Area of a TriangleArea of a Triangle 7.37.3

JMerrill, 2009JMerrill, 2009

Area of a Triangle Area of a Triangle (Formula)(Formula) When the lengths of 2 sides of a When the lengths of 2 sides of a

triangle and the measure of the triangle and the measure of the included angle are known, the included angle are known, the triangle is uniquely determined. triangle is uniquely determined. Use:Use:

S = ½ ab sin CS = ½ ab sin C S = ½ bc sin AS = ½ bc sin A S = ½ ac sin BS = ½ ac sin B

Do not memorize all the individual formulas, memorize the pattern:

S = ½ (one side)(2nd side)(sine of incl. angle)

ExampleExample

Two sides of a triangle have Two sides of a triangle have lengths 7cm and 4cm. The angle lengths 7cm and 4cm. The angle between the sides measures 73between the sides measures 73oo. . Find the area of the triangle.Find the area of the triangle.

S = ½ (7)(4)sin 73S = ½ (7)(4)sin 73o o

S = 13.388cmS = 13.388cm22

You Do #1You Do #1

Given the triangle ABC with Given the triangle ABC with measures of b = 3, c = 8, <A = measures of b = 3, c = 8, <A = 120120oo, find the area:, find the area:

10.392units10.392units22

ExampleExample

Find the area of a Find the area of a regularregular hexagon hexagon inscribed in a unit inscribed in a unit circle (means the circle (means the radius is 1 unit). radius is 1 unit). Then approximate Then approximate the area to 3 the area to 3 significant digits.significant digits.

First, divide the hexagon into six congruent triangles.

Flashback to geometry…what does “regular”

mean?

ExampleExample

Second, label the Second, label the known quantitiesknown quantities

S=6(½)(1)S=6(½)(1)(1)sin60(1)sin60

S=2.60 unitsS=2.60 units22

Where did the 6 come from?

1 160o

You Do #2You Do #2

Find the area of a regular octagon Find the area of a regular octagon inscribed in a circle with a radius inscribed in a circle with a radius of 20. Round to the nearest of 20. Round to the nearest tenth.tenth.

1131.4 units2

You Do: ChallengeYou Do: Challenge

Approximate the area Approximate the area of the irregularly-of the irregularly-shaped piece of land shaped piece of land (hint: split it into 2 (hint: split it into 2 triangles, one of which triangles, one of which is a right triangle). All is a right triangle). All measurements are measurements are given in feet. Round given in feet. Round to the nearest whole to the nearest whole number.number.

16

12

5

110o

Area of right triangle: 30ft2

Length of drawn segment: 13ft

Total area: 101ft2