = A x H
The volume of a triangular pyramid is calculated using the area of the triangle base and the height of the pyramid (the distance from the base to the peak of the pyramid, with a factor of 1/3 to account for the sloping sides.
The formula is: V = 0.3 x A x H
Here’s an illustrated lesson on prisms in general.
Here’s an illustration of how to find the volume of a triangular prism.
Circles are a special case shape, as they have no straight lines and so the perimeter and area cannot be calculated in the same way as for polygons like rectangles, squares or triangles. Formulas are the only way to measure circles.
The diameter of a circle or sphere is the length of the line between 2 points on the circle or sphere which passes through the center. You cut a pie in half and that line down the middle is the diameter.
The radius is the distance between the center and a point on the circle. The radius is always half of the diameter.
Here is an illustrated lesson on circles.
The circumference of a circle is the length measured around the outside, kind of like the perimeter of the circle. To calculate the circumference of a circle (C), multiply the diameter (D) by 3.14 (pi, which is usually written as the Greek letter π).
Fun Fact: π (Pi) is a mathematical constant originally defined as the ratio of a circle's circumference to its diameter (the circumference of a circle is equal to a little more than 3 times its diameter), roughly equal to 3.14.
The formula for circumference is: C = π x D
If the radius of the circle is known, then the formula is written as: C = 2πR
Here’s a video lesson on finding the circumference of a circle.
Area of a Circle
The area of a circle can be found by multiplying the square of the radius of the circle (that’s half of the diameter) by π. Here’s a very cool animation of how this formula works.
The formula for area of a circle is: A = π x R²
Here is an interactive lesson on how to find the area of a circle.
Volume of a Cylinder
To calculate the volume of a cylinder (V), multiply the area of the circle, by the height of the cylinder.
The formula is: V = A x H
Here’s an illustrated lesson on how to find the volume of a cylinder.
Here’s an interactive set of perimeter problems to practice on.
Here is an excellent chart showing Area and Volume for common shapes.
Complex Shapes in the Real World
Objects and shapes in real life are not always as straightforward as many of these exercises.
Most of the time, for new construction, you will have access to a set of construction drawings. You could also have a set of drawings for a renovation project. On the other hand, you might be going into a building to measure up the floor plan for an energy evaluation or to install new flooring, for example.
If you are going to work from sketches that you've made in the field, it's likely your drawing will not be to scale, so you will need to make sure you take very detailed measurements while you're in the building.
If you are working with floor plans from a builder or architect, they will be scaled.
Scale is a multiplier that defines how big an object is in real life compared to its size on a drawing. Standard scales for house plans, for example, are 1/4 or 1/8 in : 1 ft. This indicates that quarter inch or eighth inch in on the plan equates to 1 ft in reality. For smaller objects where details matter, a scale of 1/2 inch to 1 foot allows you to see the design at twice the size.
If you learn to look for familiar shapes in the plans, they can be broken up into those shapes for measurements and calculations.
The simplest shapes to work with are squares and rectangles.
On a floor plan drawing, break the complex shape down until you have simplified the plan enough so that all you have are squares and rectangles and no complex shapes. Aligning the rectangles with dimension lines will make the measuring and adding-up process easier. If you can't align all the rectangles with dimension lines, then measure each rectangle using an architect's scale.
Multiply the length times the width for each rectangle to get the area and then add up all of the rectangles to get the total square footage of the floor plan.
Often, floor plans will have bay windows and other features that have non-right angle corners. Most of these shapes can be simplified into squares, rectangles, and triangles. For example, this house with a prow front is a rectangle with two right triangles. This house is much more complex, with an angled garage and three bay window areas. Still, it can be broken down into several squares, rectangles, and right triangles. On a complex floor plan like this one, make sure you label your simplified shapes carefully so you can keep track of them as you go through the process.
Once you've got all of your simple shapes measured, find the area of each, then add them all up to get the total square footage.
For surface area and volume calculations as well as measurements like rafter length and configuring stairs, you will need to be able to determine the slope of surfaces and objects that are on a slant, like roofs and wheelchair ramps. You might deal with roof pitches as well. You will also need to be able to determine the percent grade, or grade of surfaces and objects that relate to driveways, foundation drainage, french drains, trenches, and drain pipes.
Slope, pitch, and grade are all ways to describe the angle of a slanted surface.
Some terms to become familiar with are:
Rise: the vertical distance from the lowest to the highest point of the slant, or, the vertical distance from the rooftop to wherever the run is being measured.
Run: the horizontal distance that follows the rise from the lowest to the highest point of the roof or other slanted surface
Span: The span of a roof is the distance from one outside wall to the opposite outside wall. House plans will have detailed markers for where to measure, as there might be several layers of material. The run of a gable roof is half the span.
Pitch and slope are often used interchangeably, but they are not the same. In most of the building industry, slope provides more valuable information than pitch.
Slope is expressed as a ratio of vertical distance (rise) to horizontal distance (run). Slope is measured in inches per foot. For instance, if you travel 3 inches vertically and 3 feet (36 inches) horizontally, the slope would be 3:36 , simplified as 1:12, or one inch of rise per twelve inches of run.
The measurements for rise and run must always be converted to the same units. It's easiest to begin with converting the rise and run into inches.
Here is an excellent video example of how to find slope, ratios, and proportions (and a good example of using the words ‘slope’ and ‘pitch’ to mean the same thing).
Pitch refers to a ratio of the ridge height to the entire span/width of the building, with the ridge in the middle of the span.
Pitch is expressed as a fraction. For example, a roof with a rise of 4 feet and a span of 24 feet, has a pitch of “1 to 6”, which can be expressed as the fraction of 1/6. A “12 to 24” pitch is expressed as 1/2. Here are a couple more examples: a roof that rises 8 feet over a 24-foot span has a “1 to 3” pitch. If, instead, the rise were 4 feet over a 24-foot span, then the roof pitch is “1 to 6.”
Here's an example of how slope and pitch compare. Say you have a roof that rises 3 inches for every foot of span:
The slope is 3 in 12 (also written 3/12)
The pitch is 1 to 4 (also written 1/ 4)
Percentage grade is the slope, written as a percent. For instance, a slope of 1 in 12 (also written 1/12) is one divided by twelve. The decimal fraction is .083 , multiply by 100 for an 8.3% grade.
Here’s a video lesson on how to do a grade calculation.
If you feel you need more tutoring in complex geometry, or more practice, go to udemy.com and search for ‘geometry’. These courses are not free, but they are very reasonable in price.