Chapter 16

Geometry

Geometry is the study of points, lines, angles, shapes on the plane, and solids in space. In this chapter, you hone your geometry skills with a variety of problems that ask you to calculate the measurement of angles, shapes, and solids.

The Problems You’ll Work On

Here’s a list of the types of problems you work on in this chapter:

Measuring angles

Finding the area and perimeter of squares and rectangles

Calculating the area of parallelograms and trapezoids

Knowing the formulas for the area and circumference of circles

Using the area formula for triangles

Working with right triangles using the Pythagorean theorem

Finding the volume of some common solids

What to Watch Out For

The following information will be useful to you as you work on the problems in this chapter:

You need to know basic geometric formulas to find the area and perimeter of squares and rectangles, and the area of parallelograms, trapezoids, and triangles.

You need to know the formulas to find the diameter, circumference, and area of circles.

You need to know the formulas to find the volume of cubes, rectangular solids, cylinders, spheres, pyramids, and cones.

You should be familiar with the Pythagorean theorem as well as the formulas associated with right triangles.

Angles

631–645

631. Find the value of n.

632. Find the value of n.

633. Find the value of n.

634. Find the value of n.

635. ABCD is a square. Find the value of n.

636. Find the value of n.

637. Find the value of n.

638. ABC is a right triangle. Find the value of n.

639. ABCD is a rectangle. Find the value of n.

640. Find the value of n.

641. Find the value of n.

642. Find the value of n.

643. ABC is isosceles. Find the value of n.

644. AC is a diameter of the circle. Find the value of n.

645. BCDE is a parallelogram and BE = AE. Find the value of n.

Squares

646–655 Use the formulas for the area of a square () and the perimeter of a square (P = 4s) to answer the questions.

646. What’s the area of a square whose side is 6 inches in length?

647. What’s the perimeter of a square that has a side that’s 7 meters in length?

648. What’s the area of a square with a side that’s 101 miles long?

649. If the side of a square is 3.4 centimeters, what is its perimeter?

650. If the perimeter of a square is 84 feet long, what is the length of its side?

651. If the area of a square is 144 square feet, what is its perimeter?

652. What is the area of a square room that has a perimeter of 62 feet?

653. A square room requires 25 square yards of carpeting to cover its floor. What is the perimeter of the room, in feet? (1 yard = 3 feet)

654. If each side of a square field is exactly 3 miles, what is its area in square feet? (1 mile = 5,280 feet)

655. The perimeter of a square park, in kilometers, is 10 times greater than its area in square kilometers. What is the length of one side of this park?

Rectangles

656–665 Use the formulas for the area of a rectangle (A = lw) and the perimeter of a rectangle (A = 2l + 2w) to answer the questions.

656. What is the area of a rectangle with a length of 8 centimeters and a width of 3 centimeters?

657. If a rectangle has a length of 16 meters and a width of 2 meters, what’s its perimeter?

658. If the length of a rectangle is 4.3 feet and its width is 2.7 feet, what is its area?

659. What is the perimeter of a rectangle whose length is inch and whose width is inch?

660. What is the area of the rectangle below?

661. What is the area of the rectangle below?

662. If the area of a rectangle is 100 square feet and the width is 5 feet, what is its perimeter?

663. If the area of a rectangle is 30 square inches and its length is 8 inches, what is its perimeter?

664. A rectangular picture frame has a length of 2 feet. If the area of the picture in the frame is 156 square inches, what is the perimeter of the frame?

665. If the perimeter of a rectangle is 54 and its area is 72, what is the length of the rectangle. (Hint: The length and width are both whole numbers.)

Parallelograms and Trapezoids

666–675 Use the formulas for the area of a parallelogram (A = bh) and the area of a trapezoid () to answer the questions.

666. What’s the area of the parallelogram below?

667. What’s the area of the parallelogram below?

668. What’s the area of the parallelogram below?

669. What’s the area of the trapezoid below?

670. What’s the area of the trapezoid below?

671. What’s the area of the trapezoid below?

672. If the area of a parallelogram is 94.5 square centimeters and its base is 7 centimeters, what is its height?

673. What is the height of a trapezoid that has an area of 180 and bases of lengths 9 and 21?

674. Suppose a parallelogram has an area of and a height of . What is the length of its base?

675. If a trapezoid has an area of 45, a height of 3, and one base of length 4.5, what is the length of the other base?

Area of Triangles

676–685 Use the formula for the area of a triangle () to answer the questions.

676. What is the area of a triangle with a base of 9 inches and a height of 8 inches?

677. If a triangle has a base that’s 3 meters long and a height that’s 23 meters in length, what is its area?

678. A triangle has a base of length and a height of length . What is its area?

679. What is the area of the triangle below?

680. What is the area of the triangle below?

681. A right triangle has two legs of length 4 centimeters and 12 centimeters. What is its area?

682. What is the base of a triangle with an area of 60 square meters and a height of 4 meters?

683. What is the height of a triangle with an area of 78 square inches and a base that’s 1 foot long?

684. What is the height of a triangle with a base of in length and an area of in2?

685. If the area of a triangle is 84.5 and the height and base are both the same length, what is the height of the triangle?

The Pythagorean Theorem

686–695 Use the Pythagorean theorem () to answer the questions.

686. If the two legs of a right triangle are 3 feet and 4 feet, what is the length of the hypotenuse?

687. What is the length of the hypotenuse of a right triangle whose two legs are 10 centimeters and 24 centimeters?

688. If a right triangle has two legs of length 4 and 8, what is the length of its hypotenuse?

689. What is the length of the hypotenuse of the following triangle?

690. What is the length of the hypotenuse of the triangle below?

691. What is the length of the hypotenuse of the triangle below?

692. If a right triangle has two legs of lengths and , what is the length of the hypotenuse?

693. If a right triangle has two legs of lengths and , what is the length of the hypotenuse?

694. What is the length of the shorter leg of the triangle below?

695. What is the length of the longer leg of the triangle below?

Circles

696–709 Use the formulas for the diameter of a circle (D = 2r), the area of a circle (), and the circumference of a circle () to answer the questions.

696. What’s the diameter of a circle with a radius of 8?

697. What’s the area of a circle with a radius of 11?

698. If a circle has a radius of 20, what’s the length of its circumference?

699. What’s the area of the circle below?

700. What’s the circumference of the circle below?

701. If a circle has a diameter of 99, what’s its circumference?

702. What’s the circumference of a circle whose diameter is ?

703. If a circle has a diameter of 100, what is its area?

704. What’s the radius of a circle whose area equals ?

705. If a circle has a circumference of , what is its radius?

706. What’s the area of a circle that has a circumference of ?

707. If a circle has an area of , what is its circumference?

708. What’s the radius of a circle whose area is 16?

709. What’s the area of a circle whose circumference is 18.5?

Volume

710–730 Use the solid geometry formulas:

Volume of a cube:

Volume of a box:

Volume of a cylinder:

Volume of a sphere:

Volume of a pyramid (with a square base):

Volume of a cone:

710. What’s the volume of a cube that has a side of 12 inches in length?

711. What is the volume of the cube below?

712. If a cube has a volume of 1,000,000 cubic inches, what is the length of its side?

713. What is the volume of a box that’s 15 inches long, 4 inches wide, and 10 inches tall?

714. If a box is 8.5 inches in width, 11 inches in length, and 3.5 inches in height, what’s its volume?

715. What is the volume of a box whose three dimensions are inch, inch, and inch?

716. Suppose a box with a volume of 20,000 cubic centimeters has a length of 80 centimeters and a width of 50 centimeters. What is the height of the box?

717. If a box has a volume of 45.6 cubic inches, a length of 10 inches, and a height of 100 inches, what is its width?

718. If a cylinder has a radius of 2 feet and a height of 6 feet, what’s its volume?

719. Suppose a cylinder has a radius of 45 and a height of 110. What is its volume?

720. What’s the volume of a cylinder whose radius is 0.4 meter and whose height is 1.1 meters?

721. A cylinder has a radius of inch and a height of inch. What’s its volume?

722. Suppose a cylinder has a volume of cubic feet and a radius of 3 feet. What is its height?

723. What is the volume of a sphere with a radius of 3 centimeters?

724. If a sphere has a radius of inch, what’s its volume?

725. What is the volume of a sphere that has a radius of 1.2 meters?

726. Suppose a sphere has a volume of cubic feet. What is its radius?

727. A pyramid has a square base whose side has a length of 4 inches. If its height is 6 inches, what is the volume of the pyramid?

728. Suppose a pyramid with a square base has a volume of 80 cubic meters and a height of 15 meters. What is the length of the side of its base?

729. What is the volume of a cone that’s 10 inches high and whose circular base has a radius of 30 inches?

730. If a cone has a volume of and a radius of 6, what is its height?