Area of Triangles & Quadrilaterals — Challenge
Every area formula in this unit comes from a rectangle that has been cut up or rearranged.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perimeter — Distance all the way around the outside.
- Area — Space inside, measured in square units.
- Parallel — Same slope, different intercept.
- Parallelogram — A quadrilateral with both pairs of opposite sides parallel.
- Trapezoid — A quadrilateral with exactly one pair of parallel sides.
- Base — Any side of a triangle, chosen as the one to measure from.
- Height — The perpendicular distance from the base to the opposite corner.
- Cubic unit — A cube one unit on every edge.
Problem 1
What is the area of a triangle with base 6 and height 4?
- 10
- 12
- 24
- 20
- 6
Worked answer
12
The area of a triangle is half the base times the height.
12· 6· 4 = 12 square units.
Leaving out the half gives 24, the area of the surrounding rectangle.
Problem 2
Rectangle ABCD has area 100. Point M is the midpoint of AB. What is the combined area of triangles AMD and MBC?
- 25
- 50
- 75
- 100
- 40
Worked answer
50
Each triangle has half the rectangle width as its base and the full height.
Each area is one quarter of the rectangle, so together they have area 50.
Problem 3
What is the area of a triangle with base 10 and height 10?
- 20
- 40
- 50
- 100
- 25
Worked answer
50
Half the base times the height: 12· 10· 10.
1002 = 50 square units.
Problem 4
A right triangle has legs 6 and 4. What is its area?
- 3π
- 9
- 12
- 4π
- 5π
Worked answer
12
The legs are perpendicular, so they can be used as base and height.
The area is 12(6)(4)=12.
Problem 5
What is the area of a triangle with base 5 and height 8?
- 13
- 20
- 40
- 26
- 10
Worked answer
20
Half the base times the height: 12· 5· 8.
402 = 20 square units.
Adding the base and height gives 13, which measures nothing useful here.
Problem 6
A triangle has an area of 36 and a height of 9. What is its base?
- 4
- 6
- 8
- 9
- 18
Worked answer
8
Start from the formula: 36 = 12· b· 9 = 4.5b.
Divide both sides by 4.5: b = 8.
Dividing 36 by 9 and stopping gives 4, which forgets the half.
Problem 7
What is the area of a triangle with base 12 and height 3?
- 15
- 18
- 36
- 21
- 9
Worked answer
18
Half the base times the height: 12· 12· 3.
362 = 18 square units.
Problem 8
A right triangle has legs of 7 and 9. What is its area?
- 16
- 31.5
- 63
- 28
- 35
Worked answer
31.5
In a right triangle the two legs meet at a right angle, so one can serve as the base and the other as the height.
The area is 12· 7· 9 = 632 = 31.5 square units.
No third side is needed, because the height is already given.