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Basic Geometry : Triangles

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Example Questions

Example Question #471 : Triangles

Find the area of a right triangle with base 4 and height 5.

Possible Answers:

$\displaystyle 10$

$\displaystyle 9$

$\displaystyle 20$

$\displaystyle 18$

Correct answer:

$\displaystyle 10$

Explanation:

To solve, simply use the formula for the area of a triangle. Thus,

$A=\frac{1}{2}Bh=\frac{1}{2}*4*5=10$ $\displaystyle A=\frac{1}{2}Bh=\frac{1}{2}*4*5=10$

If the formula escapes you, simply remember that two equivalent triangles put together equal a rectangle. So, the area of a triangle must be half the area of a rectangle.

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Example Question #1455 : Plane Geometry

The diameter of the circle is $\displaystyle 2$, find the area of the shaded region.

Possible Answers:

20.09 $\displaystyle 20.09$

19.21 $\displaystyle 19.21$

20.86 $\displaystyle 20.86$

21.28 $\displaystyle 21.28$

Correct answer:

20.86 $\displaystyle 20.86$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{2}{2}$ $\displaystyle \text{Radius}=\frac{2}{2}$

$\text{Radius}=1$ $\displaystyle \text{Radius}=1$

Now, substitute in the radius to find the area of the circle.

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

$\text{Area of Circle}=\pi$ $\displaystyle \text{Area of Circle}=\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{4 \times 12}{2}$ $\displaystyle \text{Area of Triangle}=\frac{4 \times 12}{2}$

$\text{Area of Triangle}=24$ $\displaystyle \text{Area of Triangle}=24$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=24-\pi$ $\displaystyle \text{Area of Shaded Region}=24-\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=20.86$ $\displaystyle \text{Area of Shaded Region}=20.86$

Report an Error

Example Question #1456 : Plane Geometry

The diameter of the circle is $\displaystyle 3$, find the area of the shaded region.

Possible Answers:

12.93 $\displaystyle 12.93$

12.48 $\displaystyle 12.48$

13.04 $\displaystyle 13.04$

11.82 $\displaystyle 11.82$

Correct answer:

12.93 $\displaystyle 12.93$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{3}{2}$ $\displaystyle \text{Radius}=\frac{3}{2}$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times\left( \frac{3}{2}\right)^2$ $\displaystyle \text{Area of Circle}=\pi\times\left( \frac{3}{2}\right)^2$

$\text{Area of Circle}=\frac{9}{4}\pi$ $\displaystyle \text{Area of Circle}=\frac{9}{4}\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{4 \times 10}{2}$ $\displaystyle \text{Area of Triangle}=\frac{4 \times 10}{2}$

$\text{Area of Triangle}=20$ $\displaystyle \text{Area of Triangle}=20$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=20-\frac{9}{4}\pi$ $\displaystyle \text{Area of Shaded Region}=20-\frac{9}{4}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=12.93$ $\displaystyle \text{Area of Shaded Region}=12.93$

Report an Error

Example Question #1457 : Plane Geometry

The diameter of the circle is 1.2 $\displaystyle 1.2$, find the area of the shaded region.

Possible Answers:

13.87 $\displaystyle 13.87$

10.71 $\displaystyle 10.71$

14.90 $\displaystyle 14.90$

12.13 $\displaystyle 12.13$

Correct answer:

13.87 $\displaystyle 13.87$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{1.2}{2}$ $\displaystyle \text{Radius}=\frac{1.2}{2}$

$\text{Radius}=0.6$ $\displaystyle \text{Radius}=0.6$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 0.6^2$ $\displaystyle \text{Area of Circle}=\pi\times 0.6^2$

$\text{Area of Circle}=0.36\pi$ $\displaystyle \text{Area of Circle}=0.36\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{2 \times 15}{2}$ $\displaystyle \text{Area of Triangle}=\frac{2 \times 15}{2}$

$\text{Area of Triangle}=15$ $\displaystyle \text{Area of Triangle}=15$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=15-0.36\pi$ $\displaystyle \text{Area of Shaded Region}=15-0.36\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=13.87$ $\displaystyle \text{Area of Shaded Region}=13.87$

Report an Error

Example Question #1458 : Plane Geometry

The diameter of the circle is $\displaystyle 6$, find the area of the shaded region.

Possible Answers:

38.71 $\displaystyle 38.71$

39.73 $\displaystyle 39.73$

40.82 $\displaystyle 40.82$

42.51 $\displaystyle 42.51$

Correct answer:

39.73 $\displaystyle 39.73$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{6}{2}$ $\displaystyle \text{Radius}=\frac{6}{2}$

$\text{Radius}=3$ $\displaystyle \text{Radius}=3$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 3^2$ $\displaystyle \text{Area of Circle}=\pi\times 3^2$

$\text{Area of Circle}=9\pi$ $\displaystyle \text{Area of Circle}=9\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{8 \times 17}{2}$ $\displaystyle \text{Area of Triangle}=\frac{8 \times 17}{2}$

$\text{Area of Triangle}=68$ $\displaystyle \text{Area of Triangle}=68$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=68-9\pi$ $\displaystyle \text{Area of Shaded Region}=68-9\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=39.73$ $\displaystyle \text{Area of Shaded Region}=39.73$

Report an Error

Example Question #1459 : Plane Geometry

The diameter of the circle is 6.2 $\displaystyle 6.2$, what is the area of the shaded region?

Possible Answers:

17.81 $\displaystyle 17.81$

15.00 $\displaystyle 15.00$

16.89 $\displaystyle 16.89$

18.45 $\displaystyle 18.45$

Correct answer:

17.81 $\displaystyle 17.81$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{6.2}{2}$ $\displaystyle \text{Radius}=\frac{6.2}{2}$

$\text{Radius}=3.1$ $\displaystyle \text{Radius}=3.1$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 3.1^2$ $\displaystyle \text{Area of Circle}=\pi\times 3.1^2$

$\text{Area of Circle}=9.61\pi$ $\displaystyle \text{Area of Circle}=9.61\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{8 \times 12}{2}$ $\displaystyle \text{Area of Triangle}=\frac{8 \times 12}{2}$

$\text{Area of Triangle}=48$ $\displaystyle \text{Area of Triangle}=48$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=48-9.61\pi$ $\displaystyle \text{Area of Shaded Region}=48-9.61\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=17.81$ $\displaystyle \text{Area of Shaded Region}=17.81$

Report an Error

Example Question #1460 : Plane Geometry

The diameter of the circle is $\displaystyle 8$, what is the area of the shaded region?

Possible Answers:

14.44 $\displaystyle 14.44$

12.73 $\displaystyle 12.73$

13.82 $\displaystyle 13.82$

14.60 $\displaystyle 14.60$

Correct answer:

12.73 $\displaystyle 12.73$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{8}{2}$ $\displaystyle \text{Radius}=\frac{8}{2}$

$\text{Radius}=4$ $\displaystyle \text{Radius}=4$

Now, substitute in the radius to find the area of the circle.

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

$\text{Area of Circle}=16\pi$ $\displaystyle \text{Area of Circle}=16\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{9 \times 14}{2}$ $\displaystyle \text{Area of Triangle}=\frac{9 \times 14}{2}$

$\text{Area of Triangle}=63$ $\displaystyle \text{Area of Triangle}=63$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=63-16\pi$ $\displaystyle \text{Area of Shaded Region}=63-16\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=12.73$ $\displaystyle \text{Area of Shaded Region}=12.73$

Report an Error

Example Question #1461 : Plane Geometry

The diameter of the circle is 3.2 $\displaystyle 3.2$, what is the area of the shaded region?

Possible Answers:

49.96 $\displaystyle 49.96$

50.95 $\displaystyle 50.95$

49.71 $\displaystyle 49.71$

45.09 $\displaystyle 45.09$

Correct answer:

49.96 $\displaystyle 49.96$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{3.2}{2}$ $\displaystyle \text{Radius}=\frac{3.2}{2}$

$\text{Radius}=1.6$ $\displaystyle \text{Radius}=1.6$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 1.6^2$ $\displaystyle \text{Area of Circle}=\pi\times 1.6^2$

$\text{Area of Circle}=2.56\pi$ $\displaystyle \text{Area of Circle}=2.56\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{4 \times 29}{2}$ $\displaystyle \text{Area of Triangle}=\frac{4 \times 29}{2}$

$\text{Area of Triangle}=58$ $\displaystyle \text{Area of Triangle}=58$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=58-2.56\pi$ $\displaystyle \text{Area of Shaded Region}=58-2.56\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=49.96$ $\displaystyle \text{Area of Shaded Region}=49.96$

Report an Error

Example Question #1462 : Plane Geometry

The diameter of the circle is $\displaystyle 10$, what is the area of the shaded region?

Possible Answers:

111.09 $\displaystyle 111.09$

112.45 $\displaystyle 112.45$

113.46 $\displaystyle 113.46$

110.95 $\displaystyle 110.95$

Correct answer:

113.46 $\displaystyle 113.46$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{10}{2}$ $\displaystyle \text{Radius}=\frac{10}{2}$

$\text{Radius}=5$ $\displaystyle \text{Radius}=5$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times 5^2$ $\displaystyle \text{Area of Circle}=\pi\times 5^2$

$\text{Area of Circle}=25\pi$ $\displaystyle \text{Area of Circle}=25\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{12 \times 32}{2}$ $\displaystyle \text{Area of Triangle}=\frac{12 \times 32}{2}$

$\text{Area of Triangle}=192$ $\displaystyle \text{Area of Triangle}=192$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=192-25\pi$ $\displaystyle \text{Area of Shaded Region}=192-25\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=113.46$ $\displaystyle \text{Area of Shaded Region}=113.46$

Report an Error

Example Question #1463 : Plane Geometry

The diameter of the circle is $\displaystyle 5$, what is the area of the shaded region?

Possible Answers:

34.12 $\displaystyle 34.12$

33.06 $\displaystyle 33.06$

35.98 $\displaystyle 35.98$

34.37 $\displaystyle 34.37$

Correct answer:

34.37 $\displaystyle 34.37$

Explanation:

To find the area of the shaded region, we will first need to find the area of the right triangle and the area of the circle.

Recall how to find the area of a circle:

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

Now, recall how to find the length of the radius from the length of the diameter.

$\text{Radius}=\frac{\text{Diameter}}{2}$ $\displaystyle \text{Radius}=\frac{\text{Diameter}}{2}$

Substitute in the given diameter to find the radius.

$\text{Radius}=\frac{5}{2}$ $\displaystyle \text{Radius}=\frac{5}{2}$

Now, substitute in the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times (\frac{5}{2})^2$ $\displaystyle \text{Area of Circle}=\pi\times (\frac{5}{2})^2$

$\text{Area of Circle}=\frac{25}{4}\pi$ $\displaystyle \text{Area of Circle}=\frac{25}{4}\pi$

Next, recall how to find the area of a right triangle.

$\text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$ $\displaystyle \text{Area of Triangle}=\frac{\text{base}\times\text{height}}{2}$

Substitute in the given base and height to find the area.

$\text{Area of Triangle}=\frac{6 \times 18}{2}$ $\displaystyle \text{Area of Triangle}=\frac{6 \times 18}{2}$

$\text{Area of Triangle}=54$ $\displaystyle \text{Area of Triangle}=54$

We can now find the area of the shaded region:

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

$\text{Area of Shaded Region}=54-\frac{25}{4}\pi$ $\displaystyle \text{Area of Shaded Region}=54-\frac{25}{4}\pi$

Solve and round to two decimal places.

$\text{Area of Shaded Region}=34.37$ $\displaystyle \text{Area of Shaded Region}=34.37$

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Basic Geometry : Triangles

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #471 : Triangles

Example Question #1455 : Plane Geometry

Example Question #1456 : Plane Geometry

Example Question #1457 : Plane Geometry

Example Question #1458 : Plane Geometry

Example Question #1459 : Plane Geometry

Example Question #1460 : Plane Geometry

Example Question #1461 : Plane Geometry

Example Question #1462 : Plane Geometry

Example Question #1463 : Plane Geometry

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