$Area = \\frac{1}{2} ab \\sin C$

$Area = \\frac{1}{2} \\times base \\times height$

Because $\\sin C$ gives the ratio of the height to side $b$

Use Law of Sines to find $a$ or $b$, then use $Area = \\frac{1}{2} ab \\sin C$

Question 1

Given a triangle with sides $a = 8$, $b = 12$, and included angle $C = 60^\degree$, what is the area of the triangle?

Accepted Answer

$41.6$

Question 2

Which formula is used to find the area of a triangle when two sides and the included angle are known (SAS)?

Accepted Answer

$Area = \frac{1}{2} ab \sin C$

Question 3

A surveyor needs to find the area of a triangular plot with sides $a = 15$ m, $b = 20$ m, and included angle $C = 45^\degree$. What is the area?

Accepted Answer

$106.1$

Question 4

If you know two angles and the included side (ASA) of a triangle, which law do you use to find the missing sides before calculating the area?

Accepted Answer

Law of Sines

Question 5

Given triangle $ABC$ with $A = 40^\degree$, $C = 60^\degree$, and $b = 6$, what is the first step to find the area?

Accepted Answer

Find the missing side using Law of Sines

Question 6

A triangle has sides $a = 10$, $b = 7$, and included angle $C = 120^\degree$. What is the area?

Accepted Answer

$60.6$

Question 7

Which of the following is the correct formula for the area of a triangle when the height is known?

Accepted Answer

$Area = \frac{1}{2} \times base \times height$

Question 8

A triangle has $A = 30^\degree$, $b = 10$, $B = 80^\degree$. What is the area of the triangle?

Accepted Answer

$29.2$

Question 9

Why is the formula $Area = \frac{1}{2} ab \sin C$ valid for any triangle, not just right triangles?

Accepted Answer

Because $\sin C$ gives the ratio of the height to side $b$

Question 10

A triangle has sides $a = 9$, $b = 12$, and included angle $C = 75^\degree$. What is the area?

Accepted Answer

$52.1$

Question 11

If you are given $A = 50^\degree$, $B = 60^\degree$, and $c = 8$, what is the correct sequence to find the area?

Accepted Answer

Use Law of Sines to find $a$ or $b$, then use $Area = \frac{1}{2} ab \sin C$

Question 12

A triangle has sides $a = 5$, $b = 7$, and included angle $C = 90^\degree$. What is the area?

Accepted Answer

$17.5$

Question 13

Which of the following is NOT a valid method for finding the area of a triangle given two angles and the included side?

Accepted Answer

Using Heron's formula directly

Question 14

A triangle has $A = 45^\degree$, $B = 55^\degree$, and $c = 10$. What is the area?

Accepted Answer

$28.7$

Question 15

In a triangle, if $a = 6$, $b = 8$, and $C = 30^\degree$, what is the area?

Accepted Answer

$24.0$

Area of SAS & ASA Triangles: Trigonometric Applications

Study Guide - Practice Questions