Worksheet: Finding the Area of Triangle Using Trigonometry

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In this worksheet, we will practice finding the area of a triangle without having to calculate its perpendicular height.

Q1:

𝐴 𝐡 𝐢 is a triangle, where 𝐡𝐢=15cm, 𝐴𝐢=25cm, and π‘šβˆ πΆ=41∘. Find the area of 𝐴𝐡𝐢, giving your answer to three decimal places.

Q2:

Find the area of the figure below giving the answer to three decimal places.

Q3:

𝐴 𝐡 𝐢 is a triangle where 𝐴𝐡=5cm, 𝐴𝐢=11cm, and π‘šβˆ π΄=107∘. Find the area of the triangle giving the answer to two decimal places.

Q4:

In the given figure, work out the area of the triangle to 2 decimal places.

Q5:

Benjamin’s house is on a corner lot. Find the area of the front yard given that the edges measure 40 ft and 56 ft, as shown in the diagram.

Q6:

The figure shows a triangular field with sides 670 m, 510 m and 330 m. Find the area of the field giving the answer to the nearest square metre.

Q7:

Which of the following is a formula that can be used to find the area of a triangle?

  • A 1 4 π‘Ž 𝑏 𝐢 c o s
  • B 1 3 π‘Ž 𝑏 𝐢 s i n
  • C 1 2 π‘Ž 𝑏 𝐢 s i n
  • D 1 2 π‘Ž 𝑏 𝐢 c o s
  • E 1 4 π‘Ž 𝑏 𝐢 s i n

Q8:

Find the area of a triangle whose two side lengths are 7 cm and 23 cm, and the angle included between them is 101∘.Give the answer to three decimal places.

Q9:

An isosceles triangle has a side length of 48 cm and a base angle of 75∘. Find the area of the triangle giving the answer to three decimal places.

Q10:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝐴𝐡=𝐴𝐢=11cm and π‘šβˆ π΄ is 158∘. Find the area of 𝐴𝐡𝐢 giving the answer correct to two decimal places.

Q11:

𝐴 𝐡 𝐢 is a triangle, where ∠𝐴𝐢𝐡 is obtuse, π‘Ž=20cm, 𝑏=14cm, and the area is 106 cm2. Find the length of 𝐴𝐡, giving the answer to two decimal places.

Q12:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝐴𝐡=9.3cm, π‘šβˆ πΆπ΄π΅=2648β€²βˆ˜ and π‘šβˆ π·π΅π΄=6630β€²βˆ˜. Find the area giving the answer to two decimal places.

Q13:

𝐴 𝐡 𝐢 is a triangle where π‘Ž=23cm, 𝑏=21cm, the area is 167 cm2 and the radius of the circumcircle is 17 cm. Find length 𝑐 giving the answer to two decimal places.

Q14:

𝑋 π‘Œ 𝑍 is a triangle where π‘šβˆ π‘‹=35∘, π‘šβˆ π‘Œ=62∘ and the area is 480 cm2. Find the length of π‘Œπ‘ giving the answer to the nearest centimetre.

Q15:

𝐴 𝐡 𝐢 is an acute-angled triangle where 𝐴𝐢=30cm, sin𝐴=0.7 and the area is 315 cm2. Find π‘šβˆ π΅ , giving the answer to the nearest minute.

  • A 4 4 2 6 β€² ∘
  • B 2 2 1 3 β€² ∘
  • C 6 7 4 7 β€² ∘
  • D 4 5 3 4 β€² ∘

Q16:

𝐴 𝐡 𝐢 is a triangle where π‘Ž=24.9cm, 𝑏=27.9cm, 𝑐=22.5cm and the radius of the circumcircle is 22.4 cm. Find the area of the triangle giving the answer to one decimal place.

Q17:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΄=64∘, π‘šβˆ πΆ=71∘ and the perimeter is 59.34 cm. Find the area of the triangle giving the answer to two decimal places.

Q18:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΄=47∘, π‘šβˆ π΅=66∘ and the area is 197 cm2. Find the perimeter of 𝐴𝐡𝐢 and the circumference of the circumcircle giving the answer to two decimal places.

  • A perimeter is 64.96 cm and circumference is 39.75 cm.
  • B perimeter is 41.65 cm and circumference is 39.75 cm.
  • C perimeter is 64.96 cm and circumference is 79.5 cm.
  • D perimeter is 41.65 cm and circumference is 79.5 cm.

Q19:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΅=68∘ and π‘šβˆ πΆ=40∘. The point 𝐷 lies on 𝐡𝐢 where 𝐴𝐷=26cm and bisects ∠𝐴. Find the area of △𝐴𝐡𝐢 giving the answer to the nearest whole number.

Q20:

The area of the triangle 𝐴𝐡𝐢 is π‘₯ and the radius of the circumcircle is π‘Ÿ. Find the value of 4π‘Ÿπ‘₯π‘Žπ‘π‘.

  • A4
  • B1
  • C2
  • D 1 2

Q21:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΅=66∘, π‘šβˆ πΆ=63∘ and the diameter of the circumcircle is 27 cm. Find the area of the triangle 𝐴𝐡𝐢 giving the answer to two decimal places.

Q22:

𝐴 𝐡 𝐢 is a triangle where π‘šβˆ π΅=36∘, π‘šβˆ πΆ=60∘ and the area of 𝐴𝐡𝐢 is 153 cm2. Find π‘Ž giving the answer to two decimal places.

Q23:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝑏=𝑐, π‘šβˆ π΄=42∘ and the perimeter is 31.31 cm. Find the area of the triangle giving the answer to two decimal places.

Q24:

Consider the following figure.

Find an expression for the height β„Ž in terms of π‘Ž, 𝑏, and 𝐢.

  • A 𝑏 𝐢 s i n
  • B π‘Ž 𝐢 s i n
  • C π‘Ž 𝐢 c o s
  • D 𝑏 𝐢 t a n
  • E 𝑏 𝐢 c o s

Find an expression for the area of the triangle in terms of π‘Ž, 𝑏, and 𝐢.

  • A π‘Ž 𝑏 𝐢 s i n
  • B π‘Ž 𝑏 𝐢 c o s
  • C 1 4 π‘Ž 𝑏 𝐢 s i n
  • D 1 2 π‘Ž 𝑏 𝐢 s i n
  • E 1 2 π‘Ž 𝑏 𝐢 c o s

Q25:

The Bermuda triangle is a region of the Atlantic Ocean whose boundary is made up of straight lines connecting Bermuda, Florida, and Puerto Rico. Given that Florida is 1030 miles from Bermuda in the direction 24 degrees south of west and Puerto Rico is 980 miles from Bermuda in the direction 4 degrees west of south, find the area of the Bermuda triangle.

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