Ray's New Higher Arithmetic · Art. 404 · Unit 28: Mensuration
120. Areas of triangles and quadrilaterals
Goal: Your student finds the areas of parallelograms, triangles (including from three sides) and trapezoids.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
- Paper, scissors and a ruler
What's in the book
Areas of triangles and quadrilaterals: why the rules work (with figures), four general rules, notes on trapeziums, polygons and the rhombus, and examples (answers printed).
The lesson, step by step
Warm-up
Ask: A rectangle is 8 by 5. What is its area? If you cut it along a diagonal, what is the area of each triangle?
The idea
Read:
The area of a rectangle is equal to the product of its length by its breadth.
Cut a paper parallelogram and move a triangle from one end to the other to make a rectangle. Its area is base × height.
The rules
Read:
Multiply one of two parallel sides by the perpendicular distance between them.
Take half the product of the base by the altitude.
Multiply half the sum of the parallel sides by the altitude.
Triangle from three sides
Read:
Add the three sides together and take half the sum; from the half sum take the sides separately; multiply the half sum and the three remainders together, and extract the square root of the product.
Try with sides 3, 4, 5: half sum 6; 6 × 3 × 2 × 1 = 36; √36 = 6. Check by half of 3 × 4.
Guided practice
Find the area of a parallelogram whose base is 9 ft. 4 in. and altitude 2 ft. 5 in.
How many tiles 8 in. square in a floor 48 ft. by 10 ft.?
Independent practice
Your student works more examples, including a land problem in rods and acres. (A rod is 16½ feet; 160 square rods make an acre.)
Check and wrap-up
Check the printed answers. Ask: Two triangles have the same base and the same height. Must they have the same area? (Yes.)