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Intermediate Geometry : How to find the area of an acute / obtuse isosceles triangle

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Example Questions

Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle has two legs of length 10" with a base of unknown length, and has a height of 6". Find the area.

Possible Answers:

$36\ in^2$

$60\ in^2$

$32\ in^2$

$48\ in^2$

$64\ in^2$

Correct answer:

$48\ in^2$

Explanation:

To find the area:

$\frac{1}{2}\cdot base\cdot height$

The information giving us the two sides helps us find the base.

Using the fact that an isosceles triangle can be split vertically down the middle (note: the length of this extra line will be equal to the height) to form two identical right triangles, we then use the Pythagorean Theorem to find the base:

a^2 + b^2 = c^2

For our problem, is the height, is the base (of ONE of the right triangles; the base of the isosceles triangle will be twice as big) and is the hypotenuse, or the leg of length 10".

We find that the base is 8", so the base of the isosceles triangle is 16".

Plugging in our numbers we get:

$16\cdot 6\cdot \frac{1}{2}=48\ in^{2}$

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Example Question #2 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If diameter of the circle is , find the area of the shaded region.

Possible Answers:

4.22

2.14

3.19

1.93

Correct answer:

2.14

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2 \times 1)=1$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 1^2= \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=\pi-1=2.14$

Make sure to round to places after the decimal.

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Example Question #3 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

60.92

77.10

75.59

91.21

Correct answer:

77.10

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(12 \times 6)=36$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 6^2= 36\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=36\pi-36=77.10$

Make sure to round to places after the decimal.

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Example Question #4 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

98.87

112.52

104.94

102.05

Correct answer:

104.94

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(14 \times 7)=49$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 7^2=49 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=49\pi-49=104.94$

Make sure to round to places after the decimal.

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Example Question #5 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

29.95

34.27

40.29

31.02

Correct answer:

34.27

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(8 \times 4)=16$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 4^2= 16\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=16\pi-16=34.27$

Make sure to round to places after the decimal.

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Example Question #6 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

190.06

328.52

222.49

214.16

Correct answer:

214.16

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(20 \times 10)=100$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 10^2=100 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=100\pi-100=214.16$

Make sure to round to places after the decimal.

Report an Error

Example Question #7 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

420.94

458.61

398.50

419.75

Correct answer:

419.75

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(28 \times 14)=196$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 14^2=196 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=196\pi-196=419.75$

Make sure to round to places after the decimal.

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Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

173.47

198.52

189.62

174.55

Correct answer:

173.47

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(18 \times 9)=81$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 9^2= 81\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=81\pi-81=173.47$

Make sure to round to places after the decimal.

Report an Error

Example Question #21 : Triangles

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

401.25

372.90

355.21

361.93

Correct answer:

361.93

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(26 \times 13)=169$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 13^2= 169\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=169\pi-169=361.93$

Make sure to round to places after the decimal.

Report an Error

Example Question #21 : Acute / Obtuse Isosceles Triangles

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is , find the area of the shaded region.

Possible Answers:

1121.77

980.27

1036.53

1092.14

Correct answer:

1036.53

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(44 \times 22)=484$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 22^2= 484\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=484\pi-484=1036.53$

Make sure to round to places after the decimal.

Report an Error

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Intermediate Geometry : How to find the area of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #2 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #3 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #4 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #5 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #6 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #7 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #21 : Triangles

Example Question #21 : Acute / Obtuse Isosceles Triangles

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