Worksheet: Finding the Area of Triangle Using Trigonometry

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

Q1:

𝐴𝐡𝐢 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:

𝐴𝐡𝐢 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:

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 meter.

Q6:

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

  • A12π‘Žπ‘πΆsin
  • B14π‘Žπ‘πΆcos
  • C13π‘Žπ‘πΆsin
  • D14π‘Žπ‘πΆsin
  • E12π‘Žπ‘πΆcos

Q7:

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.

Q8:

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.

Q9:

𝐴𝐡𝐢 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.

Q10:

𝐴𝐡𝐢 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.

Q11:

𝐴𝐡𝐢𝐷 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.

Q12:

𝐴𝐡𝐢 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.

Q13:

π‘‹π‘Œπ‘ 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 centimeter.

Q14:

𝐴𝐡𝐢 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.

  • A6747β€²βˆ˜
  • B2213β€²βˆ˜
  • C4534β€²βˆ˜
  • D4426β€²βˆ˜

Q15:

𝐴𝐡𝐢 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.

Q16:

𝐴𝐡𝐢 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.

Q17:

𝐴𝐡𝐢 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.

  • Aperimeter is 41.65 cm and circumference is 39.75 cm.
  • Bperimeter is 41.65 cm and circumference is 79.5 cm.
  • Cperimeter is 64.96 cm and circumference is 79.5 cm.
  • Dperimeter is 64.96 cm and circumference is 39.75 cm.

Q18:

𝐴𝐡𝐢 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 to two decimal places.

Q19:

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‘.

  • A12
  • B1
  • C4
  • D2

Q20:

𝐴𝐡𝐢 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.

Q21:

𝐴𝐡𝐢 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.

Q22:

𝐴𝐡𝐢 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.

Q23:

Consider the following figure.

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

  • Aπ‘ŽπΆsin
  • Bπ‘ŽπΆcos
  • C𝑏𝐢tan
  • D𝑏𝐢cos
  • E𝑏𝐢sin

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

  • A12π‘Žπ‘πΆcos
  • Bπ‘Žπ‘πΆsin
  • C12π‘Žπ‘πΆsin
  • D14π‘Žπ‘πΆsin
  • Eπ‘Žπ‘πΆcos

Q24:

𝐴𝐡𝐢 is a triangle where 𝐴𝐡=18cm, π‘šβˆ π΅=60∘ and the area of the triangle is 74√3 cm2. Find the length of 𝐡𝐢 giving the answer to two decimal places.

Q25:

In the triangle 𝐴𝐡𝐢, 𝐴𝐢=8, 𝐡𝐢=5, and π‘šβˆ π΅πΆπ΄=44∘. Work out the area of the triangle to 2 decimal places.

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