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Area of any triangle practice questions with answers

Higher · Grade 7 · Calculator

A second set of area of any triangle questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

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Question 1 Easy

Find the area of a parallelogram with sides 6 and 8 and included angle 30.

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Answer: 24

  1. Area = ab sin C.
  2. 6 × 8 × 1/2 = 24.
  • Method (1): Area = ab sin C.
  • Accuracy (1): Answer 24

Question 2 Easy

The area is 24, with one side 8 and included angle 30 degrees. What is the other side?

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Answer: 12

  1. 24 = 1/2 × 8 × b × 1/2.
  2. The other side is 12.
  • Method (1): 24 = 1/2 × 8 × b × 1/2.
  • Accuracy (1): Answer 12

Question 3 Medium

The area is 30, one side is 10 cm and the included angle is 40 degrees. What is the other side to 3 significant figures?

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Answer: 9.33

  1. 30 = 1/2 × 10 × b × sin 40.
  2. Rearrange for b.
  • Method (1): 30 = 1/2 × 10 × b × sin 40.
  • Accuracy (1): Answer 9.33

Question 4 Medium

The area of a triangle is 24 and the base is 8 cm. What is the perpendicular height?

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Answer: 6

  1. Area = 1/2 × base × height.
  2. Height = 2 × 24 / 8.
  • Method (1): Area = 1/2 × base × height.
  • Accuracy (1): Answer 6

Question 5 Hard

A right-angled triangle has perpendicular sides 6 cm and 8 cm. What is its area?

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Answer: 24

  1. The included angle is 90 degrees and sin 90 = 1.
  2. Area = 1/2 × 6 × 8.
  • Method (1): The included angle is 90 degrees and sin 90 = 1.
  • Accuracy (1): Answer 24

Question 6 Hard

The area of a parallelogram is 24, one side is 6 and the included angle is 30 degrees. What is the other side?

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Answer: 8

  1. Area = ab sin C, so 24 = 6 × b × 1/2.
  2. The other side is 8.
  • Method (1): Area = ab sin C, so 24 = 6 × b × 1/2.
  • Accuracy (1): Answer 8

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

The area is 20 and the sides are 8 and 10. What is sin of the included angle?

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Answer: 1/2

  1. sin C = 2 × area / (a × b).
  2. 2 × 20 / 80 = 1/2.
  • Method (1): sin C = 2 × area / (a × b).
  • Accuracy (1): Answer 1/2

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

State the area formula that uses sine.

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Area = 1/2 ab sin C, with C the included angle. Area = 1/2 ab sin C, with C the included angle.

Check 2

Why does angle 30 give an extra half?

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Sin 30 = 1/2. Sin 30 = 1/2.

Check 3

Area from sides 8 and 10 with included angle 30?

  1. 30
  2. 20
  3. 21
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20 1/2 × 8 × 10 × 1/2 = 20.

Check 4

Two sides of 6 enclose 60 degrees. What area does sine formula give?

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1/2 × 36 × √3/2 = 9√3. 1/2 × 36 × √3/2 = 9√3.

Check 5

Why does angle 60 leave a surd?

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Sin 60 = √3/2. Sin 60 = √3/2.

Check 6

Two sides of 4 enclose 45 degrees. What exact area follows?

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1/2 × 16 × √2/2 = 4√2. 1/2 × 16 × √2/2 = 4√2.

Check 7

What is sin 90 in this formula?

  1. 0
  2. 90
  3. 1
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1 Sin 90 = 1, giving standard 1/2 base × height.

Check 8

Area from sides 5 and 8 with a right angle included?

  1. 4
  2. 40
  3. 20
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20 1/2 × 5 × 8 × 1 = 20.

Check 9

Sides 4 and 6 with 60 degrees between them give which area?

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1/2 × 24 × √3/2 = 6√3. 1/2 × 24 × √3/2 = 6√3.

Check 10

Which side lengths must enclose the angle?

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The angle must be included between the two given sides. The angle must be included between the two given sides.

Check 11

Equal sides of 8 with 60 degrees between them give which area?

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1/2 × 64 × √3/2 = 16√3. 1/2 × 64 × √3/2 = 16√3.

Check 12

A student uses sin 45 as 1. What is lost?

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Sin 45 = √2/2, not 1. Sin 45 = √2/2, not 1.

Check 13

Area from sides 10 and 10 with included angle 30?

  1. 24
  2. 1
  3. 25
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25 1/2 × 100 × 1/2 = 25.

Check 14

The smallest 60 degree case here uses sides of 2. What is area?

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1/2 × 4 × √3/2 = √3. 1/2 × 4 × √3/2 = √3.

Check 15

Area from sides 6 and 10 with a right angle between them?

  1. 30
  2. 10
  3. 31
Show answer

30 1/2 × 60 = 30.

Keep going with Area of any triangle

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