Geometry: Get Stronger

Use the Properties of Angles

  1. Find ⓐ the supplement and ⓑ the complement of the given angle: [latex]53^∘[/latex]
  2. Find ⓐ the supplement and ⓑ the complement of the given angle: [latex]29^∘[/latex]
  3. Find the supplement of a [latex]135[/latex]∘ angle.
  4. Find the complement of a [latex]27.5[/latex]∘ angle.
  5. What is the supplement of a [latex]48^∘[/latex] angle?
  6. Two angles are supplementary. The larger angle is [latex]56[/latex]∘ more than the smaller angle. Find the measures of both angles.
  7. Two angles are complementary. The smaller angle is [latex]34[/latex]∘ less than the larger angle. Find the measures of both angles.
  8. Two angles are complementary. The smaller angle is [latex]24[/latex]∘ less than the larger angle. Find the measures of both angles.

Use the Properties of Triangles

In the following exercises, solve using properties of triangles

  1. The measures of two angles of a triangle are [latex]26[/latex]∘ and [latex]98[/latex]∘. Find the measure of the third angle.
  2. The measures of two angles of a triangle are [latex]105[/latex]∘ and [latex]31[/latex]∘. Find the measure of the third angle.
  3. The measures of two angles of a triangle are [latex]22[/latex] and [latex]85[/latex] degrees. Find the measure of the third angle.
  4. One angle of a right triangle measures [latex]33[/latex]∘. What is the measure of the other angle?
  5. One angle of a right triangle measures [latex]22.5[/latex]∘. What is the measure of the other angle?
  6. The measure of the smallest angle of a right triangle is [latex]20[/latex]∘ less than the measure of the other small angle. Find the measures of all three angles.
  7. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles.
  8. One angle of a triangle is [latex]30^∘[/latex] more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

  1. A right triangle. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.
  2. A right triangle is shown. The base is labeled 10, the height is labeled 24.
  3. A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.
  4. A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 17.
  5. A right triangle is shown. The height is labeled 7, the base is labeled 4.
  6. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.
  7. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.
  8. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.
  9. A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

  1. A [latex]13[/latex]-foot string of lights will be attached to the top of a [latex]12[/latex]-foot pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored?
    A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.
  2. Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of [latex]10[/latex] feet. What will the length of the path be?
    A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.
  3. Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is [latex]30[/latex] feet wide and [latex]35[/latex] feet tall at the highest point of the roof. If the dollhouse will be [latex]2.5[/latex] feet wide, how tall will its highest point be?
  4. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is [latex]8[/latex] feet tall and Sergio has [latex]10[/latex] feet of wire. How far from the base of the antenna can he attach the wire?
    An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet.

Use the Properties of Rectangles

In the following exercises, find the perimeter and area of each rectangle.

  1. The length of a rectangle is [latex]42[/latex] meters and the width is [latex]28[/latex] meters.
  2. A sidewalk in front of Kathy’s house is in the shape of a rectangle [latex]44[/latex] feet wide by [latex]45[/latex] feet long.
  3. The length of a rectangle is [latex]85[/latex] feet and the width is [latex]45[/latex] feet.
  4. A rectangular room is [latex]15[/latex] feet wide by [latex]14[/latex] feet long.

In the following exercises, solve.

  1. Find the length of a rectangle with perimeter of [latex]220[/latex] centimeters and width of [latex]85[/latex] centimeters.
  2. The area of a rectangle is [latex]2356[/latex] square meters. The length is [latex]38[/latex] meters. What is the width?
  3. The length of a rectangle is [latex]12[/latex] centimeters more than the width. The perimeter is [latex]74[/latex]centimeters. Find the length and the width.
  4. Find the length of a rectangle with perimeter [latex]124[/latex] inches and width [latex]38[/latex] inches.
  5. Find the width of a rectangle with perimeter [latex]92[/latex] meters and length [latex]19[/latex] meters.
  6. The area of a rectangle is [latex]414[/latex] square meters. The length is [latex]18[/latex] meters. What is the width?
  7. The length of a rectangle is [latex]9[/latex] inches more than the width. The perimeter is [latex]46[/latex] inches. Find the length and the width.
  8. The perimeter of a rectangle is [latex]58[/latex] meters. The width of the rectangle is [latex]5[/latex] meters less than the length. Find the length and the width of the rectangle.
  9. The width of the rectangle is [latex]0.7[/latex] meters less than the length. The perimeter of a rectangle is [latex]52.6[/latex] meters. Find the dimensions of the rectangle.
  10. The perimeter of a rectangle of [latex]150[/latex] feet. The length of the rectangle is twice the width. Find the length and width of the rectangle.
  11. The length of a rectangle is [latex]3[/latex] meters less than twice the width. The perimeter is [latex]36[/latex] meters. Find the length and width.
  12. The width of a rectangular window is [latex]24[/latex] inches. The area is [latex]624[/latex] square inches. What is the length?
  13. The area of a rectangular roof is [latex]2310[/latex] square meters. The length is [latex]42[/latex] meters. What is the width?
  14. The perimeter of a rectangular courtyard is [latex]160[/latex] feet. The length is [latex]10[/latex] feet more than the width. Find the length and the width.
  15. The width of a rectangular window is [latex]40[/latex] inches less than the height. The perimeter of the doorway is [latex]224[/latex] inches. Find the length and the width.

Use the Properties of Triangles

In the following exercises, solve using the properties of triangles.

  1. Find the area of a triangle with base [latex]18[/latex] inches and height [latex]15[/latex] inches.
  2. A triangular road sign has base [latex]30[/latex] inches and height [latex]40[/latex] inches. What is its area?
  3. A tile in the shape of an isosceles triangle has a base of [latex]6[/latex] inches. If the perimeter is [latex]20[/latex] inches, find the length of each of the other sides.
  4. The perimeter of a triangle is [latex]59[/latex] feet. One side of the triangle is [latex]3[/latex] feet longer than the shortest side. The third side is [latex]5[/latex] feet longer than the shortest side. Find the length of each side.
  5. Find the area of a triangle with base [latex]12[/latex] inches and height [latex]5[/latex] inches.
  6. Find the area of a triangle with base [latex]8.3[/latex] meters and height [latex]6.1[/latex] meters.
  7. A triangular flag has base of [latex]11[/latex] foot and height of [latex]1.5[/latex] feet. What is its area?
  8. If a triangle has sides of [latex]6[/latex] feet and [latex]9[/latex] feet and the perimeter is [latex]23[/latex] feet, how long is the third side?
  9. What is the base of a triangle with an area of [latex]207[/latex] square inches and height of [latex]18[/latex] inches?
  10. The perimeter of a triangular reflecting pool is [latex]36[/latex] yards. The lengths of two sides are [latex]10[/latex] yards and [latex]15[/latex] yards. How long is the third side?
  11. An isosceles triangle has a base of [latex]20[/latex] centimeters. If the perimeter is [latex]76[/latex] centimeters, find the length of each of the other sides.
  12. Find the length of each side of an equilateral triangle with a perimeter of [latex]51[/latex] yards.
  13. The perimeter of an equilateral triangle is [latex]18[/latex] meters. Find the length of each side.
  14. The perimeter of an isosceles triangle is [latex]42[/latex] feet. The length of the shortest side is [latex]12[/latex] feet. Find the length of the other two sides.
  15. A dish is in the shape of an equilateral triangle. Each side is [latex]8[/latex] inches long. Find the perimeter.
  16. A road sign in the shape of an isosceles triangle has a base of [latex]36[/latex] inches. If the perimeter is [latex]91[/latex] inches, find the length of each of the other sides.
  17. The perimeter of a triangle is [latex]39[/latex] feet. One side of the triangle is [latex]1[/latex] foot longer than the second side. The third side is [latex]2[/latex] feet longer than the second side. Find the length of each side.
  18. One side of a triangle is twice the smallest side. The third side is [latex]5[/latex] feet more than the shortest side. The perimeter is [latex]17[/latex] feet. Find the lengths of all three sides.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

  1. The height of a trapezoid is [latex]8[/latex] feet and the bases are [latex]11[/latex] and [latex]14[/latex] feet. What is the area?
  2. Find the area of the trapezoid with height [latex]25[/latex] meters and bases [latex]32.5[/latex] and [latex]21.5[/latex] meters.
  3. The height of a trapezoid is [latex]12[/latex] feet and the bases are [latex]9[/latex] and [latex]15[/latex] feet. What is the area?
  4. Find the area of a trapezoid with a height of [latex]51[/latex] meters and bases of [latex]43[/latex] and [latex]67[/latex] meters.
  5. The height of a trapezoid is [latex]15[/latex] centimeters and the bases are [latex]12.5[/latex] and [latex]18.3[/latex] centimeters. What is the area?
  6. Find the area of a trapezoid with a height of [latex]4.2[/latex] meters and bases of [latex]8.1[/latex] and [latex]5.5[/latex] meters.
  7. Laurel is making a banner shaped like a trapezoid. The height of the banner is [latex]3[/latex] feet and the bases are [latex]4[/latex] and [latex]5[/latex] feet. What is the area of the banner?
  8. Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width [latex]18.5[/latex] inches and lengths [latex]62[/latex] and [latex]50[/latex] inches. What is the area of the counter?

Use Properties of Circles

In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth.

  1. A circular mosaic has radius [latex]3[/latex] meters. Find the circumference and the area of the mosaic.
  2. Find the diameter of a circle with circumference [latex]150.72[/latex] inches.
  3. The lid of a paint bucket is a circle with radius [latex]7[/latex] inches. Find the circumference and area of the lid.
  4. A farm sprinkler spreads water in a circle with radius of [latex]8.5[/latex] feet. Find the circumference and area of the watered circle.
  5. A reflecting pool is in the shape of a circle with diameter of [latex]20[/latex] feet. What is the circumference of the pool?
  6. A circular saw has a diameter of [latex]12[/latex] inches. What is the circumference of the saw?
  7. A barbecue grill is a circle with a diameter of [latex]2.2[/latex] feet. What is the circumference of the grill?
  8. A circle has a circumference of [latex]163.28[/latex] inches. Find the diameter.
  9. A circle has a circumference of [latex]17.27[/latex] meters. Find the diameter.

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

  1. A geometric shape is shown, formed by two rectangles. The top is labeled 8. The width of the top rectangle is labeled 3. The right side of the figure is labeled 5. The width of the bottom rectangle is labeled 3.
  2. A geometric shape is shown. It is formed by two triangles. The shared base of the two triangles is labeled 20. The height of each triangle is labeled 15.
  3. A geometric shape is shown. It is a rectangle with a semi-circle attached to the top. The base of the rectangle, also the diameter of the semi-circle, is labeled 10. The height of the rectangle is labeled 16.
  4. A geometric shape is shown. It is a horizontal rectangle attached to a vertical rectangle. The top is labeled 6, the height of the horizontal rectangle is labeled 2, the distance from the edge of the horizontal rectangle to the start of the vertical rectangle is 4, the base of the vertical rectangle is 2, the right side of the shape is 4.
  5. A geometric shape is shown. It is a sideways U-shape. The top is labeled 6, the left side is labeled 6. An inside horizontal piece is labeled 3. Each of the vertical pieces on the right are labeled 2.
  6. A geometric shape is shown. It is a rectangle with a triangle attached to the bottom left side. The top is labeled 4. The right side is labeled 10. The base is labeled 9. The vertical line from the top of the triangle to the top of the rectangle is labeled 3.
  7. Two triangles are shown. They appear to be right triangles. The bases are labeled 3, the heights 4, and the longest sides 5.
  8. A geometric shape is shown. It is composed of two trapezoids. The base is labeled 10. The height of one trapezoid is 2. The horizontal and vertical sides are all labeled 5.
  9. A geometric shape is shown. It is a rectangle with a triangle and another rectangle attached. The left side is labeled 8, the bottom is 8, the right side is 13, and the width of the smaller rectangle is 2.
  10. A geometric shape is shown. A triangle is attached to a semi-circle. The base of the triangle is labeled 4. The height of the triangle and the diameter of the circle are 8.
  11. A geometric shape is shown. It is a rectangle attached to a semi-circle. The base of the rectangle is labeled 5, the height is 7.
  12. A geometric shape is shown. It is a rectangle with a triangle attached to the top on the left side and a circle attached to the top right corner. The diameter of the circle is labeled 5. The height of the triangle is labeled 5, the base is labeled 4. The height of the rectangle is labeled 6, the base 11.

In the following exercises, solve.

  1. A city park covers one block plus parts of four more blocks, as shown. The block is a square with sides [latex]250[/latex] feet long, and the triangles are isosceles right triangles. Find the area of the park.
    A square is shown with four triangles coming off each side.
  2. Perry needs to put in a new lawn. His lot is a rectangle with a length of [latex]120[/latex] feet and a width of [latex]100[/latex] feet. The house is rectangular and measures [latex]50[/latex] feet by [latex]40[/latex] feet. His driveway is rectangular and measures [latex]20[/latex] feet by [latex]30[/latex] feet, as shown. Find the area of Perry’s lawn.
    A rectangular lot is shown. In it is a home shaped like a rectangle attached to a rectangular driveway.
  3. Area of a Tabletop Yuki bought a drop-leaf kitchen table. The rectangular part of the table is a [latex]1[/latex]-ft by [latex]3[/latex]-ft rectangle with a semicircle at each end, as shown. Find the area of the table with one leaf up. Find the area of the table with both leaves up.
    An image of a table is shown. There is a rectangular portion attached to a semi-circular portion. There is another semi-circular leaf folded down on the other side of the rectangle.

Find Volume and Surface Area of Rectangular Solids

In the following exercises, find the volume surface area of the rectangular solid.

  1. a rectangular solid with length [latex]14[/latex] centimeters, width [latex]4.5[/latex] centimeters, and height [latex]10[/latex] centimeters
  2. a rectangular solid with length [latex]2[/latex] meters, width [latex]1.5[/latex] meters, height [latex]3[/latex] meters
  3. a rectangular solid with length [latex]3.5[/latex] yards, width [latex]2.1[/latex] yards, height [latex]2.4[/latex] yards
  4. a cube of tofu with sides [latex]2.5[/latex] inches
  5. a cube with side lengths of [latex]5[/latex] centimeters
  6. a cube with side lengths of [latex]10.4[/latex] feet

In the following exercises, solve.

  1. Moving van A rectangular moving van has length [latex]16[/latex] feet, width [latex]8[/latex] feet, and height [latex]8[/latex] feet. Find its volume and surface area.
  2. Carton A rectangular carton has length [latex]21.3[/latex] cm, width [latex]24.2[/latex] cm, and height [latex]6.5[/latex] cm. Find its volume and surface area.
  3. Science center Each side of the cube at the Discovery Science Center in Santa Ana is [latex]64[/latex] feet long. Find its volume and surface area.
  4. Base of statue The base of a statue is a cube with sides [latex]2.8[/latex] meters long. Find its volume and surface area.

Find Volume and Surface Area of Spheres

In the following exercises, find the volume and the surface area of the sphere. Round your answer to two decimal places.

  1. a sphere with radius [latex]4[/latex] yards
  2. a sphere with radius [latex]3[/latex] centimeters
  3. a sphere with radius [latex]7.5[/latex] feet
  4. a baseball with radius [latex]1.45[/latex] inches
  5. An exercise ball has a radius of [latex]15[/latex] inches.
  6. A golf ball has a radius of [latex]4.5[/latex] centimeters.

Find Volume and Surface Area of Cylinders

In the following exercises, find the volume and the surface area of the cylinder. Round answers to two decimal places.

  1. a cylinder with radius [latex]2[/latex] yards and height [latex]6[/latex] yards
  2. a cylinder with radius [latex]3[/latex] feet and height [latex]9[/latex] feet
  3. a cylinder with radius [latex]1.5[/latex] meters, height [latex]4.2[/latex] meters
  4. a juice can with diameter [latex]8[/latex] centimeters and height [latex]15[/latex] centimeters
  5. a can of coffee has a radius of [latex]5[/latex] cm and a height of [latex]13[/latex] cm.
  6. a cylindrical barber shop pole has a diameter of [latex]6[/latex] inches and height of [latex]24[/latex] inches.

Find Volume of Cones

In the following exercises, find the volume of the cone. Round answers to two decimal places.

  1. a cone with height [latex]5[/latex] meters and radius [latex]1[/latex] meter
  2. a cone with height [latex]9[/latex] feet and radius [latex]2[/latex] feet
  3. a cone with height [latex]12.4[/latex] centimeters and radius [latex]5[/latex] cm
  4. a cone-shaped water cup with diameter [latex]2.6[/latex] inches and height [latex]2.6[/latex] inches
  5. a cone-shaped tee-pee tent that is [latex]10[/latex] feet tall and [latex]10[/latex] feet across at the base
  6. a cone-shaped silo that is [latex]50[/latex] feet tall and [latex]70[/latex] feet across at the base