Math 1 · Unit 11 · Lesson 3 · part 3 of 5
Area: rectangles and triangles
Area is how much flat space a shape covers, in square units.
Area: rectangles and triangles
Area is how much flat space a shape covers, in square units. Math 1 asks for rectangles and triangles.
Rectangle: length times width
Use the distance formula for the length and the width, then multiply. The rectangle A(1, 2), B(6, 0), C(10, 10), D(5, 12) is tilted, so count nothing: AB = √5² + 2² = √29 ≈ 5.39 and BC = √4² + 10² = √116 ≈ 10.77. Area ≈ 5.39 × 10.77 ≈ 58 square units.
Triangle: one half of base times height
Area = 12 × base × height. The catch: the height must be perpendicular to the base, meeting it at a right angle. A segment from a vertex that does that is called an altitude.
Triangle ABC with A(−4, −2), B(8, 7), C(1, 8), and a point D(4, 4) on AB. Is CD the altitude? Check it the Unit 4 way before trusting it.
- 1Is D on AB?Slope of AB: 7 − (−2)8 − (−4) = 912 = 34. Slope of AD: 4 − (−2)4 − (−4) = 68 = 34. Same slope from the same point A, so D is on line AB.
- 2Is CD perpendicular to AB?Slope of CD: 4 − 84 − 1 = −43. That is the negative reciprocal of 34 (flip it, change the sign), so CD meets AB at a right angle. CD is the altitude, the height.
- 3Base and height.AB = √12² + 9² = √225 = 15. CD = √3² + 4² = √25 = 5. Area = 12 × 15 × 5 = 37.5 square units.