Boston Planning and Development Agency directors gave the go-ahead for a three-tower project in Back Bay and 1.4 million square foot increase in previously approved commercial and residential space in the heart of the Seaport District.

Boston Properties plans nearly 1.3 million square feet of development on a 5.2-acre site above and next to the MBTA’s Back Bay Station, its 100 Clarendon St. parking garage and air rights above the Massachusetts Turnpike. Designs call for partial demolition and reconstruction of the garage to make room for a 26-story office tower at the corner of Dartmouth and Stuart streets.

The development also includes a 35-story residential tower with 360 units on Clarendon Street at the existing MBTA bus drop-off area, a 28-story residential building with 240 units and a 33,000-square-foot expansion of the existing train station for retail space. Developers agreed to pay $4.6 million in housing linkage fees and $3 million to offset the effects on nearby historic properties including Trinity Church, the New Old South Church and YWCA Boston.

In the Seaport, Newton-based WS Development received approval to increase total development in the master-planned Seaport Square district to 7.7 million square feet, up from 6.3 million square feet approved in 2010.

The bulk of the redesign focuses on four blocks at the center of the project site off Congress Street currently used as surface parking. Density would increase from 1.1 million to nearly 2 million square feet for offices, multifamily housing and a hotel with ground-floor retail or cultural spaces in all buildings.

“Blocks L3-L6 represent some of the last remaining opportunities to attract major innovation economy employers to the Seaport, especially so close to the kind of dense residential and retail environment that has proven compelling to new economy employers and employees alike,” a BPDA staff memo provided to directors stated. “The diverse mix of residential, commercial and retail uses will ensure an active, pedestrian-oriented 18-hour environment that will be unlike anything else in Boston.”

As public benefits, WS agreed to build a 500-seat Seaport Performing Arts Center, a 100-seat social space and 150-seat black box theater while contributing $7.5 million for cultural programming, $2 million toward the Northern Avenue bridge replacement project and $1 million for maintenance and operation of Martin’s Park on Fort Point Channel.

WS also agreed to pay $2.5 million over 10 years to subsidize transportation upgrades in the Seaport, including a potential commuter ferry to North Station and additional Silver Line capacity, and $400,000 for Silver Line upgrades and/or a bus rapid transit study.